Conference on Research in Mathematics Education...

The Special Interest Group of the Mathematical Association of America on Research in Undergraduate

Mathematics Education presents its nineteenth

Conference on Research in Undergraduate Mathematics

Education

Program Marriott City Center

Pittsburgh, Pennsylvania February 25-27, 2016

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The Special Interest Group of the Mathematical Association of America

on Research in Undergraduate Mathematics Education presents its Eighteenth

Conference on Research in Undergraduate

Mathematics Education Program

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!Marriott City Center


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!!!!

The Special Interest Group of the Mathematical Association of America

on Research in Undergraduate Mathematics Education presents its Eighteenth

Conference on Research in Undergraduate

Mathematics Education Program

!!!

!Marriott City Center


!!

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Conference Planning Committee

Timothy Fukawa-Connelly (Program Chair) Temple University

Nicole Engelke (Local Organizer)

West Virginia University

Jason Belnap (Working Group Coordinator) University of Wisconsin, Oshkosh

Christine Andrews-Larson

Florida State University

Stacy Brown California State Polytechnic

University, Pomona

Paul Dawkins Northern Illinois University

Jessica Ellis

Colorado State University

Estrella Johnson Virginia Polytechnic Institute and

State University

Karen Keene North Carolina State University

Elise Lockwood

Oregon State University

Ami Mamolo University of Ontario Institute of

Technology

Jason Martin University of Central Arkansas --???

Pablo Mejia-Ramos Rutgers University

Kevin Moore

The University of Georgia

Annie Selden New Mexico State University

Megan Wawro

Virginia Polytechnic Institute and State University

Keith Weber

Rutgers University

Aaron Weinberg Ithaca College

Michelle Zandieh

Arizona State University

SIGMAA on

RUME

On behalf of the Mathematical Association of America, West Virginia University’s Department of Mathematics, and the RUME Conference Planning Committee, we welcome you to the 18th Annual Conference on Research in Undergraduate Mathematics Education! We have assembled an outstanding program of talks and events for this year’s conference. We hope that you find the conference intellectually stimulating, productive in making connections with colleagues, and socially rewarding. All talks in this year’s program have online links to their accompanying short papers, which we refer to as the RUME Conference Reports. Authors who present at the conference also have the option of submitting a 15-page paper for peer-review consideration for publication in the RUME Conference Proceedings, to be published online later this spring. Guidelines for the RUME Conference Proceedings long papers are available on the conference webpage. Submissions to the RUME Conference Proceedings are eligible for the Best Paper Award. On behalf of the RUME Executive Committee, we would like to thank the RUME Conference Planning Committee for their efforts towards making this a successful conference. Our sincere thanks goes to Christine Andrews-Larson, Jason Belnap, Stacy Brown, Paul Dawkins, Jessica Ellis, Estrella Johnson, Karen Allen Keene, Elise Lockwood, Ami Mamolo, Jason Martin, Pablo Mejia-Ramos, Kevin Moore, Annie Selden, Keith Weber, Megan Wawro, Aaron Weinberg, and Michelle Zandieh. We want to give special thanks to Jason Belnap for organizing the working group process. We also wish to thank the WVU group for their work in making this conference successful. This year we received and reviewed 169 preliminary, poster, contributed, and theoretical report proposals. This work required the time and energy of over 90 reviewers, who prepared evaluations and feedback for authors. We could not carry out the conference without their contribution. Thank you to the 2016 reviewers! If you have comments and suggestions about the submission or review process, please feel free to share them with members of the planning committee (wearing blue nametags) during the conference or send them to Tim at [email protected]. Local organizers will be wearing gold “host committee” name badges, and feel free to ask any of them questions you may have about the conference or Pittsburgh. Should you have any questions, concerns, or compliments about the conference, please do not hesitate to talk to any of us. You will receive an online Conference Evaluation form after the conference and be sure to share your suggestions with us, as we’re always looking for ways to improve. Enjoy and have a wonderful conference! Sincerely,

Nicole Engelke Tim Fukawa-Connelly Local Organizer Program Committee Chairperson

! 1 !

On behalf of the Mathematical Association of America, West Virginia University’s Department of Mathematics, and the RUME Conference Planning Committee, we welcome you to the 18th Annual Conference on Research in Undergraduate Mathematics Education! We have assembled an outstanding program of talks and events for this year’s conference. We hope that you find the conference intellectually stimulating, productive in making connections with colleagues, and socially rewarding.

All talks in this year’s program have online links to their accompanying short papers, which we refer to as the RUME Conference Reports. Authors who present at the conference also have the option of submitting a 15-page paper for peer-review consideration for publication in the RUME Conference Proceedings, to be published online later this spring. Guidelines for the RUME Conference Proceedings long papers are available on the conference webpage. Submissions to the RUME Conference Proceedings are eligible for the Best Paper Award.

On behalf of the RUME Executive Committee, we would like to thank the RUME Conference Planning Committee for their efforts towards making this a successful conference. Our sincere thanks goes to, Jason Belnap, Stacy Brown, Nicole Engelke, Estrella Johnson, Karen Allen Keene, Elise Lockwood, Mike Oehrtman, Kyeong Hah Roh, Ami Mamolo, Jason Martin, Vicki Sealey, Allison Toney, Keith Weber, Megan Wawro, Aaron Weinberg, and Michelle Zandieh. We want to give special thanks to Jason Belnap for organizing the working group process. We also wish to thank the WVU group for their work in making this conference successful.

This year we received and reviewed over 171 preliminary, poster, contributed, and theoretical report proposals. This work required the time and energy of over 90 reviewers, who prepared evaluations and feedback for authors. We could not carry out the conference without their contribution. Thank you to the 2015 reviewers!

If you have comments and suggestions about the submission or review process, please feel free to share them with members of the planning committee (wearing blue nametags) during the conference or send them to Tim at [email protected].

Local organizers will be wearing gold “host committee” name badges, and feel free to ask any of them questions you may have about the conference or Pittsburgh. Should you have any questions, concerns, or compliments about the conference, please do not hesitate to talk to any of us. You will receive an online Conference Evaluation form after the conference and be sure to share your suggestions with us, as we’re always looking for ways to improve. Enjoy and have a wonderful conference!

Sincerely,

Nicole Engelke Tim Fukawa-Connelly Local Organizer Program Committee Chairperson

RUME 2015 Conference Proceedings and Best Paper Award Information

All authors who present at the 2015 conference will have the option of submitting a 15-page paper for publication in the RUME Conference Proceedings. All long papers will be reviewed by 2-3 reviewers and considered for the best paper award.

All long papers must follow the long paper formatting guidelines (available on the conference website under the “Guidelines” link) and will be reviewed according to the proposal review criteria.

LONG PAPER REVIEW CRITERIA

• Does the proposal explore a significant issue/questions relevant to RUME? �

• How does it relate to prior research on related topics/issues? �

• Is the theoretical perspective clearly outlined? �

• Is there an appropriate choice of research methodology? �

• Is it clear what the conclusions/main claims are? �

• Are those supported by data? �

• Does the research contribute to teaching practice/theory development? �Submission of a long paper is not required of participants but is strongly encouraged. Acceptance to the RUME Conference Proceedings is dependent upon reviewers’ recommendations. All long papers are due Sunday, March 22, 2015. �To submit a long paper, check the conference website under the “Guidelines” link: �

• Go to: https://www.easychair.org/conferences/?conf=rume18 �

• Log into EasyChair. Note: You will need to use the login name and password you used �before to submit your proposal. �

• Click “New Submission” in the header �

• Enter the title and other information for the paper �

• Select “Proceedings Long Paper” for the paper category �

• Upload either a .doc or .docx version of your paper (do not upload a .pdf file) �Questions regarding long papers should be emailed to Tim Fukawa-Connelly, the Chair of the Program Committee, at [email protected]. �

Thursday

Friday Breakfast from 7-8:30

Saturday Breakfast from 7-8:30

8:00 AM – 12:00 PM RUME Working Group Meetings

8:35 AM – 9:05 AM Session 8 – Contributed Reports


9:15 AM – 9:45 AM Session 9 – Preliminary Reports


9:45 AM – 10:15 AM Coffee Break

9:45 AM – 10:15 AM Coffee Break

10:15AM – 10:45 AM Session 10 – Preliminary Reports

10:15AM – 10:45 AM Session 21 – Contributed Reports



1:00 PM – 1:15 PM Opening Session Grand Ballroom Salons 2O4

11:35 AM – 12:05 PM Session 12 – Mixed Reports –4

11:35 AM – 12:05 PM Session 23 – Mixed Reports

1:25 PM – 1:55 PM Session 1 – Contributed Reports --4

12:05 – 1:05 PM Lunch Grand Ballroom Salons 2O4

12:05 PM – 1:10 PM Lunch Grand Ballroom Salons 2O4


1:05 PM – 1:35 PM Session 13 – Contributed Reports


2:35 PM – 3:05 PM Coffee Break 1:45 PM – 2:15 PM Session 14 – Contributed Reports





3:45 PM – 4:15 PM Session 4 – Contributed Reports 2:55 PM – 3:25 PM Coffee Break 3:00 PM – 3:30 PM Coffee Break

4:20 PM – 4:50 PM Session 5 – Preliminary Reports









5:30 PM – 6:30 PM Plenary Session Dr. David Stinson Grand Ballroom Salons 2O4

5:25 PM – 6:20 PM Poster Session 2 & Cash Bar Grand Foyer

6:10 PM – 7:00 PM Poster Session 1 & Reception Grand Foyer

7:00 PM – 9:30 PM Dinner & Plenary Session Dr. Peg Smith Grand Ballroom Salons 2O4

6:30 PM Dinner On Your Own

6:30 PM – 9:15 PM Dinner, Awards Banquet & Plenary Session Dr. Sean Larsen Grand Ballroom Salons 2O4

THURSDAY,FEBRUARY25,20161:00-1:15pmGrandBallroomSalons2-4

OPENINGSESSION

1:25-1:55pm

SESSION1–CONTRIBUTEDREPORTS

MarquisA Prospectiveteachers’evaluationsofstudents’proofsbymathematicalinduction

HyejinPark

Thisstudyexamineshowprospectivesecondaryteachersvalidateseveralproofsbymathematicalinduction(MI)fromhypotheticalstudentsandhowtheirworkwithproofvalidationsrelatestohowtheygradetheirstudents’proofs.Whenaskedtogivecriteriaforevaluatingastudent’sargument,participantswishedtoseeacorrectbasestep,inductivestep,andalgebra.However,participantsprioritizedthebasestepandinductivestepoverassessingthecorrectnessofthealgebrawhenvalidatingandgradingstudents’arguments.Alloftheparticipantsgavemorepointstoanargumentthatpresentedonlytheinductivestepthantoanargumentthatpresentedonlythebasestep.Twooftheparticipantsacceptedthestudents’argumentaddressingonlytheinductivestepasavalidproof.Furtherstudiesareneededtodeterminehowprospectiveteachersevaluatetheirstudents’argumentsbyMIifmanyalgebraicerrorsarepresent,especiallyintheinductivestep.

1Paper

MarquisB Analyzingstudents’interpretationsofthedefiniteintegralasconceptprojections

JosephF.Wagner

Thisstudyofbeginningandupper-levelundergraduatephysicsstudentsextendsearlierresearchonstudents’interpretationsofthedefiniteintegral.UsingWagner’s(2006)transfer-in-piecesframeworkandthenotionofaconceptprojection,fine-grainedanalysesofstudents’understandingsofthedefiniteintegralrevealagreatervarietyandsophisticationinsomestudents’useofintegrationthanpreviousresearchershavereported.Thedualpurposeofthisworkistodemonstrateanddeveloptheutilityofconceptprojectionsasameansofinvestigatingknowledgetransfer,andtocritiqueandbuildontheexistingliteratureonstudents’conceptionsofintegration.

9

PaperMarquisC Organizationalfeaturesthatinfluencedepartments’uptakeof

student-centeredinstruction:Casestudiesfrominquiry-basedlearningincollegemathematics

SandraLaursen

ActivelearningapproachestoteachingmathematicsandscienceareknowntoincreasestudentlearningandpersistenceinSTEMdisciplines,butdonotyetreachmostundergraduates.Tobroadlyengagecollegeinstructorsinusingtheseresearch-supportedmethodswillrequirenotonlyprofessionaldevelopmentandsupportforindividuals,buttheengagementofdepartmentsandinstitutionsasorganizations.Thisstudyexaminesfourdepartmentsthatimplementedinquiry-basedlearning(IBL)incollegemathematics,focusingonthequestion,“WhatexplicitstrategiesandimplicitdepartmentalcontextshelporhindertheuptakeofIBL?”Basedoninterviewdataanddocuments,thefourdepartmentalcasestudiesrevealstrategiesusedtosupportIBLinstructorsandengagecolleaguesnotactivelyinvolved.Comparativeanalysishighlightshowcontextualfeaturessupported(ornot)thespreadandsustainabilityoftheseteachingreforms.WeuseBolmanandDeal’s(1991)frameworktoanalyzethestructural,political,humanresourceandsymbolicelementsoftheseorganizationalstrategiesandcontexts.

Paper

62

GrandBallroom5

Aninterconnectedframeworkforcharacterizingsymbolsense

MargaretKinzel

Algebraicnotationcanbeapowerfulmathematicaltool,butnotallseemtodevelop“symbolsense,”theabilitytousethattooleffectivelyacrosssituations.Analysisofinterviewdataidentifiedthreeinterconnectedviewpoints:lookingat,with,andthroughthenotation.Theframeworkandimplicationsforinstructionwillbepresented.

Paper27

2:05-2:35pm


MarquisA Students’explicit,unwarrantedassumptionsin“proofs”offalseconjectures

KellyBubp

Althoughevaluating,refining,proving,andrefutingconjecturesareimportantaspectsofdoingmathematics,manystudentshavelimitedexperienceswiththeseactivities.Inthisstudy,undergraduatestudentscompletedprove-or-disprovetasksduringtask-basedinterviews.Thispaperexplorestheexplicit,unwarrantedassumptionsmadebysixstudentsontasksinvolvingfalsestatements.Ineachcase,thestudentexplicitlyassumedanexactconditionnecessaryforthestatementinthetasktobetruealthoughitwasnotagivenhypothesis.Theneedforanungivenassumptiondidnotpromptanyofthesestudentstothinkthestatementmaybefalse.Throughpromptingfromtheinterviewer,twostudentsovercametheirassumptionandcorrectlysolvedthetaskandtwostudentspartiallyovercameitbyconstructingasolutionofcases.However,twootherstudentswereunabletoovercometheirassumptions.Studentsmakingexplicit,unwarrantedassumptionsseemstoberelatedtotheirlimitedexperiencewithconjectures.

Paper

11

MarquisB Physics:Bridgingthesymbolicandembodiedworldsofmathematicalthinking

ClarissaThompson,SepidehStewartandBruceMason

Physicsspansunderstandinginthreedomains–theEmbodied(Real)World,theFormal(Laws)World,andtheSymbolic(Math)World.Expertphysicistsfluidlymoveamongthesedomains.Deep,conceptualunderstandingandproblemsolvingthriveinfluencyinallthreeworldsandthefacilitytomakeconnectionsamongthem.However,novicestudentsstruggletoembodythesymbolsorsymbolicallyexpresstheembodiments.Thecurrentresearchfocusedonhowaphysicsinstructoruseddrawingsandmodelstohelphisstudentsdevelopmoreexpert-likethinkingandmoveamongtheworlds.

Paper

41MarquisC Inquiry-basedlearninginmathematics:Negotiatingthe

definitionofapedagogy

ZacharyHaberlerandSandraLaursen

Inquiry-basedlearningisoneofthepedagogiesthathasemergedinmathematicsasanalternativetotraditionallecturinginthelasttwodecades.Thereisagrowingbodyofresearchandscholarshiponinquiry-basedlearninginSTEMcourses,aswellasagrowingcommunityofpractitionersofIBLinmathematics.However,despitethegrowthofIBLresearchandpracticeinmathematics,wideuptakeofIBLremainshamstrunginpartbythelackofasophisticateddiscussionofitsdefinition.ThispaperoffersafirststeptowardaddressingthisproblembydescribinghowagroupofIBLpractitionersdefineIBL,howtheyadoptIBLtofittheirspecificteachingneeds,andhowdifferencesindefinitionsandperceptionsofIBLhaveconstrainedandenableditsdiffusiontonewinstructors.Paper

55

GrandBallroom5

Studentresourcespertainingtofunctionandrateofchangeindifferentialequations

GeorgeKuster

Whiletheimportanceofstudentunderstandingoffunctionandrateofchangearethemesacrosstheresearchliteratureindifferentialequations,fewstudieshaveexplicitlyfocusedonhowstudentunderstandingofthesetwotopicsgrowandinterfacewitheachotherwhilestudentslearndifferentialequations.ExtendingtheperspectiveofKnowledgeinPieces(diSessa,1993)tostudentlearningindifferentialequations,thisresearchexplorestheresourcesrelatingtofunctionandrateofchangethatstudentsusetosolvedifferentialequationstasks.Thefindingsreportedhereinarepartofalargerstudyinwhichmultiplestudentsenrolledindifferentialequationswereinterviewedperiodicallythroughoutthesemester.Theresultsculminatewithtwosetsofresourcesastudentusedrelatingtofunctionandrateofchangeandimplicationsforhowtheseconceptsmaycometogethertoaffordanunderstandingofdifferentialequations.

Paper124

2:35–3:05pm

COFFEEBREAK

3:05–3:35pm


MarquisA Anewperspectivetoanalyzeargumentationandknowledgeconstructioninundergraduateclassrooms

KarenKeene,DerekWilliamsandCelethiaMcNeil

Usingargumentationtohelpunderstandhowlearninginaclassroomoccursisacompellingandcomplextask.Weshowhoweducationresearcherscanuseanargumentationknowledgeconstructionframework(Weinberger&Fischer,2006)fromresearchinonlineinstructiontomakesenseofthelearninginaninquiryorienteddifferentialequationsclassroom.Thelongtermgoalisseeiftherearerelationshipsamongclassroomparticipationandstudentoutcomes.Theresearchreportedhereisthefirststep:analyzingthediscourseintermsofepistemic,social,andargumentativedimensions.Theresultsshowthattheepistemicdimensioncanbebetterunderstoodbyidentifyinghowstudentsverbalizeunderstandingaboutaproblem,theconceptualspacearoundtheproblem,theconnectionsbetweenthetwoandtheconnectionstopriorknowledge.Inthesocialdimension,wecanidentifyifstudentsarebuildingontheirlearningpartners’ideas,orusingtheirownideas,andorboth.Paper

42MarquisB Prototypeimagesofthedefiniteintegral

StevenJones

Researchonstudentunderstandingofdefiniteintegralshasrevealedanapparentpreferenceamongstudentsforgraphicalrepresentationsofthedefiniteintegral.Sincegraphicalrepresentationscanpotentiallybebothbeneficialandproblematic,itisimportanttounderstandthekindsofgraphicalimagesstudentsuseinthinkingaboutdefiniteintegrals.Thisreportusestheconstructof“prototypes”toinvestigatehowalargesampleofstudentsdepicteddefiniteintegralsthroughthegraphicalrepresentation.Aclear“prototype”groupofimagesappearedinthedata,aswellasrelated“almostprototype”imagegroups.

15Paper

MarquisC Thegraphicalrepresentationofanoptimizingfunction

ReneeLarueandNicoleInfante

Optimizationproblemsinfirstsemestercalculuspresentmanychallengesforstudents.Inparticular,studentsarerequiredtodrawonpreviouslylearnedcontentandintegrateitwithnewcalculusconceptsandtechniques.Whilethiscanbedonecorrectlywithoutconsideringthegraphicalrepresentationofsuchanoptimizingfunction,wearguethatconsistentlyconsideringthegraphicalrepresentationprovidesthestudentswithtoolsforbetterunderstandinganddevelopingtheiroptimizationproblem-solvingprocess.Weexaminesevenstudents’conceptimagesoftheoptimizingfunction,specificallyfocusingonthegraphicalrepresentation,andconsiderhowthisinfluencestheirproblem-solvingactivities.Paper

93GrandBallroom5

Supportforproofasaclusterconcept:Anempiricalinvestigationintomathematicians’practice

KeithWeber

InapreviousRUMEpaper,Iarguedthatproofinmathematicalpracticecanprofitablybeviewedasaclusterconceptinmathematicalpractice.IalsooutlinedseveralpredictionsthatwewouldexpecttoholdifproofwereaclusterconceptInthispaper,Iempiricallyinvestigatetheviabilityofsomeofthesepredictions.Theresultsofthestudiesconfirmedthesepredictions.Inparticular,prototypicalproofssatisfyallcriteriaoftheclusterconceptandtheirvalidityisagreeduponbymostmathematicians.Argumentsthatsatisfyonlysomeofthecriteriaoftheclusterconceptgeneratedisagreementamongstmathematicianswithmanybelievingtheirvaliditydependsuponcontext.Finally,mathematiciansdonotagreeonwhattheessenceofproofis.

Paper116

3:45–4:15pm


MarquisA AStudyofCommonStudentPracticesforDeterminingtheDomainandRangeofGraphs

PeterCho,BenjaminNorrisandDeborahMoore-Russo

Thisstudyfocusesonhowstudentsindifferentpostsecondarymathematicscoursesperformondomainandrangetasksregardinggraphsoffunctions.Studentsoftenfocusonnotableaspectsofagraphandfailtoseethegraphinitsentirety.Manystudentsstrugglewithpiecewisefunctions,especiallythoseinvolvinghorizontalsegments.FindingsindicatethatCalculusIstudentsperformedbetterondomaintasksthanstudentsinlowermathcoursestudents;however,theydidnotoutperformstudentsinlowermathcoursesonrangetasks.Ingeneral,studentperformancedidnotprovideevidenceofadeepunderstandingofdomainandrange.

Paper5

MarquisB Onsymbols,reciprocalsandinversefunctions

RinaZazkisandIgorKontorovich

Inmathematicsthesamesymbol–superscript(-1)–isusedtoindicateaninverseofafunctionandareciprocalofarationalnumber.Isthereareasonforusingthesamesymbolinbothcases?Weanalyzetheresponsestothisquestionofprospectivesecondaryschoolteacherspresentedinaformofadialoguebetweenateacherandastudent.Thedatashowthatthemajorityofparticipantstreatthesymbol☐-1asahomonym,thatis,thesymbolisassigneddifferentandunrelatedmeaningsdependingonacontext.Weexemplifyhowknowledgeofadvancedmathematicscanguideinstructionalinteraction

Paper39

MarquisC Interpretingprooffeedback:Doourstudentsknowwhatwe’resaying?

RobertC.Moore,MarthaByrne,TimothyFukawa-ConnellyandSarahHanusch

Instructorsoftenwritefeedbackonstudents’proofsevenifthereisnoexpectationforthestudentstoreviseandresubmitthework.However,itisnotknownwhatstudentsdowiththatfeedbackoriftheyunderstandtheprofessor’sintentions.Tothisend,weaskedeightadvancedmathematicsundergraduatestorespondtoprofessorcommentsonfourwrittenproofsbyinterpretingandimplementingthecomments.Weanalyzedthestudent’sresponsesthroughthelensesofcommunitiesofpracticeandlegitimateperipheralparticipation.Thispaperpresentstheanalysisoftheresponsesfromoneproof.Paper

87GrandBallroom5

Studentresponsestoinstructioninrationaltrigonometry

JamesFanning

Idiscussaninvestigationonstudents’responsestolessonsinWildberger’s(2005a)rationaltrigonometry.FirstIdetailbackgroundinformationonstudents’struggleswithtrigonometryanditsrootsinthehistoryoftrigonometry.AfterdetailingwhatrationaltrigonometryisandwhatothermathematiciansthinkofitIdescribeapre-interview,intervention,postinterviewexperiment.Inthisstudytwostudentsgothroughclinicalinterviewpertainingtosolvingtrianglesbeforeandafterinstructioninrationaltrigonometry.Thefindingsofthisstudyshowpotentialbenefitsofstudentsstudyingrationaltrigonometrybutalsohighlightpotentialdetrimentstothematerial.Paper

64:20–4:50pm

SESSION5–PRELIMINARYREPORTS

MarquisA Useofstrategicknowledgeinamathematicalbridgecourse:Differencesbetweenanundergraduateandgraduate

DarrylChamberlainJr.andDragaVidakovic

Theabilitytoconstructproofshasbecomeoneof,ifnotthe,paramountcognitivegoalofeverymathematicalsciencemajor.However,studentscontinuetostrugglewithproofconstructionand,particularly,withproofbycontradictionconstruction.Thispaperissituatedinalargerresearchprojectonthedevelopmentofanindividual’sunderstandingofproofbycontradictioninatransition-to-proofcourse.Thepurposeofthispaperistocompareproofconstructionbetweentwostudents,onegraduateandoneundergraduate,inthesametransition-to-proofcourse.TheanalysisutilizesKeithWeber’sframeworkforStrategicKnowledgeandshowsthatwhilebothstudentsreadilyusedsymbolicmanipulationtoprovestatements,thegraduatestudentutilizedinternalandflexibleprocedurestobeginproofsasopposedtotheexternalandrigidproceduresutilizedbytheundergraduate.Paper

7MarquisB Classifyingcombinations:Dostudentsdistinguishbetween

differenttypesofcombinationproblems?

EliseLockwood,NicholasWassermanandWilliamMcGuffey

Inthispaperwereportonasurveydesignedtotestwhetherornotstudentsdifferentiatedbetweentwodifferenttypesofproblemsinvolvingcombinations–problemsinwhichcombinationsareusedtocountunorderedsetsofdistinctobjects(anatural,commonwaytousecombinations),andproblemsinwhichcombinationsareusedtocountorderedsequencesoftwo(ormore)indistinguishableobjects(alessobviousapplicationofcombinations).Wehypothesizedthatnovicestudentsmayrecognizecombinationsasappropriateforthefirsttypebutnotforthesecondtype,andourresultssupportthishypothesis.Webrieflydiscussthemathematics,sharetheresults,andofferimplicationsanddirectionsforfutureresearch.

Paper30

MarquisC Acasestudyofdevelopingself-efficacyinwritingproofframeworks

AhmedBenkhalti,AnnieSeldenandJohnSelden

Thiscasestudydocumentstheprogressionofonenon-traditionalindividual’sproof-writingthroughasemester.Weanalyzedthevideotapesofthisindividual’sone-on-onesessionsworkingthroughourcoursenotesforaninquiry-basedtransition-to-proofcourse.Ourtheoreticalperspectiveinformedourworkwiththisindividualandincludedtheviewthatproofconstructionisasequenceof(mental,aswellasphysical)actions.Italsoincludedtheuseofproofframeworksasameansofinitiatingawrittenproof.Thisindividual’searlyreluctancetouseproofframeworks,afteraninitialintroductiontothem,wasdocumented,aswellasherlateracceptanceof,andproficiencywith,them.Bytheendofthefirstsemester,shehaddevelopedconsiderablefacilitywithboththeformal-rhetoricalandproblem-centeredpartsofproofsandasenseofself-efficacy.Paper

44GrandBallroom5

ResultsfromtheGroupConceptInventory:Exploringtheroleofbinaryoperationinintroductorygrouptheorytaskperformance

KathleenMelhuishandJodiFasteen

Binaryoperationsareanessential,butoftenoverlookedtopicinadvancedmathematics.WepresentresultsrelatedtostudentunderstandingofoperationfromtheGroupConceptInventory,aconceptuallyfocused,grouptheorymultiple-choicetest.Wepairresultsfromover400studentresponseswith30follow-upinterviewstoillustratetherolebinaryoperationunderstandingplayedintasksrelatedtoamultitudeofgrouptheoryconcepts.Weconcludebyhypothesizingpotentialdirectionsforthecreationofaholisticbinaryoperationunderstandingframework.Paper

56

CityCenterA Onlinecalculushomework:Thestudentexperience

AndrewKrause

TheMAAadvertisesthattheonlinehomeworksystemWeBWorKisusedsuccessfullyatover700collegesanduniversities,andtheinstitutionselectedformystudyhasimplementedWeBWorKuniversallyacrossallcalculuscourses.Iusedamixedmethodapproachtoexaminehowstudentsexperienceonlinecalculushomeworkinordertoprovideinsightsastohowonlinehomeworkmightbeimproved.Inparticular,Iexaminedthebehaviors,perceptions,andresourcesassociatedwithonlinehomework.Asurveywasadministeredtoallstudentsinthemainstreamcalculuscoursethatprovidesquantitativeinformationaboutgeneraltrendsandinformsfurtherquestioning.Forexample,morethanhalfofstudentsreportedthattheyneverstudycalculuswithclassmatesnorinofficehours.Intandemwiththelargesurvey,Ialsocloselystudiedtheonlinehomeworkexperienceof4studentsthroughscreenrecordingsandinterviews.Paper

68CityCenterB Students’symmetricabilityinrelationtotheiruseand

preferenceforsymmetryheuristicsinproblemsolving

MeredithMullerandEricPandiscio

Advancedmathematicalproblemsolvingismarkedbyefficientandfluiduseofmultiplesolutionstrategies.Symmetricargumentsareaptheuristicsandeminentlyusefulinmathematicsandsciencefields.Researchsuggeststhatmathematicsproficiencyiscorrelatedwithspatialreasoning.Wedefinesymmetricabilityasfluencywithmentallyvisualizing,manipulating,andmakingcomparisonsamong2Dobjectsunderrotationandreflection.Wehypothesizethatsymmetricabilityisadistinctsub-abilityofspatialreasoningwhichismoreaccessibletostudentsduetoinherentculturalbiasesforsymmetricbalance.Dostudentswithvaryinglevelsofsymmetricabilityuseorprefersymmetricargumentsinproblemsolving?Howdoessymmetricabilityrelatetoinsightinproblemsolving?Resultsfromapilotstudyindicatethat,amongundergraduates,thereishighvariationinsymmetricability.Further,studentswithhighersymmetricabilitytendtowardsmorepositiveattitudesaboutmathematics.Methods,futureresearch,andimplicationsarediscussed.

Paper49

5:00–5:30pm


MarquisA Waysofunderstandingandwaysofthinkinginusingthederivativeconceptinapplied(non-kinematic)contexts

StevenJones

Muchresearchonstudents’understandingofderivativesinappliedcontextshasbeendoneinkinematics-basedcontexts(i.e.position,velocity,acceleration).However,giventhewiderangeofappliedderivativesinotherfieldsofstudythatarenotbasedonkinematics,thisstudyfocusesonhowstudentsinterpretandreasonaboutappliedderivativesinnon-kinematicscontexts.Threemainwaysofunderstandingorwaysofthinkingaredescribedinthispaper,including(1)invokingtime,(2)overgeneralizationofimplicitdifferentiation,and(3)confusionbetweenderivativeexpressionandoriginalformula.

Paper31

MarquisB Graphsofinequalitiesintwovariables

KyungheeMoon

Inthisstudy,Ianalyzehowpreservicesecondaryteachersrepresentedandexplainedgraphsofthreeinequalities—alinear,acircular,andaparabolic—intwovariables.Ithensuggestnewwaystoexplaingraphsofinequalities,i.e.somealternativestothesolutiontest,basedonthepreserviceteachers’thoughtprocessesandbyincorporatingtheideaofvariation.Thesealternativesexplaingraphsofinequalitiesascollectionsofraysorcurves,whichissimilartographsoffunctionsascollectionsofpointsinonevariablefunctionsandascollectionsofcurvesintwovariablefunctions.Iconcludethestudybyapplyingthealternativestothesolvingofoptimizationproblemsanddiscussingtheimplicationsofthesealternativesforfuturepracticeandresearch.Paper

34

MarquisC Mathematicians’gradingofproofswithgaps

DavidMiller,NicoleEngelke-InfanteandKeithWeber

Inthisstudy,wepresentedninemathematicsprofessorswiththreeproofscontaininggapsandaskedtheprofessorstoassigntheproofsagradeinthecontextofatransition-to-proofcourse.Wefoundthattheparticipantsfrequentlydeductedpointsfromproofsthatwerecorrectandassignedgradesbasedontheirperceptionsofhowwellstudentsunderstoodtheproofs.Theprofessorsalsoindicatedthattheyexpectedlectureproofsinthetransition-to-proofcoursetohavethesamerigorasthosedemandedofstudents,butlectureproofscouldbelessrigorousthantherigordemandedofstudentsinadvancedmathematicscourses.Thispresentationwillfocusonparticipants’rationalesforthesebeliefs.

Paper92

6:10–6:40pm


MarquisA Covariationalandparametricreasoning

TeoPaolettiandKevinMoore

Researchershavearguedthatstudentscandevelopfoundationalmeaningsforavarietyofmathematicstopicsviaquantitativeandcovariationalreasoning.Weextendthisresearchbyexaminingtwostudents’reasoningthatweconjecturedcreatedanintellectualneedforparametricfunctions.Wefirstdescribeourtheoreticalbackgroundincludingdifferentconceptionsofcovariationresearchershavefoundusefulwhenanalyzingstudents’activitiesconstructingandrepresentingrelationshipsbetweencovaryingquantities.Wethenpresenttwostudents’activitiesduringateachingexperimentinwhichtheyconstructedandreasonedaboutcovaryingquantitiesandhighlightaspectsofthestudents’reasoningthatweconjecturecreatedanintellectualneedforparametricfunctions.Weconcludewithimplicationsthestudents’activitiesandreasoninghaveforfutureresearchandcurriculumdesign.Paper

20

MarquisB Re-claimingduringproofproduction

DavidPlaxco

Inthisresearch,IsetouttoelucidatetheconstructofRe-Claiming-awayinwhichstudents’conceptualunderstandingrelatestotheirproofactivity.ThisconstructemergedduringabroaderresearchprojectinwhichIanalyzeddatafromindividualinterviewswiththreestudentsfromajunior-levelModernAlgebracourseinordertomodelthestudents’understandingofinverseandidentity,modeltheirproofactivity,andexploreconnectionsbetweenthetwomodels.Eachstageofanalysisconsistedofiterativecoding,drawingongroundedtheorymethodology(Charmaz,2006;Glaser&Strauss,1967).Inordertomodelconceptualunderstanding,Idrawontheform/functionframework(Saxe,etal.,1998).IanalyzeproofactivityusingAberdein’s(2006a,2006b)extensionofToulmin’s(1969)modelofargumentation.ReflectionacrossthesetwoanalysescontributedtothedevelopmentoftheconstructofRe-Claiming,whichIdescribeandexploreinthisarticle.Paper

125MarquisC Limitationsofa"chunky"meaningforslope

CameronByerley,HyunkyoungYoonandPatrickThompson

Thispaperwillinvestigatethequestion“Whatmathematicalmeaningsdohighschoolmathematicsteachersholdforslope?”Itwillalsoinvestigatetowhatextentthesemeaningsbuildonanimageofquotientasameasureofrelativesize.Thedatacomesfromtheadministrationofthediagnosticinstrumentnamed“MeaningsforMathematicsTeachingSecondaryMath”(MMTsm).Paper

956:10–7:00pmGrandFoyer

POSTERSESSION&RECEPTION

Clearingthewayformindsetchangedthroughformativeassessment

RebeccaDibbsandJenniePatterson

OneofthereasonsfortheexodusinSTEMmajorsistheintroductorycalculuscurriculum.AlthoughthereisevidencethatcurriculalikeCLEARcalculuspromotedsignificantgainsinstudents’growthmindset,itisunclearhowthiscurriculumpromotesmindsetchanges.ThepurposeofthiscasestudywastoinvestigatewhichfeaturesofCLEARCalculuspromotedpositivechangesinstudents’mindsets.AfteradministeringthePatternsofAdaptiveLearningScaletoassessstudents’initialmindsetinonesectionofcalculus,fourstudentswereselectedforinterviews.Althoughparticipantswereselectedformaximalvariationintheirmindsetatthebeginningofthecourse,therewerealotofsimilarthemesintheirinterviews.StudentscitedthatCLEARCalculuscurriculumchallengestheminwaysthatfacilitatesdeepercomprehensivelearningthanthatofatraditionalcalculuscourse.

Paper

128 StudentinterestincalculusI

DerekWilliams

ThisreportsonasecondaryanalysisofdatacollectedbytheMathematicalAssociationofAmerica’sCharacteristicsofSuccessfulProgramsinCollegeCalculus(2015).Surveydatawerecollectedfrommorethan700instructors,androughly14,000studentsmakingthesedataidealformultiplelevelanalysistechniques(Raudenbush&Bryk,2002).Here,thesedataareusedtoanalyzestudents’interestinCalculusI.Resultssuggestthatstudentswithhigherfrequenciesofpresentingtotheirclassmates,collaboratingwithpeers,workingindividually,explainingtheirwork,andtakingCalculusIwithanexperiencedinstructortendtobemoreinterestedinclass

Paper

Usingreadingjournalsincalculus

TaraDavisandAnnelieseSpaeth

InparallelstudiesduringtheFall2015semester,weexaminedtheeffectsofassigningreadingjournalsinafirstsemestercalculuscourse.Atthebeginningofthesemester,studentsweregiveninstructionsabouthowtoreadthetextbook.Onalternatingweeks,studentswereaskedtocompletejournalassignments-theseincludedtakingreadingnotes,respondingtoapromptquestion,andreflectinguponanyconfusingportionsofthereading.Acomparisonbetweenstudentquizscoresfromweeksduringwhichjournalswereassignedandquizscoresfromweeksduringwhichnojournalswereassignedwillbegiven,andimplicationsforteachingwillbediscussed.

Paper

Enrichingstudent’sonlinehomeworkexperienceinpre-

calculuscourses:Hintsandcognitivesupports

NathanWakefieldandWendySmith

AspartofreformingourPre-Calculuscourses,werealizedthatreformstoinstructionneededtobeaccompaniedbyreformstothehomework.WeutilizedtheonlinehomeworksystemWeBWorKbutrecognizedourstudentswantedmoresupportonmissedquestions.WebWorK“hints”providedusanavenuetoaskstudentsleadingquestionstopromptthinkingoverprocedures.Preliminarydatashowmanystudentsareusingthesehintsandthehintsareworkingasintended.WeplantoexpandhintsbeyondourPre-Calculuscourses.TheopensourcenatureofWeBWorKprovidesanopportunityforhintstobeimplementedonawidescale.

Paper

134

Calculusstudents’understandingofthevertexofthequadraticfunctioninrelationtotheconceptofderivative

AnnieChildersandDragaVidakovic

ThepurposeofthisstudywastogaininsightintothirtyCalculusIstudents’understandingoftherelationshipbetweentheconceptofvertexofaquadraticfunctionandtheconceptofthederivative.APOS(action-process-object-schema)theory(Asialaetal.,1996)wasusedinanalysisonstudentwrittenwork,think-aloud,andfollowupgroupinterviews.Students’personalmeaningsofthevertex,includingmisconceptions,wereexplored,andhowstudentsrelatethevertextotheunderstandingofthederivative.Resultsgiveevidenceofstudents’lackofconnectionbetweendifferentproblemtypeswhichusethederivativetofindthevertex.Implicationsandsuggestionsforteachingaremadebasedontheresults.Futureresearchissuggestedasacontinuationtoimprovestudentunderstandingofthevertexofquadraticfunctionsandthederivative.Paper

Aninsightfromadevelopmentalmathematicsworkshop

EddieFuller,JessicaDeshler,MarjorieDarrah,XiangmingWuandMarcelaMeraTrujillo

Inthisreport,wepresentdatafrom404studentsinadevelopmentalmathematicscourseatalargeresearchuniversityandtrytobetterunderstandacademicandnon-academicfactorsthatpredicttheirsuccess.Thisworkisthefirststepinalargerprojecttounderstandwhenscience,technology,engineering,andmathematics(STEM)intendingstudentswhobeginindevelopmentalmathematicscoursesaresuccessfulandcontinuetobesuccessfulinhigher-levelmathematicscourses.Togainsomepreliminaryinsight,weanalyzeSATandACTmathematicsscoresforSTEMandnon-STEMmajorswhosucceededinourdevelopmentalmathematicscourseandalsolookatpersonalitytraitsandanxietylevelsinthesestudents.Specifically,wesoughttoanswerthefollowingquestionsforSTEMintendingstudents:(i)whatSATandACTmathematicsscorescorrelatewithsuccessindevelopmentalmathematics?and(ii)whatothernon-academicfactorspredictsuccessindevelopmentalmathematics?

Paper

138

College-educatedadultsontheautismspectrumandmathematicalthinking

JeffreyTruman

Thisstudyexaminesthemathematicallearningofadultsontheautismspectrum,currentlyorformerlyundergraduatestudents.Iaimtoexpandonpreviousresearch,whichoftenfocusesonyoungerstudentsintheK-12schoolsystem.Ihaveconductedvariousinterviewswithcurrentandformerstudents.Theinterviewsinvolvedacombinationofaskingfortheinterviewee'sviewsonlearningmathematics,self-reportsofexperiences(bothdirectlyrelatedtocoursesandnot),andsomeparticularmathematicaltasks.Ipresentsomepreliminaryfindingsfromtheseinterviewsandideasforfurtherresearch.

Paper

Colloquialmathematicsinmathematicslectures

KristenLew,VictoriaKrupnik,JoeOlsen,TimothyFukawa-ConnellyandKeithWeber

Inthisposter,wefocusonmathematicsprofessors’useofcolloquialmathematicswheretheyexpressmathematicalideasusinginformalEnglish.Weanalyzed80-minutelecturesinadvancedmathematicsfrom11differentmathematicsprofessors.Weidentifiedeachinstancewheremathematiciansexpressedamathematicalideausinginformallanguage.Intheposter,weusethisasabasistopresentcategoriesofthemetaphoricalimagesthatprofessorsusetohelpstudentscomprehendthemathematicsthattheyareteaching.

Paper

Talkingaboutteaching:Socialnetworksofinstructorsofundergraduatemathematics

NanehApkarian

TheRUMEcommunityhasfocusedonstudents’understandingsofandexperienceswithmathematics.Thisprojectshedslightonanotherpartofthehighereducationsystem–thedepartmentalculturesurroundingundergraduatemathematicsinstruction.Thispaperreportsontheinteractionsofmembersofasinglemathematicsdepartment,centeredontheirconversationsaboutundergraduatemathematicsinstruction.Socialnetworkanalysisofthisgroupshedsimportantlightontheinformalstructureofthedepartment.

Paper

Preliminarygeneticdecompositionforimplicitdifferentiation

anditsconnectionstomultivariablecalculus

SarahKerrigan

DerivativesareanimportantconceptinundergraduatemathematicsandacrosstheSTEMfields.Therehavebeenmanystudiesonstudentunderstandingofderivatives,fromgraphingderivativestoapplyingthemindifferentscientificareas.However,thereislittleresearchonhowstudentsconstructanunderstandingofmultivariablecalculusfromtheirunderstandingofsinglevariablecalculus.ThisposterusesAPOStheorytohypothesizethementalreflectionsandconstructionsstudentsneedtomakeinordertosolveandinterpretanimplicitdifferentiationproblemandexaminetheconnectionstomultivariablecalculus.Implicitdifferentiationisoftenthefirsttimestudentsareintroducedtothenotionofafunctiondefinedbytwodependentvariables,aconceptvitalinmultivariablecalculus.Investigatinghowstudentsinitiallyreconcilethisnewideaoftwovariablefunctionscanprovideknowledgeofhowstudentsthingaboutmultivariablecalculus.

Paper

PhysicsStudents'ConstructionandUseofDifferentialElementsinMultivariableCoordinateSystems

BenjaminSchermerhornandJohnThompson

Aspartofanefforttoexaminestudents’understandingofnon-Cartesiancoordinatesystemswhenusingvectorcalculusinthephysicstopicsofelectricityandmagnetism,weinterviewedfourpairsofstudents.Inonetask,developedtoforcethemtobeexplicitaboutthecomponentsofspecificcoordinatesystems,studentsconstructdifferentiallengthandvolumeelementsforanunconventionalsphericalcoordinatesystem.Whileallpairseventuallyarrivedatthecorrectelements,someunsuccessfullyattemptedtoreasonthroughsphericalorCartesiancoordinates,butrecognizedtheerrorwhencheckingtheirwork.Thissuggestsstudents’difficultywithdifferentialelementscomesfromanincompleteunderstandingofthesystems.

Paper

Supportingundergraduateteachers’instructionalchange

GeorgeKusterandWilliamHall

TeachingInquiry-orientedMathematics:EstablishingSupports(TIMES)isanNSF-fundedprojectdesignedtostudyhowwecansupportundergraduateinstructorsastheyimplementchangesintheirinstruction.Onefactorinthedisconnectbetweenthedevelopmentanddisseminationofstudent-centeredcurriculaarethechallengesthatinstructorsfaceastheyworktoimplementthesecurricularinnovations.Forinstance,researchersinvestigatingmathematicians’effortstoteachinstudent-centeredwayshaveidentifiedanumberofchallengesincluding:developinganunderstandingofstudentthinking,planningforandleadingwholeclassdiscussions,andbuildingonstudents’solutionstrategiesandcontributions.Thisresearchsuggestsacriticalcomponentneededtotakecurricularinnovationstoscale:supportsforinstructionalchange.Inthisposterweaddressourcurrentresearcheffortstosupportundergraduateteachers’instructionalchange.

Paper

RUME-andNon-RUME-trackstudents’motivationsofenrollinginaRUMEgraduatecourse

AshleyBerger,RebeccaGrider,JulianaBucher,MollieMills-Weis,FatmaBozkurt,andMilosSavic

Thepurposeofthisongoingstudyistoinvestigatestudents’motivationsintakingagraduate-levelRUMEcourse.Sevenindividualsemi-structuredinterviewswereconductedwithgraduatestudentsenrolledinaRUMEcourseatalargeMidwesternuniversitythathasaRUMEPh.D.optioninthemathematicsdepartment.Ouranalysisofthoseinterviewsutilizedtwotheoreticalframeworks:Self-DeterminationTheory(Ryan&Deci,2000)andHannula’s(2006)needsandgoalsstructure.Preliminaryanalysisoftheinterviewsindicatesthatnon-RUME-trackstudentsareextrinsically,need-motivated,whileRUME-trackstudentsareintrinsically,goal-motivatedwhentakingaRUMEcourse.Theresearchersconjecturethatknowingwhatinfluencesnon-RUME-trackstudentsmayaidinclosingthegapbetweenthemathematicalandRUMEcommunities.

Paper

Teachingandlearninglinearalgebraintermsofcommunityof

practice

DenizKardesBirinci,KarenBogardGivvinandJamesW.Stigler

Communitiesofpractice(CoP)aredefinedasgroupsofpeoplewhoshareaconcern,asetofproblems,orapassionaboutatopic,andwhointeractinanongoingbasistodeepentheirknowledgeandexpertise.Thepurposeofthisstudyistoexaminetheprocessofteachingandlearninglinearalgebrawithinthistheoreticalframework.Inthisresearch,weusedanethnographiccasestudydesigntostudythreelinearalgebrainstructorsandtheirstudentsatalargepublicuniversity.Theinstructorshavedifferenteducationalandculturalbackgrounds.Dataincludedobservations,aLinearAlgebraQuestionnaire,andsemi-structuredinterviews.Weobservedsignificantdifferencesinteachingmethodsbetweentheinstructors.

Paper

Usingthechainruletodevelopsecondaryschoolteachers’MathematicalKnowledgeforTeaching,focusedontherateofchange

ZareenRahman,DebasmitaBasu,KarmenYuandAminataAdewumi

Theunitdescribedinthisstudywasdesignedtoconnectsecondaryandadvancedmathematicaltopics.Itfocusedonhowtheknowledgeofchainruleimpactssecondaryteachers’understandingandteachingofrateofchangesothattheycanaddressstudents’misconceptions.ThisprojectisinformedbytheideaofMathematicalKnowledgeforTeaching,whichencompassesbothsubject-matterknowledgeandpedagogicalcontentknowledgeofteachers.Thegoalwastoenhancesecondaryschoolteachers’teachingoftherateofchangeandtheunitfeaturedtasksconnectingrateofchangeproblemsasseeninhighschoolalgebratotheconceptofchainrule.Theunitwasdesignedtoengagemathematicsteachersindiscourseaboutthecontentlearnedatthecollegeleveltocontentthatistaughtatthesecondaryschoollevel.

Paper

Exploringthefactorsthatsupportlearningwithdigitally-

deliveredactivitiesandtestingincommunitycollegealgebra

ShandyHaukandBryanMatlen

Avarietyofcomputerizedinteractivelearningplatformsexist.Mostincludeinstructionalsupportsintheformofproblemsets.Feedbacktousersrangesfrom“Correct!”tooffersofhintsandpartiallytofullyworkedexamples.Behind-the-scenesdesignofsuchsystemsvariesaswell–fromstaticdictionariesofproblemsto“intelligent”andresponsiveprogrammingthatadaptsassignmentstousers’demonstratedskills,timing,andanarrayofotherlearningtheory-informeddatacollectionwithinthecomputerizedenvironment.Thisposterpresentsbackgroundondigitallearningcontextsandinviteslivelyconversationwithattendeesontheresearchdesignofastudyaimedatassessingthefactorsthatinfluenceteachingandlearningwithsuchsystemsincommunitycollegeelementaryalgebraclasses.

Paper

Whatwouldtheresearchlooklike?Knowledgeforteachingmathematicscapstonecoursesforfuturesecondaryteachers

ShandyHauk,EricHsuandNatashaSpeer

MathematicsCapstoneCourseResourcesisa14-monthproof-of-conceptdevelopmentproject.Collaboratorsacrossthreesitesaimto:(1)developandpilottwomulti-mediaactivitiesforadvancedpre-servicesecondarymathematicsteacherlearning,(2)createguidanceforcollegemathematicsfacultyforeffectiveuseofthematerialswithtargetaudiences,and(3)gatherinformationfrominstructorsandstudentstoinformfutureworktodevelopadditionalmodulesandtoguidesubsequentresearchontheimplementationofthematerials.Thegoalofthisposterpresentationistoprovideinformationaboutcapstonemoduledevelopmentandbrainstormresearchdesignsuggestionswiththelongtermaimofdevelopingagrantproposaltoresearchtheknowledgecollegemathematicsfacultyusetoeffectivelyteachmathematicstofutureteachers.

Paper

Students’experiencesandperceptionsofaninquiry-based

modelofsupplementalinstructionforcalculus

KarmenYu

TheInquiry-BasedInstructionalSupport(IBIS)workshopmodelispartofaninnovativedegreeprogramdesignedtoprepareelementarymathematicsteachers.ThereasonbehindIBISworkshopswastosupportstudentsenrolledin“historicallydifficult”mathematicscourses,suchasCalculusIandCalculusII.ThedesignofIBISworkshopwasframedandguidedbyPeer-LedTeamLearning(PLTL)(Gosser&Roth,1998)andComplexInstruction(Cohen,1994).Duringworkshop,studentsworkinsmallgroupsandengagein“groupworthy”mathematicaltasksthatpromotetheirconceptualunderstandingofCalculustopics(Cohen,1994).Apilotstudywasconductedtoevaluatetheworkshopstructureandthesetasks.Inordertoassessstudents’workshopexperiences,follow-upinterviewswereconducted.Students’responsesindicatedthattheirworkshopexperienceshelpedtopromotethedevelopmentoftheirproblemsolvingskillsandhighlightedthecriticalrolesofthinkingandreasoninginlearningCalculuswithunderstanding.

Paper

Anoverviewofresearchonthearithmeticmeaninuniversityintroductorystatisticscourses

SamCook

Thereisadearthofresearchonthearithmeticmeanattheuniversitylevel.Thisposterwillcoveroverlapofseveralstudies(someunpublished)onuniversitystudents’understandingofthemeananduniversitystatisticsinstructors’beliefsabouttheirstudents’understandingsofthemean.

Paper

Reasoningaboutchanges:aframeofreferenceapproach

SuraniJoshua

InaRUME18TheoreticalReportmyco-authorsandIpresentedourcognitivedescriptionofaconceptualizedframeofreference,consistingofmentalcommitmentstounits,referencepoints,anddirectionalityofcomparisonwhenthinkingaboutmeasures.HereIpresentapilotstudyonhowafocusonconceptualizingaframeofreferenceimpactsstudents’abilitytoreasonquantitativelyaboutchanges.Thetwo-partempiricalstudyconsistedofclinicalinterviewswithseveralstudentsfollowedbyteachinginterviewswiththreestudentschosenbecauseoftheirvaryingabilitiestoconceptualizeaframeofreference.Myinitialevidenceshowsthattheabilitytoconceptualizeaframeofreferencegreatlybenefitsstudentsastheyattempttoreasonwithchanges.

Paper

7:00–9:30pmGrandBallroomSalons2-4

DINNERANDPLENARY

PegSmith

FRIDAY,FEBRUARY26,20168:35–9:05am


MarquisA Developmentalmathematicsstudents’useofrepresentationstodescribetheinterceptsoflinearfunctions.

AnneCawley

Thispaperreportsfindingsfromapilotstudythatinvestigatedthewaythatsixcollegestudentsenrolledinadevelopmentalworkshopworkedthroughataskofnineproblemsonlinearfunctions.Specifically,Iinvestigatedtwoaspects,theorderthatstudentscompletedtheproblemsandwhatsourcesofinformationthestudentsusedtofindtherequestedfeatures,andalsothetypesofrepresentations(symbolic,graphical,ornumerical)studentsusedtodescribetheinterceptsofthefunction.Findingssuggestthatstudentshaveanoverwhelmingrelianceonthegraphofthelinearfunctionandthatthereisvariationinthenumberofrepresentationsusedtodescribetheintercepts(Single,Transitional,andMultiUsers).Becausethegraphicalrepresentationisapreferredrepresentation,itmaybewisetobuildstudentknowledgefromthisrepresentation,makingconnectionstootherrepresentations.Thisstudycontributestounderstandingthemathematicalknowledgethatdevelopmentalmathematicsstudentsbringtotheclassroom.Paper

24MarquisB Acasestudyofamathematicteachereducator’suseof

technology

KevinLaforest

Theuseoftechnologyinmathematicsclassroomsremainsanimportantfocusinmathematicseducationduetotheproliferationoftechnologyinsocietyandalagintheimplementationoftechnologyinclassrooms.Inthispaper,Ipresentdatafromclinicalinterviewswithamathematicsteachereducator(MTE)andobservationsfromthatMTE’sclassinordertodiscusshisuseoftechnology.Specifically,IdescribethreethemesthatemergedfromtheMTE’stechnologyuseandhowtheyrelatetohisepistemologicalstance.Thesethemesare:(a)hisdevelopingaclassroomenvironmentaroundtheuseoftechnology,(b)technologyprovidingapreciseanddynamicenvironment,and(c)hisusingtechnologytohelpengenderstudents’mentalimagery.Finally,Idiscusshowtheideasemergingfromthispapercanbehelpfulforthemathematicseducationcommunity.Paper

72

MarquisC Exampleconstructioninthetransition-to-proofclassroom

SarahHanusch

Accuratelyconstructingexamplesandcounterexamplesisanimportantcomponentoflearninghowtowriteproofs.Thisstudyinvestigateshowoneinstructorofatransition-to-proofcoursetaughtstudentstoconstructexamples,andhowherstudentsreactedtotheinstruction.

Paper

1009:15–9:45am


MarquisA Perturbingpractices:Theeffectsofnoveldidacticobjectsoninstruction

KrystenPampel

Theadvancementoftechnologyhassignificantlychangedthepracticesofnumerousprofessions,includingteaching.Whenaschoolfirstadoptsanewtechnology,establishedclassroompracticesareperturbed.Theseperturbationscanhavebothpositiveandnegativeeffectsonteachers’abilitiestoteachmathematicalconceptswiththenewtechnology.Therefore,beforenewtechnologycanbeintroducedintomathematicsclassrooms,weneedtobetterunderstandhowtechnologyeffectsinstruction.Usinginterviewsandclassroomobservations,Iexploredperturbationsinclassroompracticeasaninstructorimplementednoveldidacticobjects.Inparticular,theinstructorwasusingdidacticobjectsdesignedtolaythefoundationfordevelopingaconceptualunderstandingofrationalfunctionsthroughthecoordinationofrelativemagnitudesofthenumeratoranddenominator.Theresultsareorganizedaccordingtoaframeworkthatcapturesleaderactions,communication,expectationsoftechnology,roles,timing,studentengagement,andmathematicalconceptions.

Paper

8

MarquisB Investigatingstudent-learninggainsfromvideolessonsinaflippedcalculuscourse

CassieWilliamsandJohnSiegfried

Theflippedclassroomhasgarneredattentioninpost-secondarymathematicsinthepastfewyears,butmuchoftheresearchonthismodelhasbeenonstudentperceptionsratherthanitseffectontheattainmentoflearninggoals.Insteadofcomparingtoa“traditional”model,inthisstudyweinvestigatedstudent-learninggainsintwoflippedsectionsofCalculusI.Inthissession,wewillfocusonthequestionofdetermininglearninggainsfromdeliveringcontentviavideooutsideoftheclassroom.Inparticular,wewillcomparestudent-learninggainsafterwatchingmoreconceptualvideosversusmoreproceduralones.Wewillsharequalitativeandquantitativedatagatheredfromsurveysandquizzes,aswellasresultsfromin-classassessments.

Paper

17MarquisC Students’formalizationofpre-packagedinformalarguments

MelissaMillsandDovZazkis

Wegavepairsofstudentsenrolledinagraduateanalysisclasstasksinwhichtheywereprovidedwithavideotapedinformalargumentforwhyaresultheldandaskedtoproducearigorousproofofthisresult.Thisprovidedalensintostudents’formalizationprocessandthevariousrolestheseinformalargumentsplayedineachpair’sprovingprocess.Comparingacrossseveralpairsofparticipantsrevealed3distinctrolesinformalargumentscanplayinproving,1)solelyasastartingpoint,2)asareferencethatcanbecontinuallyreturnedtoduringtheprovingprocess,and3)asaconvincingargumentthatdoesnotinformtheprovingprocess.Paper

43

GrandBallroom5

Students’PerceptionsofLearningCollegeAlgebraOnlineusingAdaptiveLearningTechnology

LoriOgden

Adaptivelearningtechnologywasusedintheteachingofanonlinecollegealgebracourse.Asstudentsworkedonthemasterygoalssetforthem,thetechnologyhelpedstudentsidentifycontentthattheyalreadyunderstoodandothercontentthattheyhadyettomaster.Goalorientationtheorysuggeststhatwhenlearningismastery-oriented,astudent’smotivationtolearnmayimprove(Ames&Archer,1988).Qualitativemethodologywasusedtodescribehowstudentsperceivedtheinstructionoftheircollegealgebracourseandtheirlearninginthecourse.Preliminaryfindingssuggestedthatanadaptiveteachingapproachmayhelpbuildstudents’confidencebecausetheycancontrolthepaceofinstructionandchosewheretofocustheireffortwithoutdrawingnegativeattentiontothemselves.

Paper

57CityCenterA Effectofemphasizingadynamicperspectiveontheformal

definitionoflimit

JeremySylvestreandWilliamHackborn

Weattempttodeterminetheefficacyofusinganalternate,equivalentformulationoftheformaldefinitionofthelimitinafirst-yearuniversitycalculuscourseinaidingtheunderstandingofthedefinitionandofalleviatingthedevelopmentofcommonmisconceptionsconcerningthelimit.

Paper

60

CityCenterB Elicitingmathematicians’pedagogicalreasoning

ChristineAndrews-Larson,ValeriePetersonandRachelKeller

GiventheprevalenceofworkintheRUMEcommunitytoexaminestudentthinkinganddevelopinstructionalmaterialsbasedonthisresearch,weargueitisimportanttodocumentthewaysinwhichundergraduatemathematicsinstructorsmakesenseofthisresearchtoinformtheirownteaching.WedrawonHorn’snotionofpedagogicalreasoninginordertoanalyzevideorecordedconversationsofovertwentymathematicianswhoelectedtoattendaworkshoponinquiry-orientedinstructionatalargenationalmathematicsconference.Inthiscontext,weexaminethequestions:(1)Howdoundergraduatemathematicsinstructorsengageineffortstomakesenseofinquiry-orientedinstruction?(2)Howdoesvariationinfacilitationrelatetoinstructors’reasoningabouttheseissues?Preliminaryfindingssuggestthatdifferencesinfacilitationrelatetohowparticipantsengagedinthemathematics,andthatthenatureofparticipants’engagementwiththemathematicswasrelatedtotheirsubsequentpedagogicalreasoning.

Paper85

9:45–10:15am

COFFEEBREAK

10:15–10:45am


MarquisA Doesitconverge?Alookatsecondsemestercalculusstudents'strugglesdeterminingconvergenceofseries

DavidEarlsandEyobDemeke

Despitethemultitudeofresearchthatexistsonstudentdifficultyinfirstsemestercalculuscourses,littleisknownaboutstudentdifficultydeterminingconvergenceofsequencesandseriesinsecondsemestercalculuscourses.Inourpreliminaryreport,weattempttoaddressthisgapspecificallybyanalyzingstudentworkfromanexamquestionthatasksstudentstodeterminetheconvergenceofaseries.Wedevelopaframeworkthatcanbeusedtohelpanalyzethemistakesstudentsmakewhendeterminingtheconvergenceofseries.Inaddition,weanalyzehowstudenterrorsrelatetoprerequisitestheyareexpectedtohaveenteringthecourse,andhowtheseerrorsareuniquetoknowledgeaboutseries.

Paper108

MarquisB Anexampleofalinguisticobstacletoproofconstruction:Doriandthehiddendoublenegative

AnnieSeldenandJohnSelden

Thispaperconsidersthedifficultythatuniversitystudents’mayhavewhenunpackinganinformallywordedtheoremstatementintoitsformalequivalentinordertounderstanditslogicalstructure,andhence,constructaproof.ThissituationisillustratedwiththecaseofDoriwhoencounteredjustsuchadifficultywithahiddendoublenegative.Shewastakingatransition-to-proofcoursethatbeganbyhavingstudentsfirstproveformallyworded“if-then”theoremstatementsthatenabledthemtoconstructproofframeworks,andthereby,makeinitialprogressonconstructingproofs.Butlater,studentswerepresentedwithsomeinformallywordedtheoremstatementstoprove.Wegoontoconsiderthequestionofwhen,andhow,toenculturatestudentsintotheofteninformalwaythattheoremstatementsarenormallywritten,whilestillenablingthemtoprogressintheirproofconstructionabilities.Paper

63MarquisC Studentcharacteristicsandonlineretention:Preliminary

investigationoffactorsrelevanttomathematicscourseoutcomes

ClaireWladis,AlyseHacheyandKatherineConway

Thereisevidencethatstudentsdropoutathigherratesfromonlinethanface-to-facecourses,yetitisnotwellunderstoodwhichstudentsareparticularlyatriskonline.Inparticular,onlinemathematics(andotherSTEM)courseshavenotbeenwell-studiedinthecontextoflarger-scaleanalysesofonlinedropout.Thisstudysurveyedonlineandface-to-facestudentsfromalargeU.S.universitysystem.Resultssuggestthatforonlinecoursesgenerally,gradesaresignificantpredictorsofdifferentialonlineversusface-to-faceperformanceandthatstudentparentsmaybeparticularlyvulnerabletopooronlinecourseoutcomes.Native-bornstudentswerealsovulnerableonline.Thenextstageofthisresearchwillbetoanalyzethefactorsthatarerelevanttoonlineversusface-to-faceretentioninmathematics(andotherSTEM)coursesspecifically.

Paper69

GrandBallroom5

Studentinterpretationandjustificationof“backward”definiteintegrals

VickiSealeyandJohnThompson

Thedefiniteintegralisanimportantconceptincalculus,withapplicationsthroughoutmathematicsandscience.Studiesofstudentunderstandingofdefiniteintegralsrevealseveralstudentdifficulties,someofwhicharerelatedtodeterminingthesignofanintegral.Clinicalinterviewsof5studentsgleanedtheirunderstandingof“backward”definiteintegrals,i.e.,integralsforwhichthelowerlimitisgreaterthantheupperlimitandthedifferentialisnegative.StudentsinitiallyinvokedtheFundamentalTheoremofCalculustojustifythenegativesign.SomestudentseventuallyaccessedtheRiemannsumappropriatelybutcouldnotdeterminehowtoobtainanegativequantitythisway.Weseetheprimaryobstaclehereasinterpretingthedifferentialasawidth,andthusanunsignedquantity,ratherthanadifferencebetweentwovalues.

Paper80

CityCenterA Divergentdefinitionsofinquiry-basedlearninginundergraduatemathematics

SamuelCook,SarahMurphyandTimFukawa-Connelly

Inquiry-basedlearningisbecomingmoreimportantandwidelypracticedinundergraduatemathematicseducation.Asaresult,researchaboutinquiry-basedlearningissimilarlybecomingmorecommon,includingquestionsoftheefficacyofsuchmethods.Yet,thusfar,therehasbeenlittleeffortonthepartofpractitionersorresearcherstocometoadescriptionoftherange(s)ofpracticethatcanorshouldbeunderstoodasinquiry-basedlearning.Asaresult,studies,comparisonsandcritiquescanbedismissedasnotusingtheappropriatedefinition,withoutadjudicatingthequalityoftheevidenceorimplicationsforresearchandteaching.Throughalarge-scaleliteraturereviewandsurveyingofexpertsinthecommunity,thisstudybeginstheconversationaboutpossibleareasofagreementthatwouldallowforaconstituentdefinitionofinquiry-basedlearningandallowfordifferentiationwithnon-inquirypedagogicalpractices.

Paper88

CityCenterB IVTasastartingpointformultiplerealanalysistopics

SteveStrand

TheproofoftheIntermediateValueTheorem(IVT)providesarichandapproachablecontextformotivatingmanyconceptscentraltorealanalysis,suchas:sequenceandfunctionconvergence,completenessoftherealnumbers,andcontinuity.Asapartofthedevelopmentoflocalinstructionaltheory,anRME-baseddesignexperimentwasconductedinwhichtwopost-calculusundergraduatestudentsdevelopedtechniquestoapproximatetherootofapolynomial.Theythenadaptedthosetechniquesintoa(rough)proofoftheIVT.

Paper118

10:55–11:25am


MarquisA SymbolizingandBrokeringinanInquiryOrientedLinearAlgebraClassroom

MichelleZandieh,MeganWawroandChrisRasmussen

Thepurposeofthispaperistoexploretheroleofsymbolizingandbrokeringinfosteringclassroominquiry.Wecharacterizeinquirybothasstudentinquiryintothemathematicsandinstructor’sinquiryintothestudents’mathematics.Disciplinarypracticesofmathematicsarethewaysthatmathematiciansgoabouttheirprofessionandincludepracticessuchasconjecturing,defining,symbolizing,andalgorithmatizing.Inthispaperwepresentexamplesofstudentsandtheirinstructorengaginginthepracticeofsymbolizinginfourways.Weintegratethisanalysiswithdetailregardinghowtheinstructorservesasabrokerbetweentheclassroomcommunityandthebroadermathematicalcommunity.

Paper98

MarquisB Mathematicians’ideaswhenformulatingproofinrealanalysis

MelissaTroudt

Thisreportpresentssomefindingsfromastudythatinvestigatedtheideasprofessionalmathematiciansfindusefulindevelopingmathematicalproofsinrealanalysis.Thisresearchsoughttodescribetheideasthemathematiciansdevelopedthattheydeemedusefulinmovingtheirargumentstowardafinalproof,thecontextsurroundingthedevelopmentoftheseideasintermsofDewey’stheoryofinquiry,andtheevolvingstructureofthepersonalargumentutilizingToulmin’sargumentationscheme.Threeresearchmathematicianscompletedtasksinrealanalysisthinkingaloudininterviewandat-homesettingsandtheirworkwascapturedviavideoandLivescribetechnology.TheresultsofopeniterativecodingaswellastheapplicationofDewey’sandToulmin’sframeworkswerethreecategoriesofideasthatemergedthroughthemathematicians’purposefulrecognitionofproblemstobesolvedandtheirreflectiveandevaluativeactionstosolvethem.Paper

114MarquisC Reinventingthemultiplicationprinciple

EliseLockwoodandBranwenSchaub

Countingproblemsofferopportunitiesforrichmathematicalthinking,yetstudentsstruggletosolvesuchproblemscorrectly.Inanefforttobetterunderstandstudents’understandingofafundamentalaspectofcombinatorialenumeration,wehadtwoundergraduatestudentsreinventastatementofthemultiplicationprincipleduringaneight-sessionteachingexperiment.Inthispresentation,wereportonthestudents’progressionfromanascenttoasophisticatedstatementofthemultiplicationprinciple,andwehighlighttwokeymathematicalissuesthatemergedforthestudentsthroughthisprocess.Weadditionallypresentpotentialimplicationsanddirectionsforfurtherresearch.Paper

1611:35–12:05pm

SESSION12–MIXEDREPORTS

MarquisA Resultsfromanationalsurveyofabstractalgebrainstructors:Mathedissolvingproblemstheydon'thave

TimFukawa-Connelly,EstrellaJohnsonandRachelKeller

Thereissignificantinterestfrompolicyboardsandfundingagenciestochangestudents’experiencesinundergraduatemathematicsclasses.Abstractalgebraspecificallyhasbeenthesubjectofreforminitiatives,includingnewcurriculaandpedagogies,sinceatleastthe1960s;yetthereislittleevidenceaboutwhetherthesechangeinitiativeshaveprovensuccessful.Pursuanttoansweringthisquestion,weconductedasurveyofabstractalgebrainstructorstogenerallyinvestigatetypicalpractices,andmorespecifically,theirknowledge,goals,andorientationstowardsteachingandlearning.Onaverage,moderatelevelsofsatisfactionwerereportedwithregardtothecourseitselforstudentoutcomes;moreover,littleinterestin,orknowledgeof,reformpracticesorcurriculawereidentified.Wefoundthat77%ofrespondentsspendthemajorityofclasstimelecturing–notsurprisingwhenconsidering82%reportedthebeliefthatlectureisthemosteffectivewaytoteach.

Paper

122MarquisB Investigatingamathematicsgraduatestudent’sconstructionof

ahypotheticallearningtrajectory

AshleyDuncan

Thisstudyreportsresultsofhowateacher’smathematicalmeaningsandinstructionalplanningdecisionstransformedwhileparticipatinginandthengeneratingahypotheticallearningtrajectoryonangles,anglemeasureandtheradiusasaunitofmeasurement.Usingateachingexperimentmethodology,aninitialclinicalinterviewwasdesignedtorevealtheteacher’smeaningsforanglesandanglemeasureandtogaininformationabouttheteacher’sinstructionalplanningdecisions.TheteacherparticipatedinaresearchergeneratedHLTdesignedtopromotetheconstructionofproductivemeaningsforanglesandanglemeasureandthenconstructedherownHLTforherstudents.TheinitialinterviewrevealedthattheteacherhadseveralunproductivemeaningsforanglesandanglemeasurethatcausedtheteacherperturbationswhileparticipatinginthetasksoftheresearchergeneratedHLT.Thisparticipationallowedhertoconstructdifferentmeaningsforanglesandanglemeasurewhichchangedherinstructionalplanningdecisions.

Paper

117

MarquisC Ontheaxiomaticformalizationofmathematicalunderstanding

DanielCheshire

Thisstudyadoptsaproperty-basedperspectivetoinvestigatetheformsofabstraction,instantiation,andrepresentationusedbyundergraduatetopologystudentswhenactingtounderstandandusetheconceptofacontinuousfunctionasitisdefinedaxiomatically.Basedonaseriesoftask-basedinterviews,profilecasesarebeingdevelopedtocompareandcontrastthedistinctwaysofthinkingandprocessesofunderstandingobservedbystudentsundergoingthistransition.Aframeworkhasbeenestablishedtointerprettheparticipants’interactionswiththeunderlyingmathematicalpropertiesofcontinuousfunctionswhiletheyreconstructedtheirconceptimagestoreflectatopological(axiomatic)structure.Thiswillprovideinsightintohowsuchpropertiescanbesuccessfullyincorporatedintostudents’conceptimagesandaccessed;andwhichobstaclespreventthis.Preliminaryresultsrevealseveralcoherentcategoriesofparticipants’progressionofunderstanding.Thisreportwilloutlinetheseprofilesandseekcriticalfeedbackonthedirectionofthedescribedresearch.Paper

101GrandBallroom5

Strugglingtocomprehendthezero-productproperty

JohnPaulCook

Thezero-productproperty(ZPP),typicallystatedas‘ifab=0thena=0orb=0,’isanimportantpropertyinschoolalgebra(asatechniqueforsolvingequations)andabstractalgebra(asthedefiningcharacteristicofintegraldomains).WhilethestrugglesofsecondarymathematicsstudentstoemploytheZPParewell-documented,itunclearhowundergraduatestudentspreparingtotakeabstractalgebraunderstandtheZPPastheyenterabstractalgebra.Tothisend,thispaperdocumentsstudents’understandingoftheZPPwhilealsoinvestigatinghowstudentsmightbeabletodevelopandharnesstheirownintuitiveunderstandingsoftheproperty.Preliminaryfindings,inadditiontocharacterizinghowstudentsreasonwithandunderstandtheZPP,indicatethattheZPPisnolesstrivialforadvancedundergraduatestudentsthanitisforsecondarymathematicsstudent�

Paper

102


LUNCH

1:05–1:35pm


MarquisA Examiningstudentattitudesandmathematicalknowledgeinsidetheflippedclassroomexperience

MatthewVoigt

Flippedclassroomsorhybridonlinecoursesarebecomingincreasinglyprevalentattheundergraduatelevelasinstitutionsseekcost-savingmeasureswhilealsodesiringtoimplementtechnologicalinnovationstoattract21stcenturylearners.Thisstudyexaminedundergraduatepre-calculusstudents’(N=427)experiences,attitudesandmathematicalknowledgeinaflippedclassroomformatcomparedtostudentsinatraditionallectureformat.Ourinitialresultsindicatestudentsintheflippedformatweremorepositiveabouttheiroverallclassroomexperiences,weremoreconfidentintheirmathematicalabilities,weremorewillingtocollaboratetosolvemathematicalproblems,andachievedslighthighergainsinmathematicalknowledge.Contrarytopriorresearch,thisstudyindicatedthatamajorityofstudentsintheflippedclassroomwouldtaketheclassagaininthesameformat,butofconcernisthegenderdisparity,indicatingthatfemalestudentsaremuchmorelikelytoresisttakingaclassinaflippedformat.

Paper26

MarquisB Students’meaningsofa(potentially)powerfultoolforgeneralizingincombinatorics

EliseLockwoodandZackeryReed

Inthispaperweprovidetwocontrastingcasesofstudentworkonaseriesofcombinatorialtasksthatweredesignedtofacilitategeneralizingactivity.Thesecontrastingcasesoffertwodifferentmeanings(Thompson,2013)thatstudentshadaboutwhatmightexternallyappeartobethesametool–ageneraloutcomestructurethatbothstudentsspontaneouslydeveloped.Byexaminingthestudents’meanings,weseewhatmadethetoolmorepowerfulforonestudentthantheotherandwhataspectsofhiscombinatorialreasoningandhisabilitytogeneralizepriorworkwereefficacious.Weconcludewithimplicationsanddirectionsforfurtherresearch.

Paper66

MarquisC FrameworkforMathematicalUnderstandingforSecondaryTeaching:AMathematicalActivityperspective

M.KathleenHeidandPatriciaWilson

Aframeworkformathematicalunderstandingforsecondaryteachingwasdevelopedfromanalysisofthemathematicsinclassroomevents.TheMathematicalActivityperspectivedescribesthemathematicalactionsthatcharacterizethenatureofthemathematicalunderstandingthatsecondaryteacherscouldproductivelyuse.Paper

781:45–2:15pm


MarquisA Integratingoralpresentationsinmathematicscontentcoursesforpre-serviceteachers

SayonitaGhoshHajraandAbeerHasan

Inthispaperwereportonastudyofassessment-basedoralpresentationtasksinamathematicscontentcourseforpre-serviceteachersatapublicuniversityinthewesternUnitedStates.Weusedstatisticalinferencetotestforthesignificanceoftheobservedimprovementinpre-serviceteachers’attitudestowardsusingoralpresentationtasksintheirmathematicallearningandtowardsteacherpreparation.Ourresultssuggestthatuseoforalpresentationimprovespre-serviceteachers’attitudesandbeliefstowardsmathematicslearning.Moreover,responsestothepost-presentationquestionnaireprovideinsightsonthebenefitsofusingoralpresentationtasksinmathematicscoursesforpre-serviceteachers.

Paper4

MarquisB Gender,switching,andstudentperceptionsofCalculusI

JessicaEllisandRebeccaCooper

Weanalyzesurveydatatoexplorehowstudents’reportedperceptionsoftheirCalculusIexperiencesrelatetotheirgenderandpersistenceincalculus.Wedrawfromstudentfree-responsesfromuniversitiesinvolvedinacomprehensiveUSnationalstudyofCalculusI.Weperformathematicanalysisonthedata,identifyingquantitativepatternswithinthemesandanalyzedstudentresponsestobetterunderstandthesepatterns.Ouranalysesindicatethatfemalestudentsreportnegativeaffecttowardsthemselvesmoreoftenthanmales,andthatfemalestudentsdiscusstheirhighschoolpreparationdifferentlythanmales.Wediscusshowthesepotentialfactorsmayinfluencestudentpersistenceincalculus.

Paper76

MarquisC Mary,Mary,isnotquitesocontrary:Unlessshe’swearingHilbert’sshoes

StacyBrown

Researchers(Leron,1985;Harel&Sowder,1998)havearguedthatstudents’lackapreferenceforindirectproofsandhavearguedthatthelackofpreferenceisduetoapreferenceforconstructivearguments.Recentempiricalresearch(author,2015),however,whichemployedacomparativeselectiontaskinvolvingadirectproofandanindirectproofofthecontrapositionform,foundnoevidenceofalackofpreferenceforindirectproof.Recognizingthatindirectproofsofthecontradictionformmaydifferfromthosethatemploythecontraposition,thisstudydocumentsstudents’proofpreferencesandselectionrationaleswhenengaginginacomparativeselectiontaskinvolvingadirectproofandanindirectproofofthecontradictionform.

Paper

812:25–2:55pm


MarquisA Graphinghabits:“Ijustdon’tlikethat”

KevinMoore,TeoPaoletti,IrmaStevensandNatalieHobson

Students’waysofthinkingforgraphsremainanimportantfocusinmathematicseducationduetoboththeprevalenceofgraphicalrepresentationsinthestudyofmathematicsandthepersistentdifficultiesstudentsencounterwithgraphs.Inthisreport,wedrawfromclinicalinterviewstoreportwaysofthinking(orhabits)undergraduatestudentsmaintainforassimilatinggraphs.Inparticular,wecharacterizeaspectsintrinsictostudents’waysofthinkingforgraphsthatinhibitedtheirabilitytorepresentcovariationalrelationshipstheyconceivedtoconstitutesomephenomenonorsituation.Asanexample,weillustratethatstudents’waysofthinkingforgraphswerenotproductivefortheirrepresentingarelationshipsuchthatneitherquantity’svalueincreasedordecreasedmonotonically.

Paper

2MarquisB TheEffectofMathematicsHybridCourseonStudents’

MathematicalBeliefs

Kuo-LiangChang,RoxanneBrinkerhoffandEllenBackus

Computer-basedcourses(e.g.,onlineorhybrid)havesignificantlychangedthedesignofpedagogyandcurriculuminthepastdecade,whichincludeonlineteachingandlearningonmathematicseducation.Asbeliefsplayanessentialroleonachievement,theimpactofcomputer-basedcoursesonmathematicalbeliefsisstillunderdeveloped.Inparticular,weareinterestedinwhethermathematicshybridclass(blendofonlineandface-to-face)hasdifferentimpactonstudents’mathematicalbeliefscomparedtoregularface-to-faceclass.Atwo-by-twodesignofinstructionmethod(hybridvs.regular)andmathematicsperformance(highvs.low)wasemployed.Theresultsshowedthatbothhybridandregularclassstudentsbelievedunderstandingandmemorizationwereequallyimportantinmathematicslearning.Hybridclassstudentsshowedmoreflexibilityinselectingsolutionmethodscomparedtoregularclassstudentsontheirbeliefsaboutproblemsolving.Paper

13

MarquisC Usingtheeffectsizesofsubtaskstocompareinstructionalmethods:Anetworkmodel

GarryJohns,ChristopherNakamuraandCurtisGrosse

Networkshavebecomeincreasinglyimportantinstudyingairpollution,energyuse,geneticsandpsychology.Thesedirectedgraphsalsohavefeaturesthatmaybeusefulinmodelingstudentlearningbyansweringquestionssuchasthefollowing:Howcanwedetermineifoneteachingapproachhasbetteroutcomesthanasecondmethod?Inthispaperwepresentaframeworkfordividinganapproachintosubtasks,assigninganumericalvalue(suchasaneffectsize)toeachsubtaskandthencombiningthesevaluestodetermineanoveralleffectivenessratingfortheoriginalapproach.Thisprocessallowsresearcherstoinvestigatepotentialcausesforstudentachievementratherthansimplecorrelations,andcancomparetheeffectivenessofamethodforvarioustypesofstudentsorinstructors.

Paper18

2:55–3:25pm

COFFEEBREAK

3:25–3:55pm


MarquisA Effectofteacherpromptsonstudentproofconstruction

MargaretMorrowandMaryShepherd

Manystudentshavedifficultylearningtoconstructmathematicalproofs.Inanupperlevelmathematicscourseusinginquirybasedmethods,whilethisissomeresearchonthetypesofverbaldiscourseinthesecourses,thereislittle,ifany,researchonteachers’writtencommentsonstudents’work.Thispaperpresentssomeverypreliminaryresultsfromongoinganalysisfromoneteacher’swrittenpromptsonstudents’roughdraftsofproofsforanAbstractAlgebracourse.TheteacherpromptswillinitiallybeanalyzedthroughaframeworkproposedbyBlanton&Stylianou(2014)forverbaldiscourseandtheframeworkwillbemodifiedinthecourseoftheanalysis.Canweunderstandifatypeofpromptis“better”insomesenseingettingstudentstoreflectontheirworkandrefinetheirproofs?Itisanticipatedthatteacherpromptsintheformoftransactivequestionsaremoreeffectiveinhelpingstudentsconstructproofs.Paper

64

MarquisB Secondaryteachersconfrontingmathematicaluncertainty:Reactionstoateacherassessmentitemonexponents

HeejooSuh,HeatherHowellandYvonneLai

Teachingisinherentlyuncertain,andteachingsecondarymathematicsisnoexception.Wetaketheviewthatuncertaintycanpresentopportunityforteacherstorefinetheirpractice,andthatundergraduatemathematicalpreparationforsecondaryteachingcanbenefitfromengagingpre-serviceteachersintaskspresentinguncertainty.Weexamined13secondaryteachers’reactionstomathematicaluncertaintywhenengagedwithconceptsaboutextendingthedomainfortheoperationofexponentiation.Thedataaredrawnfromaninterview-basedstudyofitemsdevelopedtoassessmathematicalknowledgeforteachingatthesecondarylevel.Inourfindings,wecharacterizewaysinwhichteacherseitherdeniedormathematicallyinvestigatedtheuncertainty.PotentialimplicationsforinstructorsincludeusingmathematicaluncertaintytoprovideanopportunityforundergraduatestolearnbothcontentandpracticesoftheCommonCoreStateStandards.Theproposalconcludeswithquestionsaddressinghowundergraduatemathematicsinstructorscoulduseuncertaintyasaresourcewhenteachingpreserviceteachers.

Paper70

MarquisC Students’conceptimageoftangentlinecomparedtotheirunderstandingofthedefinitionofthederivative

BrittanyVincentandVickiSealey

Ourresearchexploresfirst-semestercalculusstudents'understandingoftangentlinesandthederivativeconceptthroughaseriesofthreeinterviewsconductedoverthecourseofonesemester.UsingacombinationofZandieh's(2000)derivativeframeworkandTallandVinner's(1981)notionsofconceptimageandconceptdefinition,ouranalysisexaminestherolethatstudents'conceptimageoftangentlinesplaysintheirconceptualunderstandingofthederivativeconcept.Preliminaryresultsseemtoindicatethatstudentsaremoresuccessfulwhentheirconceptimageoftangentincludesthelimitingpositionofsecantlines,asopposedtoatangentlineasthelinethattouchesthecurveatonepoint.Paper

89

GrandBallroom5

Unraveling,synthesizingandreweaving:Approachestoconstructinggeneralstatements.

DuaneGraysay

Learningprogressionsforthedevelopmentoftheabilitytolookforandmakeuseofmathematicalstructurewouldbenefitfromunderstandinghowstudentsinmathematics-focusedmajorsmightconstructsuchstructuresintheformofgeneralstatements.Theauthorrecruitedtenuniversitystudentstointerviewsfocusedontasksthataskedforthereconstructionofageneralstatementtoaccommodateabroaderdomain.Throughcomparativeanalysisofresponses,fourmajorcategoriesofapproachestosuchtaskswereidentified.Thispreliminaryreportdescribesinbriefthosefourcategories.

Paper102

CityCenterA Studentconceptionsofdefiniteintegrationandaccumulationfunctions

BrianFisher,JasonSamuelsandAaronWangberg

Priorresearchhasshownseveralcommonstudentconceptualizationsofintegrationamongundergraduates.Thisreportfocusesondatafromawrittenassessmentofstudents’viewsondefiniteintegrationandaccumulationfunctionstocategorizestudentconceptualizationsandreportontheirprevalenceamongtheundergraduatepopulation.Analysisoftheseresultsfoundfourcategorizationsforstudentdescriptionsofdefiniteintegrals:antiderivative,area,aninfinitesumofonedimensionalpieces,andalimitofapproximations.Whenaskedaboutanaccumulationfunction,studentresponsesweregroupedintothreecategorizations:thosebasedontheprocessofcalculatingasingledefiniteintegral,thosebasedontheresultofcalculatingadefiniteintegral,andthosebasedontherelationshipbetweenchangesintheinputandoutputvariablesoftheaccumulationfunction.Theseresultswerecollectedaspartofalargerstudyonstudentlearninginmultivariablecalculus,andtheimplicationsoftheseresultsonmultivariablecalculuswillbeconsidered.Paper

113

CityCenterB SupportingstudentsinseeingsequenceconvergenceinTaylorseriesconvergence

JasonMartin,MatthewThomasandMichaelOehrtman

VirtualmanipulativesdesignedtoincreasestudentunderstandingoftheconceptsofapproximationbyTaylorpolynomialsandconvergenceofTaylorserieswereusedincalculuscoursesatmultipleinstitutions.225studentsrespondedtotasksrequiringgraphingTaylorpolynomials,graphingTaylorseries,anddescribingrelationshipsbetweendifferentnotionsofconvergence.Wedetailsignificantdifferencesobservedbetweenstudentswhousedvirtualmanipulativesandthosethatdidnot.WeproposethattheuseofthesevirtualmanipulativespromotesanunderstandingofTaylorseriessupportinganunderstandingconsistentwiththeformaldefinitionofpointwiseconvergence.Paper

1154:05–4:35pm


MarquisA Ontheuseofdynamicanimationstosupportstudentsinreasoningquantitatively

GrantSander

Thisstudyaddressesthewell-documentedissuethatstudentsstruggletowritemeaningfulexpressionsandformulastorepresentandrelatethevaluesofquantitiesinappliedproblemcontexts.Indevelopinganonlineintervention,wedrewfromresearchthatrevealedtheimportanceofandprocessesinvolvedinconceptualizingquantitativerelationshipstosupportstudentsinconceptualizingandrepresentingquantitativerelationshipsinappliedproblemcontexts.Theresultssuggestthattheuseofdynamicanimationswithpromptsthatfocusstudents’attentiononconceptualizingandrelatingquantitiescanbeeffectiveinsupportingstudentsinconstructingmeaningfulexpressionstorepresentthevalueofonequantityintermsofanother,andformulastodefinehowtwoco-varyingquantitieschangetogether.

Paper12

MarquisB Inquiry-orientedinstruction:Aconceptualizationoftheinstructionalthecomponentsandpractices

GeorgeKusterandEstrellaJohnson

Inthispaperweprovideacharacterizationofinquiry-orientedinstruction.Webeginwithadescriptionoftherolesofthetasks,thestudents,andtheteacherinadvancingthemathematicalagenda.Wethenshiftourfocustofourmaininstructionalcomponentsthatarecentraltocarryingouttheseroles:Generatingstudentwaysofreasoning,Buildingonstudentcontributions,Developingasharedunderstanding,andConnectingtostandardmathematicallanguageandnotation.Eachofthesefourcomponentsisfurtherdelineatedintoatotalofeightpractices.ThesepracticesaredefinedandexemplifiedbydrawingontheK-16researchliterature.Asaresult,thisconceptualizationofinquiry-orientedinstructionmakesconnectionsacrossresearchcommunitiesandprovidesacharacterizationthatisnotlimitedtoundergraduate,secondary,orelementarymathematicseducation.Theultimategoalforthisworkistoserveasatheoreticalfoundationforameasureofinquiry-orientedinstruction.Paper

45MarquisC Designresearchoninquiry-basedmultivariablecalculus:

Focusingonstudents’argumentationandinstructionaldesign

OhNamKwon,YounggonBaeandKukHwanOh

Inthisstudy,researchersdesignandimplementaninquiry-basedmultivariablecalculuscoursetoenhancestudents’argumentationinmathematicaldiscussions.Thisresearchaimstounderstandthestudents’argumentationinproofconstructionsactivities,andtoderivethecharacteristicsofthreesitesofintervention:instructionaldesign,classroominteraction,andtheinstructor’srole.Overthecourseof14weeks,18freshmenmathematicseducationmajorsparticipatedinthisstudy.Multiplesourcesofdatawerecollected,students’reasoningintheclassroomdiscussionswereanalyzedwithintheToulmin’sargumentationstructure,andtheinstructionalinterventionsweregraduallyrevisedaccordingtotheiterativecyclicprocessofthedesignresearch.Thestudents’argumentationstructurespresentedintheclassroomgraduallydevelopedintomorecomplicatedformsasthestudyprogressed,andtheresearchersconcludethattheinterventionswereeffectiveatimprovingstudents’arguments.Paper

674:45–5:15pm


MarquisA ThecaseofanundergraduatemathematicscohortofAfricanAmericanmalesstrivingformathematicalexcellence

ChristopherJett

HistoricallyBlackCollegesandUniversities(HBCUs)provideadifferentmilieuasitpertainstosupportingstudentsacademicallyinalldisciplines,andthisstudychampionsanHBCUeffortwithinthecontextofundergraduatemathematics.Specifically,ithighlightsthecaseofacohortof16AfricanAmericanmalemathematicsmajorsatanall-maleHBCU.Theoverarchingresearchquestionsoughttodelvedeeperintotheseparticipants’educationalexperiences.Usingqualitativeresearchmethodsgroundedincriticalracetheory,preliminarydatashowthattheseAfricanAmericanmalemathematicsmajorswereaffirmedraciallyandmathematicallyintheirundergraduatemathematicsspace.Paper

74MarquisB Developinganopen-endedlinearalgebraassessment:initial

findingsfromclinicalinterviews

MuhammadHaider,KhalidBouhjar,KellyFindley,RubyQueaandChristineAndrews-Larson

Theprimarygoalofthisstudywastodesignandvalidateaconceptualassessmentinundergraduatelinearalgebracourse.Weworktowardthisgoalbyconductingsemi-structuredclinicalinterviewswith8undergraduatestudentswhowerecurrentlyenrolledorhadpreviouslytakenlinearalgebra.Wetrytoidentifythevarietyofwaysstudentsreasonedabouttheitemswiththeintentofidentifyingwaysinwhichtheassessmentmeasuredorfailedtomeasurestudents’understandingoftheintendedtopics.StudentswereinterviewedwhiletheycompletedtheassessmentandinterviewdatawasanalyzedbyusingananalyticaltoolofconceptimageandconceptdefinitionofTallandVinner(1981).Weidentifiedtwothemesinstudents’reasoning:thefirstthemeinvolvesstudentsreasoningaboutspanintermsoflinearcombinationsofvectors,andthesecondoneinvolvesstudentsstrugglingtoresolvethenumberofvectorsgivenwiththenumberofentriesineachvector.

Paper75

MarquisC Exploringtensions:Leanne’sstoryofsupportingpre-servicemathematicsteacherswithlearningdisabilities

RobynRuttenberg-RozenandAmiMamolo

Thispaperpresentsacasestudyofamathematicsteachereducator,Leanne,andherstoryoftryingtosupportthedevelopmentoftwopre-serviceelementaryschoolteacherswithrecognizedlearningdisabilities.Weanalyzedatathroughalensofmathematicalknowledgeforteaching,focusinginparticularonconcernsandtensionsabout(i)maintainingacademicrigorwhilemeetingtheemotional,cognitiveandpedagogicalneedsofherstudents,(ii)seeminglyopposingpedagogiesbetweenspecialeducationandmathematicseducationpractices,and(iii)equitableopportunitiesforteacherswithdisabilitiesandtheconsequencesfortheirpotentialpupils.WeofferananalysisofLeanne’spersonalstruggle,highlightingimplicationsforteachereducationandofferingrecommendationsforfutureresearch.

Paper83

GrandBallroom5

TheComplementofRUME:What'sMissingFromOurResearch?

NatashaSpeerandDaveKung

TheResearchinUndergraduateMathematicsEducation(RUME)communityhasgeneratedasubstantialliteraturebaseonstudentthinkingaboutideasintheundergraduatecurriculum.However,notalltopicsinthecurriculumhavebeentheobjectofresearch.ReasonsforthisincludetherelativelyyoungageofRUMEworkandthefactthatresearchtopicsarenotnecessarilydrivenbythecontentoftheundergraduatecurriculum.Whattopicsremainlargelyuntouched?Wegiveapreliminaryanalysis,withaparticularfocusonconceptsinthestandardcalculussequence.Usesforthiskindofanalysisoftheliteraturebaseintheeducationofnoviceresearchersandpotentialfuturedirectionsforfurtheranalysesarediscussed.

Paper86

CityCenterA Exploringpre-serviceteachers’mentalmodelsofdoingmath

BenWescoatt

Thispreliminarystudyexploresthementalmodelspre-serviceteachersholdofdoingmath.Mentalmodelsarecognitivestructurespeopleusewhilereasoningabouttheworld.Thementalmodelsrelatedtomathematicswouldinfluenceateacher’spedagogicaldecisionsandthusinfluencethementalmodelofmathematicsthattheirstudentswouldconstruct.Inthisstudy,pre-serviceelementaryteachersdrewimagesofmathematiciansdoingmathandofthemselvesdoingmath.Usingcomparativejudgements,theyselectedanimagethatbestrepresentedamathematiciandoingmath.Mostimagesofmathematiciansdoingmathwereofamaninfrontofablackboardfilledwithmathematicalsymbols.Themathematiciansappearedhappy.Incontrast,manyimagesofparticipantsshowedthemtobeunhappyorinconfusedstates.Thepreliminaryresultssuggestthattheirsharedmentalmodelofdoingmathisnaïveandshapedbylimitedexperienceswithmathematicsintheclassroom.Paper

90CityCenterB ‘It’snotanEnglishclass’:Iscorrectgrammaranimportantpart

ofmathematicalproofwritingattheundergraduatelevel?

KristenLewandJuanPabloMejia-Ramos

Westudiedthegenreofmathematicalproofwritingattheundergraduatelevelbyaskingmathematiciansandundergraduatestudentstoreadsevenpartialproofsbasedonstudent-generatedworkandtoidentifyanddiscussusesofmathematicallanguagethatwereoutoftheordinarywithrespecttowhattheyconsideredstandardmathematicalproofwriting.Preliminaryresultsindicatetheuseofcorrectgrammarisnecessaryinproofwriting,butnotalwaysaddressedintransition-to-proofcourses.

Paper


PLENARYSESSION

DavidStinson

6:30pm DINNERONYOUROWNSATURDAY,FEBRUARY21,20158:35–9:05am


MarquisA AdaptationsoflearningglasssolutionsinundergraduateSTEMeducation

ShawnFirouzian,ChrisRasmussenandMatthewAnderson

OneofthemainissuesSTEMfacultyfaceispromotingstudentsuccessinlarge-enrollmentclasseswhilesimultaneouslymeetingstudents’andadministrators’demandsfortheflexibilityandeconomyofonlineandhybridclasses.TheLearningGlassisaninnovativenewinstructionaltechnologythatholdsconsiderablepromiseforengagingstudentsandimprovinglearningoutcomes.Inthisreportwesharetheresultsofanefficacystudybetweenanonlinecalculus-basedphysicscourseusingLearningGlasstechnologyandalargeauditorium-stylelecturehalltaughtviadocumentprojector.Bothcoursesweretaughtwiththesameinstructorusingidenticalcontent,includingexamsandhomework.Ourquasi-experimentaldesigninvolvedidenticalpre-andpost-courseassessmentsevaluatingstudents’attitudesandbehaviortowardsscienceandtheirconceptuallearninggains.Resultsarepromising,withequivalentlearninggainsforallstudents,includingminorityandeconomicallydisadvantagedstudents.Paper

65MarquisB Pre-serviceteachers'meaningsofarea

SayonitaGhoshHajra,BetsyMcNealandDavidBowers

Anexploratorystudywasconductedofpre-serviceteachers’understandingofareaatapublicuniversityinthewesternUnitedStates.Forty-threepre-serviceteacherstookpartinthestudy.Theirdefinitionsofareaandtheirresponsestoarea-unitstaskswererecordedthroughoutthesemester.Wefoundawidegapbetweenpre-serviceteachers’meaningofareaandtheiruseofarea-units.Initially,pre-serviceteachershadweakdefinitionsofarea.Overthesemester,thesedefinitionswererefined,butmisconceptionsaboutareaandarea-unitswereilluminatedinactivitiesinvolvingnon-standardunitsandareasofirregularregions.Weconcludethat,despitedetailedmodelsofchildren’sunderstandingofarea,muchworkisneededtounderstandthelearningtrajectoriesofpre-serviceteachers,particularlywhenmisconceptionsexist.

Paper

33

MarquisC Personificationasalensintorelationshipswithmathematics

DovZazkisandAmiMamolo

Personificationistheattributionofhumanqualitiestonon-humanentities(Inagaki&Hatano,1987).Elicitingpersonificationasaresearchmethodtakesadvantageofanaturallyoccurringmeansthroughwhich(some)peoplediscussthenuancedemotionalrelationshipstheyhavewiththoseentities.Inthispaper,weintroducetheelicitingpersonificationmethodforexploringindividuals’imagesofmathematics,aswellasdiscussaninitialsetofapproachesforanalyzingtheresultingdata.Datafrombothpre-serviceteachersandresearchmathematiciansarediscussedinordertoillustratethemethod.Paper

359:15–9:45am


MarquisA Calculusstudents'understandingoflogicalimplicationanditsrelationshiptotheirunderstandingofcalculustheorems

JoshuaCaseandNatashaSpeer

Inundergraduatemathematics,deductivereasoningisanimportantskillforlearningtheoreticalideasandisprimarilycharacterizedbytheconceptoflogicalimplication.Thisplaysroleswhenevertheoremsareapplied,i.e.,onemustfirstcheckifatheorem’shypothesesaresatisfiedandthenmakecorrectinferences.Incalculus,studentsmustlearnhowtoapplytheorems.However,mostundergraduateshavenotreceivedinstructioninpropositionallogic.Howdothesestudentscomprehendtheabstractnotionoflogicalimplicationandhowdotheyreasonconditionallywithcalculustheorems?Resultsfromourstudyindicatedthatstudentsstruggledwithnotionsoflogicalimplicationinabstractcontexts,butperformedbetterwhenworkingincalculuscontexts.Strategiesstudentsused(successfullyandunsuccessfully)werecharacterized.Findingsindicatethatsomestudentsuse“examplegenerating”strategiestosuccessfullydeterminethevalidityofcalculusimplications.Backgroundoncurrentliterature,resultsofourstudy,furtheravenuesofinquiry,andinstructionalimplicationsarediscussed.

Paper28

MarquisB Service-learninginaprecalculusclass:Tutoringimprovesthecourseperformanceofthetutor.

EkaterinaYurasovskaya

Wehaveintroducedanexperiment:aspartofaPrecalculusclass,universitystudentshavebeentutoringalgebraprerequisitestostudentsfromthecommunityviaanacademicservice-learningprogram.Thegoaloftheexperimentwastoimproveuniversitystudents’masteryofbasicalgebraandtoquantitativelydescribebenefitsofservice-learningtostudents’performanceinmathematics.Attheendoftheexperiment,weobserved59%decreaseofbasicalgebraicerrorsbetweenexperimentalandcontrolsections.Thesetupandanalysisofthestudyhavebeeninformedbythetheoreticalresearchonservice-learningandpeerlearning,bothgroundedintheconstructivisttheoryofJohnDewey.Paper

46MarquisC Howwellpreparedarepreserviceelementaryteacherstoteach

earlyalgebra?

FundaGonulates,LeslieNaborsOlah,HeejooSuh,XueyingJiandHeatherHowell

Thisstudyaimstoinvestigateundergraduatepreserviceteachers’knowledgeinthedomainofearlyalgebra.Weconducted90-minuteclinicalinterviewswith15preserviceteachersintheirfourthyearofafive-yearteacherpreparationprogram.Theseinterviewsessionscollectedpreserviceteachers’responsestoaseriesofassessmentitemsdesignedtomeasuretheircontentknowledgeforteachingearlyalgebra,withfollowupquestionsprobingtheircontent-basedreasoning.Wealsocollectedself-reportoftheirpreparationinthiscontentarea.Wefoundthattheparticipantshaddifficultyontheitemstargetingthemeaninganduseofoperationalpropertiesandalsoinevaluatingtheappropriateuseoftheequalsignwhenpresentedwithdifferentusesinstudentwork.Theyreportedthattheyhadfewopportunitiestolearnaboutearlyalgebraasmathematicalcontentandasatopictoteach.Thesefindingscaninformthedevelopmentofteacher-educationcurriculaandsupportmaterials.

Paper79

GrandBallroom5

Undergraduatestudentsproof-readingstrategies:Acasestudyatoneresearchinstitution

EyobDemekeandMateuszPacha-Sucharzewski

Weber(2015)identifiedfiveeffectiveproof-readingstrategiesthatundergraduatestudentsinproof-basedcoursescanusetofacilitatetheirproofcomprehension.FollowingWeber’s(2015)study,wedesignedasurveystudytoexaminehowundergraduatestudents’proof-readingstrategiesrelatetowhatproficientlearnersofmathematics(mathematicsprofessorsandgraduatestudentsinmathematics)sayundergraduatesshouldemploywhenreadingproofs.Ourpreliminaryfindingsare:(i)MajorityoftheprofessorsinourstudyclaimedthatundergraduatesshouldusethestrategiesidentifiedinWeber’s(2015)study,(ii)Professors’responsesignificantlydifferedfromundergraduates’inonlytwoofthefiveproof-readingstrategiesdescribedinWeber’s(2015)study(attemptingtoprovetheorembeforereadingitsproofandillustratingconfusingassertionswithexamples),andfinally(iii)Undergraduatestudents,forthemostpart,tendtoagreewiththeirprofessors’preferredproof-readingstrategies.Paper

106CityCenterA Beyondprocedures:Quantitativereasoninginupper-division

MathMethodsinPhysics

MichaelLoverude

Manyupper-divisionphysicscourseshaveasgoalsthatstudentsshould‘thinklikeaphysicist.’Whilethisisnotwell-defined,mostwouldagreethatthinkinglikeaphysicistincludesquantitativereasoningskills:consideringlimitingcases,dimensionalanalysis,andusingapproximations.However,thereisoftenrelativelylittlecurricularsupportforthesepracticesandmanyinstructorsdonotassessthemexplicitly.Aspartofaprojecttoinvestigatestudentlearninginmathmethods,wehavedevelopedanumberofwrittenquestionstestingtheextenttowhichstudentsinanupper-divisioncourseinMathematicalMethodsinPhysicscanemploytheseskills.Althoughtherearelimitationstoassessingtheseskillswithwrittenquestions,theycanprovideinsighttotheextenttowhichstudentscanapplyagivenskillwhenprompted.Paper

112

CityCenterB Acaseforwholeclassdiscussions:Twocasestudiesoftheinteractionbetweeninstructorroleandinstructorexperiencewitharesearch-informedcurriculum

AaronWangberg,ElizabethGire,BrianFisherandJasonSamuels

Thispaperpresentscasestudiesoftwoinstructorsimplementingaresearchinformedmultivariablecalculuscurriculum.Theanalysis,structuredaroundsocialconstructivistconcepts,focusesontheinteractionsbetweentherolesoftheinstructorinfacilitatingstudentdiscussionsandtheinstructors’experienceswiththeactivities.Thisstudyisapartofanefforttoevaluateandimprovetheproject’seffectivenessinsupportinginstructorsinimplementingtheactivitiestopromoterichdiscussionswithandamongstudents.Wefindtheseinstructorstobefocusedontheirrolesasfacilitatorsforstudent-centeredsmall-groupdiscussionandthattheychoosenottohaveofwholeclassdiscussions.Wearguethatinitiatingwholeclassdiscussionswouldaddressconcernsandnegativeexperiencesreportedbytheinstructors.Paper

1099:45–10:15am

COFFEEBREAK

10:15–10:45am


MarquisA Aframeworkforexaminingthe2-Dand3-Dspatialskillsneededforcalculus

NicoleEngelke,MarjorieDarrahandKristenMurphy

HavingwelldevelopedspatialskillsiscriticaltosuccessinmanySTEMfieldssuchasengineering,chemistry,andphysics;theseskillsareequallycriticalforsuccessinmathematics.Wepresentaframeworkforexamininghowspatialskillsaremanifestedinmathproblems.Weexamineestablishedspatialskillsdefinitionsandcorrelatethemwiththespatialskillsneededtosuccessfullysolveastandardcalculusproblem–findthevolumeofasolidofrevolution.Thisproblemisdeconstructedintostepsandanalyzedaccordingtowhat2-Dand3-Dspatialskillsarenecessarytovisualizetheproblem.Theresultsofapilotstudyinwhichweexaminethespatialskillsthatfirstsemestercalculusstudentspossessarepresentedalongwiththepotentialimplicationsthestudents’skilllevelcouldbringtobearontheproblem.Weconcludewithsuggestionsforremediationofthesespatialskillsinthecalculusclassroomanddirectionsforfutureresearch.Paper

29

MarquisB Waysinwhichengaginginsomeoneelse’sreasoningisproductive

NanehApkarian,ChrisRasmussen,HayleyMilbourne,TommyDreyfus,XuefenGaoandMatthewVoigt

Typicalgoalsforinquiry-orientedmathematicsclassroomsareforstudentstoexplaintheirreasoningandtomakesenseofothers’reasoning.Inthispaperweofferaframeworkforinterpretingwaysinwhichengaginginthereasoningofsomeoneelseisproductiveforthepersonwhoislistening.Theframeworkistheresultofanalysisof10individualproblem-solvinginterviewswith10mathematicseducationgraduatestudentsenrolledinamathematicscontentcourseonchaosandfractals.Thetheoreticalgroundingforthisworkistheemergentperspective(Cobb&Yackel,1996),whichviewsmathematicalprogressasaprocessofactiveindividualconstructionandaprocessofmathematicalenculturation.Theframeworkcapturestherelationshipbetweenengagingwithanother’sreasoning,decentering,elaboratingjustifications,andrefining/enrichingconceptions.

25

PaperMarquisC Transforminggraduatestudents’meaningsforaveragerateof

change

StacyMusgraveandMarilynCarlson

Thisreportoffersabriefconceptualanalysisofaveragerateofchange(AROC)andsharesevidencethatevenmathematicallysophisticatedmathematicsgraduatestudentsstruggletospeakfluentlyaboutAROC.Weofferdatafromclinicalinterviewswithgraduateteachingassistantswhoparticipatedinatleastonesemesterofaprofessionaldevelopmentinterventiondesignedtosupportmathematicsgraduatestudentsindevelopingdeepandconnectedmeaningsofkeyideasofprecalculuslevelmathematicsaspartofabroaderinterventiontosupportmathematicsgraduatestudentsinteachingideasofprecalculusmathematicsmeaningfullytostudents.Theresultsrevealedthatthepost-interventiongraduatestudentsdescribeAROCmoreconceptuallythantheirpre-interventioncounterparts,butmanystillstruggletoverbalizeameaningforAROCbeyondaveragespeed,ageometricinterpretationbasedontheslopeofasecantline,oracomputation.

Paper

9710:55–11:25am


MarquisA Supportinginstitutionalchange:Atwo-prongedapproachrelatedtograduateteachingassistantprofessionaldevelopment

JessicaEllis,JessicaDeshlerandNatashaSpeer

Graduatestudentsteachingassistants(GTAs)areresponsibleforteachingalargepercentageofundergraduatemathematicscoursesandmanyofthemwillgoontocareersaseducators.However,theyoftenreceiveminimaltrainingfortheirteachingresponsibilities,andasaresultoftenarenotsuccessfulasteachers.Inresponse,thereisincreasednationalinterestinimprovingthewaymathematicsdepartmentspreparetheirGTAs.Inthisreport,wesharetheinitialphasesofjointworkaimedatsupportinginstitutionsindevelopingorimprovingaGTAprofessionaldevelopment(PD)program.WereportonfindingsfromanalysesofabaselinesurveydesignedtoprovideinsightsintothecharacteristicsofcurrentGTAPDprogramsintermsoftheircontent,formatandduration.ResultsindicatethattherearemanyinstitutionsseekingimprovementstotheirGTAPDprogram,andthattheirneedsareinlinewiththechangestrategiesthatthejointprojectsareemploying.

Paper37

MarquisB Whystudentscannotsolveproblems:Anexplorationofcollegestudents'problemsolvingprocessesbystudyingtheirorganizationandexecutionbehaviors

KedarNepal

Thisqualitativestudyinvestigatesundergraduatestudents’mathematicalproblemsolvingprocessesbyanalyzingtheirglobalplansforsolvingtheproblems.Thestudentsinthreeundergraduatecourseswereaskedtowritetheirglobalplansbeforetheystartedtosolveproblemsintheirin-classquizzesandexams.Theexecutionbehaviorsoftheirglobalplansandtheirsuccessorfailureinproblemsolvingwereexploredbyanalyzingtheirsolutions.Onlystudentworkthatusedclearandvalidplanswasanalyzed,usingqualitativetechniquestodeterminethesuccess(orfailure)ofstudents’problemsolving,andalsotoidentifythefactorsthatwerehinderingstudents’effortstosolveproblemssuccessfully.Manycategoriesofstudenterrorswereidentified,andhowthoseerrorsaffectedstudents’problemsolvingeffortswillbediscussed.ThisstudyisbasedonGarofaloandLester’s(1985),andalsoSchoenfeld’s(2010)frameworks,whichconsistofsomecategoriesofactivitiesorbehaviorsthatareinvolvedwhileperformingamathematicaltask.Paper

40

MarquisC Supportingpreserviceteachers’useofconnectionsandtechnologyinalgebrateachingandlearning

ErynStehrandHyunyiJung

TheConferenceBoardoftheMathematicalSciencesrecentlyadvocatedformakingconnectionsandincorporatingtechnologyinsecondarymathematicsteachereducationprograms,butprogramsacrosstheUnitedStatesincorporatesuchexperiencestovaryingdegrees.Thisstudyexplorespreservicesecondarymathematicsteachers’opportunitiestoexpandtheirknowledgeofalgebrathroughconnectionsandtheuseoftechnologyandtolearnhowtousebothtosupportteachingandlearningofalgebra.Weexploretheresearchquestion:WhatopportunitiesdosecondarymathematicsteacherpreparationprogramsprovideforPSTstolearnaboutconnectionsandencountertechnologiesinlearningalgebraandlearningtoteachalgebra?WeexaminedatacollectedfromfiveteachereducationprogramschosenfromacrosstheU.S.Ourdatasuggestnotallsecondarymathematicsteacherpreparationprogramsintegrateexperienceswithmakingconnectionsofdifferenttypesandusingtechnologytoenhancelearningacrossmathematicsandmathematicseducationcourses.Wepresentoverallfindingswithexemplars.

Paper59

GrandBallroom5

EquityinDevelopmentalMathematicsStudents’AchievementataLargeMidwesternUniversity

KennethBradfield

Withsomanystudentsenteringcollegeunderpreparedforthemainstreamsequenceofmathematicscourses,mathematicsdepartmentscontinuetoofferdevelopmentalorremedialcourseswithinnovativemethodsofdelivery.Inordertosupportallstudentsintheircollegeeducation,researcherscontinuetoinvestigatetheeffectivenessofundergraduateremediationprogramswithmixedresults.ThispaperprovidesquantitativedatafromanNSF-fundedprojectfromalargeMidwesternuniversityoverthreeyearsofadevelopmentalmathematicscourse.Pre-andpost-measuresshowthatbothurbanandAfrican-Americanstudentsbenefitedthemostfromsupplementalinstructionincontrasttotheonline-onlyformat.Basedontheseresults,Iofferrecommendationsforundergraduatemathematicsdepartmentstosupportequitableopportunitiesforallstudentsensuringasuccessfuldevelopmentalmathematicsprogram.Paper

104

CityCenterB Mathematicians’rationalforpresentingproofs:Acasestudyofintroductoryabstractalgebraandrealanalysiscourses

EyobDemekeandDavidEarls

Proofsareessentialtocommunicatemathematicsinupper-levelundergraduatecourses.Inaninterviewstudywithninemathematicians,Weber(2012)describesfivereasonsforwhymathematicianspresentproofstotheirundergraduatestudents.FollowingWeber’s(2012)study,wedesignedamixedstudytospecificallyexaminewhatmathematicianssayundergraduatesshouldgainfromtheproofstheyreadorseeduringlectureinintroductoryabstractalgebraandrealanalysis.Ourpreliminaryfindingssuggestthat:(i)Asignificantnumberofmathematicianssaidundergraduatesshouldgaintheskillsneededtorecognizevariousprooftypeandprovingtechniques,(ii)consistentwithWeber’s(2012)findings,onlyonemathematiciansaidundergraduatesshouldgainconvictionfromproofs,andfinally(3)somemathematicianspresentedproofforreasonsnotdescribedinWeber’s(2012)studysuchastohelptheirstudentsdevelopappreciationforrigor.Paper

110CityCenterA Studentperformanceonproofcomprehensiontestsin

transition-to-proofcourses

JuanPabloMejiaRamosandKeithWeber

Aspartofaprojectaimedatdesigningandvalidatingthreeproofcomprehensiontestsfortheoremspresentedinatransition-to-proofcourse,weaskedbetween130and200undergraduatestudentsinseveralsectionsofoneofthesecoursestotakelongversions(20to21multiple-choicequestions)ofthesetests.Whileanalysisofthesedataisongoing,wediscusspreliminaryfindingsaboutpsychometricpropertiesofthesetestsandstudentperformanceontheseproofcomprehensionmeasures.Paper

12311:35–12:05pm

SESSION23–MIXEDREPORTS

MarquisA Opportunitytolearnsolvingcontext-basedtasksprovidedbybusinesscalculustextbooks:Anexploratorystudy

ThembinkosiMkhatshwaandHelenDoerr

Thepurposeofthisstudywastoinvestigatetheopportunitiestolearnhowtosolverealisticcontext-basedproblemsthatundergraduatebusinesscalculustextbooksintheUnitedStatesoffertobusinessand/oreconomicsstudents.Todothis,weselectedandanalyzedsixdifferenttextbooksthatarewidelyusedintheteachingofbusinesscalculusnationwide.Therearethreemajorfindingsfromthisstudy:(1)amajorityofthetasksinallthetextbooksusesacamouflagecontext,(2)allthetasksinallthetextbookshavematchinginformation,and(3)onlythreetextbookshadreflectiontasks.Thefindingsofthissuggestthatbusinesscalculustextbooksdonotofferstudentsrichandsufficientopportunitiestolearnhowtosolverealisticproblemsinabusinessand/oreconomiccontext.

Paper

3MarquisB Students’conceptualizationsandrepresentationsofhowtwo

quantities’changetogether

KristinFrank

InthisarticleIdiscussthenatureoftwouniversityprecalculusstudents’meaningsforfunctionsandgraphs.Ifocusonthewaysinwhichthesemeaningsinfluencehowthesestudentsreasonedaboutandrepresentedhowtwoquantitieschangetogether.Myanalysisrevealedthatastudentwhoviewsagraphasastaticshapeanddoesnotseeagraphasarepresentationofhowtwoquantitieschangetogetherwillnotbesuccessfulinconstructingmeaningfulgraphs,evenininstanceswhensheisabletoreasonabouttwoquantitieschangingtogether.Studentsmadeprogressinseeinggraphsasemergentrepresentationsofhowtwoquantitieschangetogetherwhentheyconceptualizedthepoint(x,y)asamultiplicativeobjectthatrepresentedtherelationshipbetweenanxandyvalue.Paper

77

MarquisC Classroomculture,technology,&modeling:Acasestudyofstudents’engagementwithstatisticalideas

DanaKirin,JenniferNollandErinGlover

Advancesintechnologieshavechangedthewaystatisticiansdotheirwork,aswellashowpeoplereceiveandprocessinformation.Thecasestudypresentedherefollowstwogroupsoftwostudentswhoparticipatedinareform-orientedcurriculumthatutilizedtechnologytoengagestudentswithmodelingandsimulationactivitiestodeveloptheirstatisticalliteracy,thinking,andreasoning.Ouranalysisappliesasocialtheoryoflearningandaframeworkforstudentengagementasameansforstudyingstudents’developmentofstatisticalreasoning.Inaddition,weinvestigatetheimpactofacurriculumfocusedonmodelingandsimulationonthedevelopmentofstudents’statisticalreasoningskills.Paper


LUNCH

1:10–1:40pm


MarquisA Studentproblemsolvinginthecontextofvolumesofrevolution

AnandBernardandStevenJones

Theliteratureonproblemsolvingindicatesthatfocusingonstrategiesforspecifictypesofproblemsmaybemorebeneficialthanseekingtodeterminegrand,generalproblemsolvingstrategiesthatworkacrosslargedomains.Giventhisguideline,weseektounderstandandmapoutdifferentstrategiesstudents’usedinthespecificcontextofvolumesofrevolutionproblemsfromcalculus.Ourstudydemonstratesthecomplexnatureofsolvingvolumesofrevolutionproblemsbasedonthemultitudeofdiversepathsthestudentsinourstudytooktoachievethedesired“epistemicform”ofanintegralexpressionforagivenvolumeproblem.Whilethelarge-grained,overarchingstrategyforthesestudentsdidnotdiffermuch,thecomplexitycameinhowthestudentcarriedouteachstepintheiroverallstrategy.

Paper

14

MarquisB Students’conceptionsoffactorialspriortoandwithincombinatorialcontexts

EliseLockwoodandSarahErickson

Countingproblemsofferrichopportunitiesforstudentstoengageinmathematicalthinking,buttheycanbedifficultforstudentstosolve.Inthispaper,wepresentastudythatexaminesstudentthinkingaboutoneconceptwithincounting,factorials,whichareakeyaspectofmanycombinatorialideas.Inanefforttobetterunderstandstudents’conceptionsoffactorials,weconductedinterviewswith20undergraduatestudents.Wepresentakeydistinctionbetweencomputationalversuscombinatorialconceptions,andweexplorethreeaspectsofdatathatshedlightonstudents’conceptions(theirinitialcharacterizations,theirdefinitionsof0!,andtheirresponsestoLikert-responsequestions).Wepresentimplicationsthismayhaveformathematicseducatorsbothwithinandseparatefromcombinatorics,andwediscusspossibledirectionsforfutureresearch.Paper

32MarquisC Whenshouldresearchonproof-orientedmathematical

behaviorattendtotheroleofparticularmathematicalcontent?

PaulChristianDawkinsandShivKarunakaran

Becauseprovingcharacterizesmuchmathematicalpractice,itcontinuestobeaprominentfocusofmathematicseducationresearch.Aspectsofproving,suchasdefinitionuse,exampleuse,andlogic,actassubdomainsforthisareaofresearch.Toyieldsuchcontent-generalclaims,studiesoftendownplayortrytocontrolfortheinfluenceofparticularmathematicalcontent(analysis,algebra,numbertheoryetc.)andstudents’mathematicalmeaningsforthiscontent.Inthispaper,weconsiderthepossiblenegativeconsequencesformathematicseducationresearchofadoptingsuchadomain-generalcharacterizationofprovingbehavior.Wedosobycomparingcontent-generalandcontent-specificanalysesoftwoprovingepisodestakenfromthepriorresearchofthetwoauthorsrespectively.Weintendtosensitizetheresearchcommunitytotheroleparticularmathematicalcontentcanandshouldplayinresearchonmathematicalproving.Paper

481:50–2:20pm


MarquisA Lackingconfidenceandresourcesdespitehavingvalue:Apotentialexplanationforlearninggoalsandinstructionaltasksusedinundergraduatemathematicscoursesforprospectivesecondaryteachers

YvonneLai

Inthispaper,Ireportonaninterview-basedstudyof9mathematicianstoinvestigatetheprocessofchoosingtasksforundergraduatemathematicscoursesforprospectivesecondaryteachers.Participantswereaskedtoprioritizecomplementarylearninggoalsandtasksforanundergraduatemathematicscourseforprospectivesecondaryteachersandtoratetheirconfidenceintheirabilitytoteachwiththosetasksandgoals.Whilethemathematicianslargelyvaluedtasktypesandgoalsthatmathematicseducationresearchershaveproposedtobebeneficialforsuchcourses,themathematiciansalsolargelyexpressedlackofconfidenceintheirabilitytoteachwiththesetasktypesandgoals.Expectancy-valuetheory,incombinationwiththesefindings,isproposedasoneaccountofwhy,despiteconsensusaboutbroadaimsofmathematicalpreparationforsecondaryteaching,theseaimsmaybeinconsistentwithlearningopportunitiesaffordedbyactualtasksandgoalsused.

Paper

94MarquisB Helpinginstructorstoadoptresearch-supportedtechniques:

LessonsfromIBLworkshops

CharlesN.HaywardandSandraL.Laursen

Inquiry-basedlearning(IBL)isaresearch-supportedformofactivelearninginmathematics.Whilestudiescontinuallyshowbenefitsofactivelearning,itisdifficulttogetfacultytoadoptthesemethods.Wepresentresultsfromasetofintensive,one-weekworkshopsdesignedtoteachuniversitymathematicsinstructorstouseIBL.Weusesurveyandinterviewdatatoexplorewhytheseworkshopssuccessfullygotmanyparticipants(atleast58%)toadoptIBL.Resultsareframedthroughathree-stagetheoryofinstructorchangedevelopedbyPaulsenandFeldman(1995).Wefocusspecificallyonthefirststage,‘unfreezing.’Inthisstage,instructorsgainthemotivationtochange,sothesefindingsmayprovidethemostusefullessonsforhelpingmoreinstructorstoadoptresearch-supportedinstructionalstrategies.OneofthekeyfactorsforthehighadoptionofIBLwasportrayingitbroadlyandinclusivelyinavarietyofcontexts,ratherthanasahighlyprescriptivemethod.Paper

36

MarquisC Students’obstaclesandresistancetoRiemannsuminterpretationsofthedefiniteintegral

JosephF.Wagner

Studentsuseavarietyofresourcestomakesenseofintegration,andinterpretingthedefiniteintegralasasumofinfinitesimalproducts(rootedintheconceptofaRiemannsum)isparticularlyusefulinmanyphysicalcontexts.Thisstudyofbeginningandupper-levelundergraduatephysicsstudentsexaminessomeobstaclesstudentsencounterwhentryingtomakesenseofintegration,aswellassomediscomfortsandskepticismsomestudentsmaintainevenafterconstructingusefulconceptionsoftheintegral.Inparticular,manystudentsattempttoexplainwhatintegrationdoesbytryingtointerpretthealgebraicmanipulationsandcomputationsinvolvedinfindingantiderivatives.Thistendency,perhapsarisingfromtheirpastexperienceofmakingsenseofalgebraicexpressionsandequations,suggestsareluctancetousetheirunderstandingof"whataRiemannsumdoes"tointerpret"whatanintegraldoes.”Paper

102:30–3:00pm


MarquisA AnationalinvestigationofPrecalculusthroughCalculus2

ChrisRasmussen,NanehApkarian,DavidBressoud,JessicaEllis,EstrellaJohnsonandSeanLarsen

Wepresentfindingsfromarecentlycompletedcensussurveyofallmathematicsdepartmentsthatofferagraduatedegree(Master’sand/orPhD)inmathematics.Thecensussurveyispartofalargerprojectinvestigatingdepartment-levelfactorsthatinfluencestudentsuccessovertheentireprogressionoftheintroductorymathematicscoursesthatarerequiredofmostSTEMmajors,beginningwithPrecalculusandcontinuingthroughthefullyearofsinglevariablecalculus.Thefindingspaintaportraitofstudents’curricularexperienceswithPrecalculusandsinglevariablecalculus,aswellastheviewpointsheldbydepartmentsofmathematicsaboutthatexperience.Weseethatdepartmentsarenotunawareofthevalueofparticularfeaturescharacteristicofmoresuccessfulcalculusprograms,butthattheyarenotalwayssuccessfulatimplementation.However,ourdataalsosuggesthopeforthefuture.Ourworknotonlyrevealswhatiscurrentlyhappening,butalsowhatischanging,how,andwhy.

Paper23

MarquisB Whennothingleadstoeverything:Novicesandexpertsworkingatthelevelofalogicaltheory

StacyBrown

BuildingonAntoniniandMariotti’s(2008)theorizationofmathematicaltheoremandresearchonstudents’meta-theoreticaldifficultieswithindirectproof,thisstudyexaminesmathematicsmajorsandmathematicians:(1)responsesandapproachestothevalidationtasksrelatedtotheassertionS*→S,whengivenaprimarystatement,S,oftheform∀n,P(n)⇒Q(n))andasecondarystatement,S*oftheformusedinproofsbycontradiction;namely,∄n,P(n)∧~Q(n));and,(2)selectionofastatementtoprovegivenS*andS.Findingsindicatethatnoviceproofwritersresponsesdifferfromadvancedstudentsandmathematiciansbothintheirapproachesandselections,withnovicestendingtobecomeentangledinnaturallanguageantonymsandtoengageinthechunkingof,ratherthanparsingof,quantifiedcompoundstatements.

Paper82

MarquisC Effectsofdynamicvisualizationsoftwareuseonstrugglingstudents’understandingofcalculus:ThecaseofDavid

JulieSuttonandJamesEpperson

Usingdynamicvisualizationsoftware(DVS)mayengageundergraduatestudentsincalculuswhileprovidinginstructorsinsightintostudentlearningandunderstanding.Resultspresentedderivefromaqualitativestudyofninestudents,eachcompletingaseriesoffourindividualinterviews.WediscussthemesarisingfrominterviewswithDavid,astudentexploringmathematicalrelationshipswithDVSwhoearnsaCincalculus.Davidpreferstovisualizewhensolvingmathematicaltasksandpreviousresearchsuggeststhatsuchstudents,whilenotthe‘stars’oftheirmathematicsclassroom,mayhaveadeeperunderstandingofmathematicalconceptsthattheirnon-visualizingpeers.Usingmodifiedgroundedtheorytechniques,weexamineevidenceofuncontrollablementalimagery,theneedtorefocusDavidonsalientaspectsoftheanimations,instanceswhenDavid’sapparentconceptualknowledgeisneitherfullyconnectedtonorsupportedbyproceduralknowledge,andDavid’sfailuretotransferknowledgewhenDVSwasnotofferedduringassessment.

Paper

1203:00–3:30pm

COFFEEBREAK

3:30–4:00pm


MarquisA Measuringstudentconceptualunderstanding:ThecaseofEuler’smethod

WilliamHall,KarenKeeneandNicholasFortune

Thispreliminarypaperreportsonearlyworkforadifferentialequationsconceptinventory,whichisbeingdevelopedforanNSF-fundedprojecttosupportmathematicsinstructorsastheyimplementinquiry-orientedcurricula.Thegoalistoassessstudentlearningofdifferentialequations.Preliminaryresultsshowthattheiterativemethodofdevelopingandfieldtestingitems,conductingstudentinterviews,andmodificationmayprovesuccessfultocompleteavalidconceptinventory.ThefieldtestingandpilotingofquestionsconcerningEuler’smethodshowthatstudentsdorespondastheresearchsuggestsbutthatEuler’smethodcanberecreatedbystudentsandthecorrectresponsecanbe“figuredout.”Paper

50MarquisB Developingmathematicalknowledgeforteachingincontent

coursesforpreserviceelementaryteachers

BillyJackson,JustinDimmel,MeredithMuller

Inrecentyears,muchattentionintheteachereducationliteraturehasbeengiventowaysinwhichinserviceteachersdevelopfacilitywiththeconstructknownasmathematicalknowledgeforteaching(MKT).MuchlessisknownabouttheabilityofpreserviceteacherstoconstructMKT.Toaddressthis,thecurrentpreliminaryreportaddstotheresearchbasebyinvestigatingtwoprimaryquestions:(1)CanteachersbuildMKTintheircontentcourses?,and(2)Canteachersengageinmeaningfulmathematicaldiscourseasaresultoftheircontentcourses?ThereportexaminestheeffectsofasemesterlongcourseonnumberandoperationsdesignedtoallowpreserviceelementaryteachersopportunitiestobuilddifferentaspectsofMKT.Verypreliminaryanalysisshowsthatmanystudentslackthisknowledgeuponenteringthecourse,butmostareabletobegintobuildadegreeoffacilityinitbycoursecompletion.

Paper

47

MarquisC Obstaclesindevelopingrobustproportionalreasoningstructures:Astoryofteachers’thinkingabouttheshapetask

MattWeber,AmiePieroneandAprilStrom

Thispaperpresentssomeinitialfindingsofaninvestigationfocusedonmathematicsteachers’waysofthinkingaboutproportionalrelationships,withanemphasisonmultiplicativereasoning.Deficienciesinproportionalreasoningamongteacherscanbeseriousimpedimentstothedevelopmentofrobustreasoningamongtheirstudents.Assuch,thisstudyfocusesonhowmathematicsteachersreasonthroughtasksthatinvolveproportionalreasoningbyaddressingthefollowingtworesearchquestions:(1)Inwhatwaysdoteachersreasonthroughaspecifictaskdesignedtoelicitproportionalreasoning?and(2)Whatdifficultiesdoteachersencounterwhilereasoningthroughsuchtasks?Thispaperdiscussestheconstructionofarobustproportionalreasoningstructureinthecontextofaspecifictaskanddiscussesoneparticularobstacle,whichimpedestheconstructionofsuchastructure.Paper

107GrandBallroom5

Changesinassessmentpracticesofcalculusinstructorswhilepilotingresearch-basedcurricularactivities

MichaelOehrtman,MatthewWilson,MichaelTallmanandJasonMartin

Wereportouranalysisofchangesinassessmentpracticesofintroductorycalculusinstructorspilotingweeklylabsdesignedtoenhancethecoherence,rigor,andaccessibilityofcentralconceptsintheirclassroomactivity.Ouranalysiscomparedallitemsonmidtermandfinalexamscreatedbysixinstructorspriortotheirparticipationintheprogram(355items)withthosetheycreatedduringtheirparticipation(417items).Priorexamsofthesixinstructorsweresimilartothenationalprofile,butduringthepilotprogramincreasedfrom11.3%ofitemsrequiringdemonstrationofunderstandingto31.7%.Theirquestionsinvolvingrepresentationsotherthansymbolicexpressionschangedfrom36.7%to58.5%oftheitems.Thefrequencyofexamquestionsrequiringexplanationsgrewfrom4%to15.1%,andtheyshiftedfrom0.8%to4.1%ofitemsrequiringanopen-endedresponse.Weexaminequalitativedatatoexploreinstructors’attributionsforthesechanges.Paper

111

CityCenterA Acriticallookatundergraduatemathematicsclassrooms:DetailingmathematicssuccessasaracializedandgenderedexperienceforLatin@collegeengineers

LuisLeyva

Latin@sdemonstratedanincreaseofnearly75%inengineeringdegreecompletionoverthelast15years(NationalScienceFoundation,2015).However,Latin@sremainlargelyunderrepresentedacrossSTEMdisciplineswithscholarscallingforanalysesoftheirundergraduateeducationexperiencestoimproveretention(Cole&Espinoza,2008;Crisp,Nora,&Taggart,2009).WithcalculusasagatekeeperintoadvancedSTEMcourses,undergraduatemathematicsmustbeexaminedasasocialexperienceforLatin@engineeringstudents.ThisreportpresentsfindingsfromaphenomenologicalstudyonmathematicssuccessasaracializedandgenderedexperienceamongfiveLatin@collegeengineersatapredominantlywhiteinstitution.Inlightofrecentcallsforequityconsiderationsinundergraduatemathematicseducation(Adiredja,Alexander,&Andrews-Larson,2015;Rasmussen&Wawro,underreview),thisreportfocusesonLatin@collegeengineers’mathematicsclassroomexperienceswithimplicationsforestablishingmorepositiveandmeaningfulmathematicslearningopportunitiesforLatin@sandotherunderrepresentedpopulationsinSTEM.

Paper

121CityCenterB Students’sense-makingpracticesforvideolectures

AaronWeinbergandMatthewThomas

Therehasbeenincreasedinterestintheuseofvideosforteachingtechniquessuchas“flipped”classrooms.However,thereislimitedevidencethatconnectstheuseofthesevideoswithactuallearning.Thus,thereisaneedtostudythewaysstudentsexperienceandlearnfromvideos.Inthispaper,weusesense-makingframesasatooltoanalyzestudent’svideo-watching.Wedescribepreliminaryresultsfrominterviewswith12studentswhowatchedshortvideosonintroductorystatisticsandprobabilityconceptsanddiscussimplicationsforstudentlearning.

Paper126

4:10–4:40pm


MarquisA Graduatestudents’pedagogicalchangesusingiterativelessonstudy

SeanYee,KimberlyRogersandSimaSharghi

Researchersattwouniversitiesimplementedaniterativelessonstudyprocesswithtengraduatestudentinstructors(GSIs),fivefromeachuniversity’smathematicsdepartment.Overthespanoftwoweeks,eachgroupofGSIsmetwithafacilitatortocollaborativelyplananundergraduatemathematicslesson,implementthelesson,revisetheirlessonplan,reteachthelessontoanotherclassofstudents,andcompleteafinalreflection.Usingamultiplecasestudyqualitativemethodology,wethematicallycodedGSIconsistenciesandrevisionstolessonplanningduringtheiterativeprocessaccordingtothePrinciplestoActionsnationalmathematicalteachingpractices.AtbothuniversitiestherewerespecificteachingpracticesthatGSIsusedthroughouttheiterativelessonstudyandspecificteachingpracticesthatGSIsrevised.Identifyingtheseteachingpracticesoffersinsightintotheutilityandvalueofiterativelessonstudywithgraduatestudentinstructors.

Paper51

MarquisB Investigatingtheroleofasecondaryteacher’simageofinstructionalconstraintsonhisenactedsubjectmatterknowledge

MichaelTallman

Ipresenttheresultsofastudydesignedtodetermineiftherewereincongruitiesbetweenasecondaryteacher’smathematicalknowledgeandthemathematicalknowledgeheleveragedinthecontextofteaching,andifso,toascertainhowtheteacher’senactedsubjectmatterknowledgewasconditionedbyhisconsciousresponsestothecircumstancesheappraisedasconstraintsonhispractice.Toaddressthisfocus,Iconductedthreesemi-structuredclinicalinterviewsthatelicitedtheteacher’srationaleforinstructionaloccasionsinwhichthemathematicalwaysofunderstandingheconveyedinhisteachingdifferedfromthewaysofunderstandinghedemonstratedduringaseriesoftask-basedclinicalinterviews.Myanalysisrevealedthatthattheoccasionsinwhichtheteacherconveyed/demonstratedinconsistentwaysofunderstandingwerenotoccasionedbyhisreactingtoinstructionalconstraints,butwereinsteadaconsequenceofhisunawarenessofthementalactivityinvolvedinconstructingparticularwaysofunderstandingmathematicalideas.

Paper52

MarquisC Learningtothink,talk,andactlikeaninstructor:Aframeworkfornovicetertiaryinstructorteachingpreparationprograms

JessicaEllis

InthisreportIpresentaframeworktocharacterizenovicetertiaryinstructorteachingpreparationprograms.Thisframeworkwasdevelopedthroughcasestudyanalysesoffourgraduatestudentteachingassistantprofessionaldevelopment(GTAPD)programsatinstitutionsidentifiedashavingmoresuccessfulcalculusprogramscomparedtootherinstitutions.Thecomponentsoftheframeworkarethestructureoftheprogram,thedepartmentalandinstitutionalcultureandcontextthattheprogramissituatedwithin,andthetypesofknowledgeandpracticesemphasizedintheprogram.InthisreportIcharacterizeoneoftheprogramsinvolvedinthedevelopmentoftheframeworkasanexampleofhowitisused.Inadditiontocharacterizingexistingprograms,thisframeworkcanbeusedtoevaluateprogramsandaidinthedevelopmentofnewnovicetertiaryinstructorteachingpreparationprograms.Paper

844:50–5:20pm


MarquisA Howshouldyouparticipate?Letmecounttheways

RachelKeller,KarenZwanchandStevenDeshong

RetentionofstudentsinSTEMmajorsisanissueofnationalstabilitybecausegovernmentprojectionsindicateournationtoneedonemillionadditionalSTEMmajorsby2022(PCAST,2012);thusly,thecurrenttrendsinattritionarealarming.StudentsleaveSTEMforvariousreasons,butpoorexperiencesinCalculusIseemtobeasignificantcontributingfactorformanyswitchers,especiallyfemalestudents.Usingdatasituatedwithinalargerstudy(CharacteristicsofSuccessfulProgramsinCollegeCalculus),thepresentreportlooksspecificallyatstudentparticipationanditsinfluenceonCalculusIsuccess.Resultsindicatethatwhileparticipationissignificantlycorrelatedwithsuccess,thiseffectisnotuniformlydistributedacrosstypesofparticipationorgendergroups.Interestingly,overallsuccessrateswereequal,butgenderdifferenceswerenotedinfrequencyofparticipatorybehaviorsanddistributionofgrades;specifically,males(whoearnedmoreAgrades)preferredin-classparticipationandfemalespreferredout-of-classparticipatoryactivities.

Paper22

MarquisB ProbabilisticThinking:Aninitiallookatstudents’meaningsforprobability

NeilHatfield

ProbabilityisthecentralcomponentthatallowsStatisticstoprovideausefultoolformanyfields.Thus,themeaningsthatstudentsdevelopforprobabilityhavethepotentialforlastingimpacts.UsingThompson’s(20015)theoryofmeanings,thisreportsharestheresultsofexamining114undergraduatestudents’conveyedmeaningsforprobabilityaftertheyreceivedinstruction.Paper

61MarquisC Fosteringteacherchangethroughincreasednoticing:Creating

authenticopportunitiesforteacherstoreflectonstudentthinking

AlanO'BryanandMarilynCarlson

Thispaperreportsresultsfromacasestudyfocusingonasecondaryteacher’ssense-makingasshewaschallengedtoreinterprethermeaningsforalgebraicsymbolsandprocesses.Buildingfromtheseopportunities,sheredesignedlessonstogatherinformationabouthowherstudentsconceptualizedquantitiesandhowtheythoughtofvariables,terms,andexpressionsasrepresentingthosequantities’values.Shethenusedthisinformationtorespondproductivelytoherunderstandingofindividualstudents’meaningsandreasoningelicitedduringtheselessons.Wearguethatthiscasestudydemonstratesthepotentialforcoordinatingquantitativereasoningwithteachernoticingasalenstosupportteacherlearningandwerecommendspecificmathematicalpracticesthatcanhelpteachersdevelopmorefocusednoticingofstudents’mathematicalmeaningsduringinstruction.Paper

715:25–6:20pmGrandFoyer

POSTERSESSION

Impactofadvancedmathematicalknowledgeontheteachingandlearningofsecondarymathematics

EileenMurray,DebasmitaBasu,MatthewWright

Thisposterpresentspreliminarydatafromanexploratorystudythataimstoadvanceourunderstandingofthenatureofmathematicsofferedtoprospectivemathematicsteachersbylookingatmathematicalconnections.Inthisstudy,weinvestigatehowin-serviceandpre-servicemiddleschoolteachersmakeconnectionsbetweentertiaryandsecondarymathematicsaswellasifandhowtheunderstandingofconnectionsinfluencesteachers’thoughtsaboutteachingandlearningmathematics.

Paper

Analyzingclassroomdevelopmentsoflanguageandnotationforinterpretingmatricesaslineartransformations.

RubyQuea,ChristineAndrews-Larson

Aspartofalargerstudyofstudentsreasoninginlinearalgebra,thisresearchanalyzeshowstudentsmakesenseoflanguageandnotationintroducedbyinstructorswhenlearningmatricesaslineartransformations.Thispaperexaminestheimplementationofaninquiry-orientedinstructionthatconsistsofstudentsgenerating,composing,andinvertingmatricesinthecontextofincreasingtheheightandleaningaletter“N”placedona2-dimensionalCartesiancoordinatesystem(Wawroet.al.,2012).Ianalyzedtwoclassroomimplementationsandnotedhowinstructorsintroducedandformalizedmathematicallanguageandnotationinthecontextofthisparticularinstructionalsequence,andthenrelatedthattothewaysthatlanguageandnotationweresubsequentlytakenupbystudents.Thisworkwasconductedinordertoenablemetobuildtheoryabouttherelationshipbetweenstudentlearningandthewaysinwhichlanguageandnotationareintroduced.

Paper

WhatdostudentsattendtowhenfirstgraphinginR3?

AllisonDorko

ThisposterconsiderswhatstudentsattendtoastheyfirstencounterR3coordinateaxesandareaskedtographfunctionswithfreevariables.Graphsarecriticalrepresentations,yetstudentsstrugglewithgraphingfunctionsofmorethanonevariable.Becausepriorworkhasrevealedthatstudents’conceptionsofmultivariablegraphareoftenrelatedtotheirconceptionsaboutsinglevariablefunctions,weusedanactor-orientedtransferperspectivetoidentifywhatstudentsseeassimilarbetweengraphingfunctionswithfreevariablesinR2andR3.Weconsideredwhatstudentsattendedtomathematically,andfoundthattheyfocusedonequidistance,parallelism,andcoordinatepoints.

Paper

Supportformathematicians'teachingreforminanonlineworkinggroupforinquiryorienteddifferentialequations

NicholasFortune

Thereismoreneedforresearchonhowmathematicianscanaltertheirteachingstyletoareformapproach(Speer,Smith,&Horvath,2010),especiallyiftheyhavealwaysbeenteachingthesameway(Speer&Wagner,2009;Wagner,Speer,&Rossa,2007).Oneparticularareathatneedsmoreworkisinvestigationsofsupportstructuresformathematicianshopingtoreformtheirteachingpractice.ThisposterfocusesonsupportsdesignedtoaidinthereformofteachingpracticeandspecificallydiscussestheTeachingInquiry-orientedMathematics:ExternalSupports(TIMES)projectandoneonlineworkinggroup(OWG)usedasamodeofsupportintheproject.ResultsindicatethatfacetsoftheOWGaresuccessfulsupportstructuresformathematicianswhodesiretoaligntheirpracticetoaninquiryoriented(IO)approachtoundergraduatedifferentialequations(Rasmussen&Kwon,2007;Rasmussen,2003).

Paper

Students’understandingofmathematicsinthecontextofchemicalkinetics

KinseyBain,AlenaMoonandMarcyTowns

Thisworkexploresgeneralchemistrystudents’useofmathematicalreasoningtosolvequantitativechemicalkineticsproblems.Personalconstructs,avariationofconstructivism,providesthetheoreticalunderpinningforthiswork,assertingthatstudentsengageinacontinuousprocessofconstructingandmodifyingtheirmentalmodelsaccordingtonewexperiences.Thestudyaimedtoanswerthefollowingresearchquestion:Howdonon-majorstudentsinasecond-semestergeneralchemistrycourseandaphysicalchemistrycourseusemathematicstosolvekineticsproblemsinvolvingratelaws?Toanswerthisquestion,semi-structuredinterviewsusingathink-aloudprotocolwereconducted.Ablendedprocessingframework,whichtargetshowproblemsolversdrawfromdifferentknowledgedomains,wasusedtointerpretstudents’problemsolving.Preliminaryfindingsdescribeinstancesinwhichstudentsblendtheirknowledgetosolvekineticsproblems.

Paper

AcomparativestudyofcalculusIatalargeresearchuniversity

XiangmingWu,JessicaDeshler,MarcelaMeraTrujillo,EddieFullerandMarjorieDarrah

Inthisreport,wedescribetheresultsofanalyzingdatacollectedfrom502CalculusIstudentsatalargeresearchuniversityintheU.S.StudentswereenrolledinoneoffivedifferentversionsofCalculusIofferedattheuniversity.Weareinterestedin(i)whetherthedifferentcharacteristicsofeachversionofthecourseaffectstudents’attitudestowardmathematicsand(ii)howeachcoursemightaffectstudents’intentionsofpursuingscience,technology,engineeringandmathematics(STEM)degrees.WeexaminedatarelatedtothesetwoissuesgatheredfromstudentsviasurveysduringonesemesterinCalculusI.

Paper

Activelearninginundergraduateprecalculusandsingle-variablecalculus

NanehApakarianandDanaKirin

ThestudypresentedhereexaminestheactivelearningstrategiescurrentlyinplaceinthePrecalculusthroughsinglevariablecalculussequence.Whilemanylamentthelackofactivelearninginundergraduatemathematics,ourworkrevealstherealitybehindthatfeeling.Resultsfromanationalsurveyofmathematicsdepartmentsallowustoreporttheproportionofcoursesinthemainstreamsequenceutilizingactivelearningstrategies,whatthosestrategiesare,andhowthosestrategiesarebeingimplemented.

Paper

AProposedFrameworkforTrackingProfessionalDevelopmentThroughGTA’s

HayleyMilbourneandSusanNickerson

Thereareseveraldifferentmodelsofgraduateteachingassistantprogramsinmathematicsdepartmentsacrossthenation(Ellis,2015).Oneparticularpublicuniversityhasrecentlyreformattedtheircalculusprogramtowardapeer-mentormodelandthisisthefirstyearofimplementation.Inthepeer-mentormodel,thereisaleadTAwhoservesasasupportfortheotherTAsintheprogram.Becauseofthis,theprofessionaldevelopmentinwhichtheTAsareengagedisformallydirectedbybothfacultyandapeer.WeareinterestedindiscussingaframeworkknownastheVygotskySpaceasamethodologyfortrackingtheappropriationandsharingofpedagogicalpracticesamongthoseresponsibleforcalculusinstruction.

Paper

HowCalculusstudentsatsuccessfulprogramstalkabouttheirinstructors

AnnieBergmanandDanaKirin

TheCSPCC(CharacteristicsofSuccessfulProgramsinCollegeCalculus)projectwasa5-yearstudyfocusedonCalculusIinstructionatcollegesanduniversitiesacrosstheUnitedStateswithoverarchinggoalsofidentifyingthefactorsthatcontributetosuccessfulprograms.Inthisposter,wedrawfromstudentfocusgroupinterviewdatacollectedfromschoolsthatwereidentifiedbytheCSPCCprojectasbeingsuccessful.Theanalyseswewillpresentinthisposterwillcharacterizethewaysinwhichcalculusstudentstalkabouttheirinstructorsinanattempttounderstandhowtheirperceptionsshapetheirexperience.

Paper

Investigatinguniversitystudentsdifficultieswithalgebra

SepidehStewartandStacyReeder

Algebraisfrequentlyreferredtoasthe“gateway”courseforhighschoolmathematics.EvenamongthosewhocompletehighschoolAlgebracourses,manystrugglewithmoreadvancedmathematicsandarefrequentlyunderpreparedforcollegelevelmathematics.Formanyyears,collegeinstructorshaveviewedthefinalproblemsolvingstepsintheirrespectivedisciplinesas“justAlgebra”,butinreality,aweakfoundationinAlgebramaybethecauseoffailureformanycollegestudents.Thepurposeofthisprojectistoidentifycommonalgebraicerrorsstudentsmakeincollegelevelmathematicscoursesthatplaguetheirabilitytosucceedinhigherlevelmathematicscourses.Theidentificationofthesecommonerrorswillaidinthecreationofamodelforintervention.Paper

Aqualitativestudyofthewaysstudentsandfacultyinthebiologicalsciencesthinkaboutandusethedefiniteintegral

WilliamHall

Inthisposter,Isharemymethodsandpilotinterviewresultsconcerningaqualitativestudyofthewaysundergraduatestudentsandfacultyfromthebiologicalsciencesthinkaboutandusethedefiniteintegral.Inthisresearch,Iutilizetask-basedinterviewsincludingfiveappliedcalculustasksinordertoexplorehowstudentsandfacultythinkaboutarea,accumulation,andthedefiniteintegral.Earlyresultsfrompilotinterviewshelpedmerevisetheinterviewprotocolsandindicatethatstudentreasoningmaybeaffectedbyexperienceandcontext.Inpresentingthisposter,Ihopetogainfeedbackfromthecommunityonmyresearchmethodologyandpotentialanalyticalstrategies.Paper

RootofMisconceptions–theIncorporationofMathematical

IdeasinHistory

Kuo-LiangChang

Theevolutionofamathematicalconceptinhistoryhasbeentheprocessofmergingdifferentideastoformamorerich,general,andrigorousconcept.Ironically,students,whenlearningsuchwell-developedconcepts,havesimilardifficultiesandmakethesamemisconceptionsagainandagain.Toillustrate,despitethewell-developedanddefinedconceptofrealnumbers,manystudentsstillhavedifficultiesincomparingfractionsordoingbasicoperationsonirrationalnumbers.Inthisposter,theincorporationofdifferentideastoformageneralandrigorousmathematicalconceptinhistoryisexamined.Students’strugglesandmisconceptionsinlearningtheconceptsareinvestigatedfromtheperspectiveoftheincorporationprocess.Finally,amodelfordifferentiatingandvalidatingthevariationsofageneralmathematicalconceptissuggestedforresolvinglearningdifficultiesandmisconceptions.Paper

StudentExperiencesinaProblem-CenteredDevelopmentalMathematicsClass

MarthaMakowski

Giventhelargenumbersofstudentswhoenrolleachyearindevelopmentalmathematicsclasses,communitycollegeshavestartedcreatingdevelopmentalclassesthatbothengagestudentsinmeaningfulmathematicsandprovidethemwithanefficientpathwaytothecollege-levelcurriculum.Thisstudyexamineshowcommunitycollegestudentsenrolledinaproblem-baseddevelopmentalalgebraclassexperiencethecurriculumandinstruction.Eightstudentsfromatargetclassroomwereinterviewedabouttheirexperiencesintheclass,focusingonhowtheyexperiencedgroupwork,problemsolving,andtheroleoftheteacher.Surveyswerealsousedtomeasuretheirattitudestowardsmathematicsalongseveraldimensionsatthebeginningandendofthesemester.Studentswholikedgroupworkwereenergizedbytheopportunitytoworkwithpeoplewhilelearningandvaluedthemultipleopportunitiesgroupworkprovidedforthemtochecktheirwork.Fewstudentssawgroupworkasanopportunitytoexploreconceptualideas.Theresultssuggestthatimplementationsofsuchclassescouldbenefitfromstructureddiscussionaboutgroupworknormsandexplicitdiscussionaboutthepurposeofsolvingproblems.Paper

Acollaborativeeffortforimprovingcalculusthroughbetter

assessmentpractices

JustinHeavilin,HodsonKyleandBrynjaKohler

Likemanyinstitutionsacrossthecountry,UtahStateUniversity’sDepartmentofMathematicsandStatisticshasembarkedonanefforttoimprovethecalculussequencewiththefollowingobjectives:(1)improveourstudents’comprehensionandapplicationofkeytopics,(2)retain/recruitmorestudentsintoSTEMmajors,and(3)providemoreconsistencyacrosssections.Afterinitialplanningandpreparationinthe2014-15academicyear,newpracticeswerereadyforimplementation.Inthefallof2015,teamsofinstructorsworkedfromcommonguidedcoursenotes,andmetweeklytodiscussinstructionanddevelopcommonassessments.ThisposterdisplaysthemethodologyoftestdesignanditemanalysisweemployedintheCalculus2course.Whileourteamisonlyatthebeginningstagesofthiswork,themethodsforcreatingreliableandrelevantmeasuresofstudentlearningholdpromiseforachievingthegoalsofourreform.

Paper

Exploringstudentunderstandingofthenegativesigninintroductoryphysicscontexts

SuzanneBrahmiaandAndrewBoudreaux

Recentstudiesinphysicseducationresearchdemonstratethatalthoughphysicsstudentsaregenerallysuccessfulexecutingmathematicalprocedures,theystrugglewiththeuseofmathematicalconceptsforsensemaking.Inthisposterweinvestigatestudentreasoningaboutnegativenumbersincontextscommonlyencounteredincalculus-basedintroductoryphysics.Wedescribealarge-scalestudy(N>900)involvingtwointroductoryphysicscourses:calculus-basedmechanicsandcalculus-basedelectricityandmagnetism(E&M).Wepresentdatafromsixassessmentitems(3inmechanicsand3inE&M)thatprobestudentunderstandingofnegativenumbersinphysicscontexts.Ourresultsrevealthatevenmathematicallywell-preparedstudentsstrugglewiththewaythatwesymbolizeinphysics,andthatthevariedusesofthenegativesigninphysicscanpresentanobstacletounderstandingthatpersiststhroughouttheintroductorysequence.

Paper

Classroomobservation,instructorinterview,andinstructorself-reportastoolsindeterminingfidelityofimplementationforanintervention

ShandyHauk,KatieSalgueroandJoyceKaser

Aweb-basedactivityandtestingsystem(WATS)hasfeaturessuchasadaptiveproblemsets,instructionalvideos,anddata-driventoolsforinstructorstousetomonitorandscaffoldstudentlearning.CentraltoWATSadoptionandusearequestionsabouttheimplementationprocess:Whatconstitutes“good”implementationandhowfarfrom“good”isgoodenough?Herewereportonastudyaboutimplementationthatispartofastate-widerandomizedcontrolledtrialexaminingstudentlearningincommunitycollegealgebrawhenaparticularWATSsuiteoftoolsisused.Discussionquestionsforconferenceparticipantsdigintothechallengesandopportunitiesinresearchingfidelityofimplementationinthecommunitycollegecontext,particularlytheroleofinstructionalpracticeasacontextualcomponentoftheresearch.

Paper

Separatingissuesinthelearningofalgebrafrommathematicalproblemsolving

R.CavenderCampbell,KathrynRhoadsandJamesA.MendozaEpperson

Students’difficultyinlearningschoolalgebrahasmotivatedaplethoraofresearchonknowledgeandskillsneededforsuccessinalgebraandsubsequentundergraduatemathematicscourses.However,ingatewaymathematicscoursesforscience,technology,engineering,andmathematicsmajors,studentsuccessratesremainlow.Onereasonforthismaybetothelackofunderstandingofthresholdsinstudentmathematicalproblemsolving(MPS)practicesnecessaryforsuccessinlatercourses.BuildingfromoursynthesisoftheliteratureinMPS,wedevelopedLikertscaleitemstoassessundergraduatestudents’MPS.Weusedthisemergingassessmentandindividual,task-basedinterviewstobetterunderstandstudents’MPS.Preliminaryresultssuggestthatstudents’issuesinalgebradonotprohibitthemfromusingtheirtypicalproblemsolvingmethods.Thus,theassessmentitemsreflectstudents’MPS,regardlessofpossiblemisconceptionsinalgebra,andprovideamechanismforexaminingMPScapacityseparatefromproceduralandconceptualissuesinalgebra.

Paper

Communicativeartifactsofproof:Transitionsfromascertaining

topersuading

DavidPlaxcoandMilosSavic

Withthisposter,wewishtohighlightanimportantaspectoftheprovingprocess.Specifically,werevisitHarelandSowder’s(1998,2007)proofschemestoextendtheauthors’constructsofascertainingandpersuading.Withthisdiscussion,wereflectontheoriginaltheoreticalframeworkinlightofmorerecentresearchinthefieldanddrawfocustoacriticalaspectoftheprovingprocessinwhichtheprovergeneratesthecommunicativeartifactsofproof(CAP)criticaltoshiftsbetweenascertainingtopersuading.WealsodiscusspossiblewaysinwhichanattentiontothepsychologicalandsocialactivitiesinvolvedinthedevelopmentoftheCAPmightinformresearchandinstruction.

Paper

Onthevarietyofthemultiplicationprinciple’spresentationincollegetexts

ZackeryReedandEliseLockwood

TheMultiplicationPrincipleisoneofthemostfoundationalprinciplesofcounting.Unlikefoundationalconceptsinotherfields,wherethereisuniformityinpresentationacrosstextandinstruction,wehavefoundthatthereismuchvarietyinthepresentationoftheMultiplicationPrinciple.Thisposterhighlightsthemultipleaspectsofthisvariety,specificallythosewithimplicationsforthecombinatorialresearchandeducationcommunity.Suchtopicsincludethestatementtypes,languageandrepresentationofstatements,andmathematicalimplications.

Paper Assessingstudents’understandingofeigenvectorsand

eigenvaluesinlinearalgebra

KevinWatson,MeganWawro,andMichelleZandieh

ManyconceptswithinLinearAlgebraareextremelyusefulinSTEMfields;inparticulararetheconceptsofeigenvectorandeigenvalue.Throughexaminingthebodyofresearchonstudentreasoninginlinearalgebraandourownunderstandingofeigenvectorsandeigenvalues,wearedevelopingpreliminaryideasaboutaframeworkforeigentheory.Basedonthesepreliminaryideas,wearealsocreatinganassessmenttoolthatwillteststudents’understandingofeigentheory.Thisposterwillpresentourpreliminaryframework,andexamplesofthemultiple-choice-extendedquestionswehavecreatedtoassessstudentunderstanding.

Paper


Usingadjacencymatricestoanalyzeaproposedlinearalgebraassessment

HayleyMilbourne,KatherineCzeranko,ChrisRasmussenandMichelleZandieh

Anassessmentofstudentlearningofmajortopicsinlinearalgebraiscurrentlybeingcreatedaspartofalargerstudyoninquiry-orientedlinearalgebra.Thisincludesboththeassessmentinstrumentandawaytounderstandtheresults.TheassessmentinstrumentismodeledoffoftheColoradoUpper-divisionElectrostatics(CUE)diagnostic(Wilcox&Pollock,2013).Therearetwopartstoeachquestion:amultiple-choicepartandanexplanationpart.Intheexplanationpart,thestudentisgivenalistofpossibleexplanationsandisaskedtoselectallthatcouldjustifytheiroriginalchoice.Thistypeofassessmentprovidesinformationontheconnectionsmadebystudents.However,analyzingtheresultsisnotstraightforward.Weproposetheuseofadjacencymatrices,asdevelopedbySelinski,Rasmussen,Wawro,&Zandieh(2014),toanalyzetheconnectionsthatstudentsdemonstrate.

Paper

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