IDENTIFYINGANDDEFININGTHECOMPUTATIONALPRACTICES OFINTRODUCTORYPHYSICS By MichaelJonathonObsniuk ADISSERTATION Submittedto MichiganStateUniversity inpartialful˝llmentoftherequirements forthedegreeof PhysicsDoctorofPhilosophy 2020 ABSTRACT IDENTIFYINGANDDEFININGTHECOMPUTATIONALPRACTICESOF INTRODUCTORYPHYSICS By MichaelJonathonObsniuk Computationisanimportantskillthatisusedinalmostallmodernscienti˝cinvesti- gations.Forthisreason,thetaskofeducatingthepopulationontheuseofcomputation inengineeringisofprimaryinteresttomanyprofessionalsfromindustrytoacademia. Althoughtherehasbeenmuchpriorresearchoncomputationineducationbroadly,prior researchwithintheparticularsub-disciplineofintroductoryphysicsstillhasmanyunan- sweredquestionsthatmustbeaddressed.Attheforefrontoftheseunansweredquestions, thereisincreasinginterestinthevariouscomputationalpracticesthatstudentsengagein andthetypesofthinkingthataccompanythem.Accordingly,thisthesisattemptstodeepen theunderstandingofcomputationbyidentifyingandde˝ningthecomputationalpractices thatareindicativeofcomputationalthinkingthatintroductoryphysicsstudentsfrequently engagein. First,weidenti˝edthecommon,lesscommon,andunobservedcomputationalpractices inanovelphysicsclassroomProjectsandPracticesinPhysics( P 3 )byusingatheoretical frameworkandtwoqualitativemethodologies.Identifyingthebroadandsometimesvague computationalpracticesde˝nedbythetheoreticalframeworkwasfacilitatedbybothatask andathematicanalysisappliedtoin-classvideodata. Next,wede˝nedthosepracticesinconcretetermsrelativetothecoursefromwhichdata wascollected.Eachpracticehasasetofcharacteristics,andeachcharacteristichasasetof qualitiesthatcanbede˝nedintermsofthephysicalconceptsthatstudentsmustgrapple withinthisandrelatedcourses. Finally,weprovidediscussiononthepossiblelinesofreasoningbehindagivenprac- tice'sfrequency.Manyofthelearninggoalsthatthecoursewasdesignedaroundinevitably in˛uencedthetypesoffrequenciesofthepracticesthatweidenti˝ed. Answeringthesetypesofquestionsisofimportancetoanyoneinterestedinintegrating computationintotheundergraduatephysicscurriculum.Abetterunderstandingofthe di˙erentcomputationalpracticesthatstudentsengageincanonlyhelptomitigatethe manychallengesassociatedwithteachingcomputation.Accordingly,thisthesisismeantto shedlightonthecomputationalpracticesthatstudentsfrequentlyengageinwhilesolving introductoryphysicsandengineeringproblems. TABLEOFCONTENTS LISTOFTABLES .................................... vi LISTOFFIGURES ................................... viii Chapter1Introduction ............................... 1 Chapter2Background ................................ 7 2.1Computationalthinking.............................7 2.2PhysicsEducationResearch...........................11 2.2.1Implementation..............................11 2.2.2Prior˝ndings...............................16 2.2.3Remainingquestions...........................20 2.3Framework.....................................21 2.4Taskanalysis...................................23 2.5Thematicanalysis.................................27 Chapter3Context .................................. 31 3.1Courseschedule..................................31 3.2VPython......................................32 3.3Pre-classwork...................................33 3.4In-classwork...................................34 3.4.1Analyticproblem.............................36 3.4.2Computationalproblem.........................36 3.4.2.1Minimallyworkingprograms.................37 3.4.2.2Tutorquestions.........................38 3.4.3Feedback/Assessment...........................40 3.5Post-classwork..................................41 Chapter4Motivation ................................ 43 4.1Debugging.....................................44 4.1.1Analysis..................................45 4.1.1.1Recognition...........................46 4.1.1.2Physicsdebugging.......................47 4.1.1.3Resolution............................49 4.1.2Discussion.................................50 4.1.3Conclusion.................................52 Chapter5Observations ............................... 53 5.1Analysis......................................53 5.1.1Datareduction..............................54 5.1.2Codingprocess..............................56 iv 5.1.3Inter-raterreliability...........................60 5.2Computationalpractices.............................62 5.2.1Creatingdata...............................64 5.2.2Analyzingdata..............................68 5.2.3Designingmodels.............................73 5.2.4Assessingmodels.............................77 5.2.5Creatingabstractions...........................82 5.2.6Troubleshootinganddebugging.....................85 5.2.7Thinkinginlevels.............................89 5.2.8Communicatinginformation.......................93 Chapter6Discussion ................................. 97 6.1Findings......................................97 6.1.1Commonpractices............................97 6.1.2Lesscommonpractices..........................101 6.1.3Unobservedpractices...........................104 6.2Limitations....................................105 6.2.1Framework................................106 6.2.1.1Data...............................106 6.2.1.2Modeling............................107 6.2.1.3Problemsolving........................108 6.2.1.4Systems.............................108 6.2.2Course...................................109 6.2.2.1Groupvs.individual......................109 6.2.2.2Sca˙oldingvs.discovery....................109 6.2.2.3Introvs.advanced.......................111 6.2.3Activity..................................111 6.2.4Analysis..................................112 Chapter7Conclusion ................................ 114 7.1Summary.....................................114 7.2Futureresearch..................................116 7.3Concludingremarks................................117 APPENDICES ...................................... 118 AppendixACommonpractices............................119 AppendixBLesscommonpractices..........................136 AppendixCUnobservedpractices...........................145 BIBLIOGRAPHY .................................... 148 v LISTOFTABLES Table2.1:TheframeworkdevelopedbyWeintropet.altodescribethecomputational practicesobservedinscienceandmathematicsclassrooms.Eachcategory containsbetween˝veandsevenindividualpractices,andeachpracticehas betweentwoandsevenfundamentalcharacteristics.............22 Table2.2:SomeofthenecessarystepsthatmustbetakenwhenconstructingaNew- toniangravitationalforceincode.Eachstepisassociatedwiththecon- struction/modi˝cationofalineofcode....................25 Table3.1:Ascheduleforthesemesterfocusingontopicscovered,homework/reading deadlines,andin-classproblems........................35 Table5.1:Thecharacteristicsandassociatedqualitiespertainingtothecomputational practiceofcreatingdata:automatingthecreationofdatathathelpsto advancetowardgoals..............................64 Table5.2:Thecharacteristicsandassociatedqualitiespertainingtothecomputa- tionalpracticeofanalyzingdata:ageneralprocessofanalysisleadingto conclusionsbasedonevidence.........................69 Table5.3:Thecharacteristicsandassociatedqualitiespertainingtothecomputational practiceofdesigningacomputationalmodel:de˝ningcomponents,relating themtooneanother,andusingthemtomakepredictions..........74 Table5.4:Thecharacteristicsandassociatedqualitiespertainingtothecomputational practiceofassessingacomputationalmodel:identifyingassumptionsand validatingthem.................................78 Table5.5:Thecharacteristicsandassociatedqualitiespertainingtothecomputational practiceofcreatingcomputationalabstractions:representingphysicalcon- cepts.......................................82 Table5.6:Thecharacteristicsandassociatedqualitiespertainingtothecomputational practiceoftroubleshootinganddebugging:isolatinganunexpectederror andcorrectingitinasystematicmanner...................85 Table5.7:Thecharacteristicsandassociatedqualitiespertainingtothecomputational practiceofthinkinginlevels:breakingaprogramintodi˙erentlevelsand attributingfeaturestothem..........................90 vi Table5.8:Thecharacteristicsandassociatedqualitiespertainingtothecomputational practiceofcommunicatinginformation:ageneralprocessofcommunication thatdemonstratesanunderstanding......................93 Table6.1:Thecomputationalpracticesthathavebeendeemedcommonareshown withthenumberoftimeseachpracticewasidenti˝ed,thepercentageof itscategorythatitoccupies(i.e.,thenumberoftimesapracticewasob- serveddividedbythetotalnumberofpracticesfromthatcategory),and thepercentageofallthepracticesthatitoccupies(i.e.,thenumberoftimes apracticewasobserveddividedbythetotalnumberofpracticesfromall categories).Horizontaldividersseparatethedi˙erentcategories(i.e.,data, modeling,problemsolving,andsystemsthinking)..............98 Table6.2:Thecomputationalpracticesthathavebeendeemedlesscommonareshown withthenumberoftimeseachpracticewasidenti˝ed,thepercentageof itscategorythatitoccupies,andthepercentageofallthepracticesthatit occupies.....................................101 vii LISTOFFIGURES Figure2.1:GraphicaluserinterfaceforBOXERshowingthegraphicsdata(e.g.,the stepinstructions)andtheresultinggraphicforaspritenamedkey12 Figure2.2:Aprogramillustratingthatthebasiccontrolstructureandintegration algorithmarepre-writtensothatstudentscanfocusonthecomputational forcemodelthatmustbeconstructedinline11...............13 Figure2.3:GraphicaluserinterfaceforanEJSillustratingthenature ofthesoftware.Elements(e.g.,apendulumbob)canbeaddedorremoved fromthedi˙erentpanels(e.g.,thedrawingpanel)inthesimulationview.14 Figure2.4:APhETsimulationillustratingthedependenceofpendulummotionon thelengthofthependulum,themassofthependulumbob,themagnitude ofthelocalaccelerationduetothegravity,andanyfrictionalforces....15 Figure2.5:Glowscriptoutputdemonstratingitsabilitytogeneratethree-dimensional visualizationsofobjects,vectors,andgraphs.Theabilitytoquicklyand accuratelygeneratethree-dimensionalvectorsallowsformore˛exibility andadeeperunderstandingofelectric˝elds.................15 Figure2.6:Asampleofthein-depthanalysisWeatherfordperformed.Eachparticular lineofcodethatthegroupisfocusingonistrackedintimeandcoded accordingtoaresearcher-developedscheme.................18 Figure2.7:Anexpectedsolutiontoacomputationalsatellite-Earthproblemwhere theNewtoniangravitationalforcehasbeenconstructedfromaseparation vectoranditsmagnitude.Theforcecalculationhasbeenincorporated intothemomentumthroughNewton'ssecondlaw,andthemomentumis incorporatedintothepositionthroughapositionupdate..........20 Figure2.8:A˝nalthematicmapshowingthecomponentsofathemenamed tationasThemaincomponentsofthisthemeareow andymen...............................30 Figure3.1:Portionofon-linenotesthatismadeavailabletothestudentsduringthe ˝rstweekofthecourse.Thesenotesintroducethefundamentalprogram- mingideasandalistofcommonerrorswithtipsandtricks........33 viii Figure3.2:Pre-classhomeworkquestionfocusingonthedi˙erentwaysthatthemag- nitudeofavectorcanbeconstructedinVPythoncode:explicitlycoding thesquarerootofthesumofthesquaresofthecomponentsandusing thepre-de˝nedPythonfunction................34 Figure3.3:TheNewtoniangravitationalforceproblemstatementdeliveredtothe studentsinthethirdweekofclass......................37 Figure3.4:TheinitialcodeandvisualizationoftheMWPthatisgiventothestudents inthethirdweekofthecourse........................38 Figure3.5:Aselectionoftutorquestionsthatfocusonthecomputationalmodeleach grouphasconstructed.............................39 Figure3.6:Asnippetofwrittenfeedbackgiventoastudentafterthethirdweek...41 Figure3.7:Aportionofapost-classhomeworkquestiondeliveredinthethirdweek ofthecourse.Thisquestionrequiresstudentstotroubleshootanddebug thecode....................................42 Figure4.1:Interactionsbetweenindividualsformagroup,andthegroupinteracts withthecomputer...............................44 Figure4.2:Thedebuggingprocessnecessarilycorrespondstoaphasebesetoneither sidebythephasesofrecognitionandresolution.Notetheabsenceofa verticalscale,astheverticalseparationmerelyactstodistinguishphases.46 Figure5.1:Aportionoftranscriptmeanttohighlighttheindicationofunspokenand inferredactions.Forexample,line367showsthisgrouplookingintheir notesforanequation.Theequationthatthey˝ndiswrittendowninline 370.......................................55 Figure5.2:Thetemplateusedforthecodingprocess.Eachexcerptisnumbered, eachlineofspeech/actionisnumberedandattributedtoanindividual memberofthegroup,andthethreetypesofrationaleareusedtojustify theclassi˝cationofaparticularpractice...................59 Figure5.3:Examplesofthethreelevelsofcon˝denceareshowningreen,yellow, andredtoindicatehigh,medium,andlowcon˝dence,respectively.Each inter-ratersuggestionisusedtomodifyorsolidifythelevelofcon˝dence giventoaparticularpractice.........................60 ix Figure5.4:Theinitialrationalegeneratedforanexcerptalongwithinter-ratersug- gestionsandsubsequentmodi˝cationovertime.Withtheadditionof somerequestedinformation,thestrengthoftherationalewasimproved andthecon˝dencewaspromotedfrommediumtohigh..........61 Figure5.5:Thefrequencyofeachpracticethatwasfoundwithinouruniquedataset.63 Figure6.1:Theiterativeprocessofmodelingphysicalsystemsthatwasdescribedto theclassonthe˝rstday............................110 x Chapter1 Introduction Sincetheadventofrelativelyinexpensiveandpowerfulcomputers,researchershavebeen interestedintheiruseasbothprofessionalandpedagogicaltools.Theirabilitytoquickly andpreciselyperformextremelylargeamountsofnumericalcalculationsmakesthemwell suitedformodelingandsolvingmodernproblemsintheSTEM˝elds(e.g.,engineeringor biostatistics).Similarly,theirabilitytoeasilygeneraterealisticvisualizationsmakesthem wellsuitedforthecommunicationofscienti˝cinformation.Forthesereasons,computationis indispensableinmodernscienti˝cpursuitsandhasincreasinglybeenthefocusofeducation research[1,2]. Computation,ortheuseofcomputerstoanalyzecomplicatedsystems,continuestogrow inmany˝elds.Givenitsutilityinthesetypesofprofessionaldomains,thetaskofe˙ectively trainingstudentsincomputationhasrisentotheforefrontofSTEMeducationresearch. However,thisimportanttaskhasbeenshowntoinvolvechallenges,astherearevariedskills andpiecesofknowledgethatstudentsmustdevelopamasteryofinordertoe˙ectively utilizecomputation.Still,thedesiretointegratecomputationintotheSTEMcurriculumis strongerthanever[3]. Whileusingcomputationtosolvecomplexphysicsandengineeringproblems,practi- tionersoftenengageinwhatarecalledcomputationalpractices.Computationalpractices canbede˝ned,inoneway,asasynthesisofcomputationalknowledgeandcomputational 1 skillhighlightingtheimportanceofbeingabletoputtheoreticalideastopracticalwork [4].Althoughknowledgeandskillaloneareimportant,beingabletocombinethetwointo ane˙ectivepracticeisevenmoreso.Althoughattemptshavebeenmadetode˝necom- putationalpracticesbroadly[4,5,6,7],theyarestilllackingclearandprecisede˝nition withinmanyparticulardomains(e.g.,computationalphysics).Accordingly,thisthesisfo- cusesonidentifyingthecommoncomputationalphysicspracticesthatstudentsengagein whilesolvingrealisticphysicsandengineeringproblemsinanintroductorymechanicscourse forengineeringstudents. Computationalpracticescanbede˝ned,inanotherway,asthethingsthatstudents willactuallybedoingwhentheyhavegraduatedand,presumably,getajobinthe˝eld. Thesepracticesareprobablydisciplinespeci˝c(e.g.,physicsmayhavedi˙erentpractices thanbiologists)andindustryspeci˝c(e.g.,manufacturingmaterialsprobablyhasdi˙erence practicesthantestingmaterials). Thereareanumberofreasonsforfocusingoncomputationalphysicsanditsassociated computationalpractices.Perhapsmostimportantisthatthereisahighdemandforcompu- tationalskillsintheworkplaceforrecentphysicsgraduates[4].Withmanystudentsentering schoolforfuturejobprospects,beingabletoe˙ectivelypreparefuturegraduatesforentering industryorcontinuingeducationrequiresin-depthresearchtodevelopbestpractices.As documentedinarecentreportfromtheAIP,thereishighdemandforcomputationalskills intheworkplaceforrecentphysicsdegreeholdersthingslikeprogramming,simulating,or modeling[8].Modernphysicscurriculashouldre˛ectthemodernpracticesofprofessional physicists,andcomputationisnowseentobejustasimportantastheoryandexperiment. Forthisreason,facultyfromphysicsdepartmentsacrossthenationcallformorecomputation inthecurriculum[3]. 2 Perhapsmostimportant,computationalskillsarebecomingincreasinglynecessary21st centuryskills,especiallyforanyoneusingphysics.Withmosttheoryonsolidfoundation, computationcanbeusedtoapplyittocomplex,non-linear,andrealisticmodernengineering problems. Additionally,itisbelievedthatstudentsofcomputationalphysicsgainadeeperunder- standingofthephysicalconcepts[9,10]alongtheway.VisualpackagessuchasVPython orGlowscript[11]allownoviceprogrammerstocreatethree-dimensionalvisualizationsthat allowthemtomoreeasilyinteractwiththefundamentalconcepts. Further,computationallowsfortheanalysisofrealisticproblemsthathavenoclosed- formsolution.Itsabilitytonumericallyintegratesupportsamoreexploratoryapproach toanalyzingphysicalsystemsandlearningphysics.Thatis,therepeatedapplicationof Newton'ssecondlawallowsforamoregeneralanalysis.Thismoreexploratoryapproachis thoughttoencouragestudentstoconstructmorerealisticandaccurate(e.g.,includingair resistance)computationalmodelsthroughcomputationalthinking[4]. Computationalthinkingisatermthathasbecomeincreasinglypopularsinceitsintro- ductionintheearly1980s[12,13,14,15].Thisterm,althoughfrequentlyusedtoday,is di˚culttoconciselyexplaingivenitsmanyandvariedde˝nitions.Evenwithinthe˝elds ofeducationandcomputerscience,manydi˙erentviewpointsexistonthetopic,andthe correspondingde˝nitionsarejustasvaried[16].However,manyofthesede˝nitionsshare onefundamentalcharacteristic:solvingcomplexproblemsthroughabstractionandanalytic thinkingwiththeaidofcomputeralgorithms.Inotherwords,thistypeofthinkingisany typeofthinkingthatfocusesonusingcomputeralgorithmstosolveproblems.Thistype ofthinkingisextremelyimportantinengineering,wheredi˙erentialequationsaresolved numericallytosolveproblems.Accordingly,computationalthinkingfocusesontheuseof 3 computeralgorithms. Thistypeofthinkingissohighlyvaluedbythemodernenterpriseofscienceeducation thattheNextGenerationScienceStandards(NGSS)includeselementsofcomputational thinkinginK-12settings.Asearlyasthe˝fthgrade,studentsareexpectedtobeableto thinkcomputationally.TheNGSSdescribescomputationalthinking,atthislevel,interms ofanalyzingdataandcomparingapproaches.Bythetimestudentsreachmiddleschool, computationalthinkingadvancestoanalyzinglargedatasetsandgeneratingexplanations. Finally,inhighschool,computationalthinkingexpandstoconstructingcomputationalmod- elsandusingthemtoanswerquestions[5].Clearly,computationalthinkingisacomplicated conceptwhichrequiressubstantialexplanation. Expertsinthe˝eldstillhaveawaystogowhenitcomestoclearlyde˝ningcomputational thinkingwithinscienceeducation,andwithinphysicseducationmorespeci˝cally.However de˝ned,though,thistypeofabstractandalgorithmicthinkingispervasiveitextends beyondcomputerscienceinto˝eldsfromgeologytoastronomy,andevenbeyondSTEM [2].Itisbecomingincreasinglyclearthatthinkingisafundamentalskillfor everyone,notjustcomputerscientists Givenrecentinterestinscienti˝cpractices,andcomputationalthinkingmorespeci˝cally, ataxonomyofthecomputationalpracticesindicativeofcomputationalthinkinghasbeen proposedbeWeintropet.al[7].Thistaxonomy,comprisedoftwenty-twoindividualyet inter-relatedpractices,˝ttingintofourdi˙erentcategories,ismeanttohelpguideinstructors andresearchersastheyattempttoteachandbetterunderstandcomputationalthinkingin scienceclassrooms.Eachpractice,accordingtothetaxonomy,isde˝nedbroadlyandfrom anexpertlevelsoastobeapplicabletoawiderangeofscienceclassrooms. However,thebroadandexpert-generatedde˝nitionsthatmakethetaxonomywidely 4 applicablealsoleaveitrelativelyvagueanddi˚culttoapplytoanyparticularsituation. Reducingthevaguenessanddi˚cultyofapplyingthistaxonomytoaspeci˝cdomainof inquiry(i.e.,introductoryphysics)isachallengingbutimportanttask.Havingataxonomy thatisbothpreciseandeasytoapplywillprovideasolidfoundationforinstructorstogen- erate/validatecomputationalproblemsandforresearcherstoanalyzethelearningprocess. Accordingly,thisthesisattemptstoanswerthefollowingquestions: 1. whatarethecomputationalpracticescommontointroductorymechanics, 2. howarethosepracticesde˝nedintermsofconcreteexamples,and 3. whydoweseethosepractices? Itcannotbeoverstatedthatitisthecultureof P 3 theactiveandsocialengagementof studentsinlearningthatisencouragedthroughcontinualtutorinteractionthatin˛uences thepracticesthatweseeinourdataheavily. Ultimately,thisthesisismeanttoillustratetheprocessofidentifyingthecommonprac- ticesthatgroupsofstudentsengageinwhilesolvingarealisticcomputationalintroductory physicsproblem.InCh.2weexplicatethepriorresearchoncomputationanditsresults, aswellasthetheoreticalandmethodologicalunderpinningsofthestudy.Thisincludesthe historicalandmorerecentresultsfromPhysicsEducationResearch(PER)andComputer ScienceEducationResearch(CSER).InCh.3,wedescribethecoursefromwhichourdata hasbeencollectedacalculus-basedintroductoryphysicscoursewithafocusonengineering, workingingroups,andcomputation.Wealsodescribethetypesofcomputationalproblems studentsareworkingonwhileinclass.InCh.4,weprovideamotivationfornotonlythe existenceofthestudy,butalsothetheoriesandmethodsthatwedecidedonusing.Finally, 5 inChs.wepresenttheanalysisandresultsofourcurrentstudywithdiscussionand concludingremarks. 6 Chapter2 Background Inordertobetterunderstandtheanalysisandresultsofthisthesis,therearethreebroad andunderlyingtopicsthatdeserveelaboration.First,theconceptofcomputationalthinking anditsde˝nition.Next,theresultsfromPhysicsEducationResearch(PER),includingthe variousimplementationsofcomputationalphysicsanditse˙ectonlearning.Finally,the qualitativemethodologiesandtheframeworkthatwehaveusedtoguideouranalysis. 2.1Computationalthinking Asmentionedintheintroduction,computationalthinkinganditsassociatedpracticeswithin introductoryphysicsareofprimaryinteresttothisthesis.Thesepracticesaretheobservables thatwecanlookforwithinourdata.Buildingonpreviousresearchthatfocusesonscienti˝c practices[4,5,7],wehaveattemptedtomoreclearlyandpreciselyde˝nethecomputational practiceswithinintroductoryphysics. Thehistoryofcomputationalthinkinganditsde˝nitionislongbutincomplete[12, 17,13,14,15,16,2].Thetermwas˝rstintroducedbySeymourPapertasitrelatedto studentsactivelyconstructingknowledgethroughtheproductionofanartifactideally, butnotnecessarily,acomputerprogram.Thisideaoflearningthroughconstruction,often calledwasbuiltonthePiagetianideaofConstructivism statesthatstudentslearnbestwhentheyareactivelyinvolvedintheconstructionoftheir 7 knowledge[18].Constructionism,ontheotherhand,believesthatitistheconstructionof atangibleobjectthatisofcriticalimportancewhenactivelyconstructingknowledge[12]. Papertwasveryinterestedinlookingathowcomputerscouldbeusedtoteach.Some ofhisearliestresearchintoaneducationalprogramminglanguage(i.e.,Logo,aptlynamed foritsfocusonreasoning)anditsuseasalearningtoolfocusedheavilyontheconstruction oftwo-dimensionalshapesonacomputerscreen[19].However,Papertdidnotinitiallyat- tempttode˝necomputationalthinkingintermsofconstructionism.Rather,hecommented thatattemptstointegratecomputationalthinkingintoeverydaylifehadfailedbecauseof theinsu˚cientde˝nitionofcomputationalthinking.Heoptimisticallyclaimedthatmore attemptstode˝necomputationalthinkingwouldbemade,andeventuallypieceswill cometogetherPapertwouldlatergoontosaythatcomputationalthinkinginvolves newthatarebothandpowerful Morerecently,buildingonPapert'spreliminaryobservations,JeanetteWingde˝ned computationalthinkingasitrelatedtotheprocessingpowerofmoderncomputerswiththe additionofhumancreativity.ThisechoedthecoresentimentsexpressedbyPapertofusing humancreativitytonewidethatarepowShestatesthat thinkinginvolvessolvingproblems,designingsystems,andunderstanding humanbehavior,bydrawingontheconceptsfundamentaltocomputerscience. Wingwascarefultoremindreadersthatcomputationalthinkingisafundamentalskill foreveryone,notjustcomputerscientists[14].Thisspeakstotherobustnatureofcompu- tationalthinking,butalsospeakstothedi˚cultyinclearlyde˝ningit.Shebelievedthat computationalthinkingshouldbetaughtattheintroductorycollegelevel,andshouldeven gosofarbackastobeintroducedatthepre-collegelevel.Wingmadesubstantialprogress inde˝ningcomputationalthinking,butstillfallsshortespeciallywithinparticularsub- 8 domainslikecomputationalphysicsorchemistry. FurtherelaborationbyAlfredAhopointedoutthattheprocessof˝ndingtheright tool(e.g.,asoftwarepackagelikeExceloramodelliketheEuler-Cromeralgorithms)for therightjobisaclearindicatorofcomputationalthinking.Heconsideredcomputational thinkingtobethetprocessesinvolvedinformulatingproblemssotheirsolutionscan berepresentedascomputationalstepsandMathematicalabstractionisatthe heartofcomputationalthinking,andbeingabletochoosebetweencompetingabstractions isofcriticalimportance[15].Ahomadeclearthatalthoughtherearemanyusefulde˝nitions ofcomputationalthinkingwithinthe˝eldofcomputerscience,newdomainsofinvestigation (e.g.,introductoryphysics)requirede˝nitionsoftheirown.Itisimportanttohavethese domain-speci˝cde˝nitionstobetterencouragetheassociatedpractices. Theoreticalde˝nitionsaside,TheNextGenerationScienceStandardshasmostrecently attemptedtooperationalizeade˝nitionofcomputationalthinkinginK-12scienceclass- rooms.Theyhaveincludedcomputationalthinkingasoneoftheircorepractices,and identifyahandfulofexpectationsforK-12studentsthatrequirecomputationalthinking. AccordingtotheNGSS,studentsshouldbeableto[5]: E1. Recognizedimensionalquantitiesanduseappropriateunitsinscienti˝capplicationof mathematicalformulasandgraphs. E2. Expressrelationshipsandquantitiesinappropriatemathematicaloralgorithmicforms forscienti˝cmodelingandinvestigations. E3. Recognizethatcomputersimulationsarebuiltonmathematicalmodelsthatincorpo- rateunderlyingassumptionsaboutthephenomenaorsystembeingstudied. E4. Usesimpletestcasesofmathematicalexpressions,computerprograms,orsimulations 9 tocheckforvalidity. E5. Usegrade-level-appropriateunderstandingofmathematicsandstatisticsinanalyzing data. Theseexpectations,thoughuseful,arestillratherbroadandcanbereasonablyapplied toanyscienceclassroom.Forexample,theexpectationofbeingabletorecognizedimensions inamathematicalformula(E1)mightshowupinachemistryclassroomfocusingonmass conservationbeforeandafterachemicalreaction.Alternatively,theexpectationofstudents understandingthatsimulationsrelyonmathematicalmodels(E3)mightshowupinabiology courseinvolvingpredator/preypredictionsbasedonanunderlyingcomputationalalgorithm (e.g.,theLotka-Volterraequations). Moreclearlyandpreciselyde˝ningtheseexpectationsisanimportanttask,especially withinaparticulardomainofinterest.Withoutpreciseanddomain-speci˝cde˝nitions, applyingthemtoaparticularclassroomisratherdi˚cultforpractitioners.Accordingly,one ˝eldwhoseprecisede˝nitionsareparticularlylacking(though,progressisbeingmadeon) isphysics. Similarly,althoughde˝ningcomputationalthinkingwithinK-12isanidealstartingpoint, itshouldalsobeextendedtomoreadvancedlevels.Therearemanyconceptsrequiring computationalthinkingthatareuniquetotheuniversitylevelandabove,andasstudents advancethroughouttheireducationalcareer,itisimportantthatwestudythem.Towit,the AAPTRecommendationsforComputationalPhysicsintheUndergraduatePhysicsCurricu- lumhasidenti˝edtheskills(physics-relatedandtechnical)andtoolsthatshouldbeincluded inamodernphysicscurriculum[4].Theserecommendationsincluderoughlytenskillslike debugging,testing,andvalidatingcodeandtoolslikeExcelorPython. 10 Still,moreresearchisneededtonotonlymoreclearlyde˝nethecomputationalpractices observedinintroductoryphysics,butalsotomoreclearlyunderstandthehabitsofmind andtypesofthinkingthatstudentsareengagingin.Itisimportantthatwefurtherde˝ne expectationsaroundcomputationalthinkingwithinaparticulardomainofinterest(i.e., introductoryphysics)andataparticularlevel(i.e.,universitycalculus-based). 2.2PhysicsEducationResearch Thissectionfocusesonthedevelopmentofthedi˙erentimplementationsofcomputational physicsproblems(e.g.,BOXER)[20,21,3,22],theresultsfromPER(e.g.,studentchal- lenges)[9,23,24,25,26,27],andmostimportantlytheremainingquestions. 2.2.1Implementation ThefocusoncomputationalthinkinginPhysicsEducationResearch(PER)hasbeenin- creasingoverthepastdecade.Historically,computationasapedagogicaltoolhastaken manyforms,butitsimplementationhasusuallyfocusedontwothings:itsabilitytohandle tediouscalculationsanditsabilitytogenerateprecisevisualizations. Forexample,oneoftheearliestformsofcomputationattheintroductorylevel,called BOXER,usedtogeneratetwo-dimensionalshapesonacomputer screen[20].Thisallowedevennoviceprogrammerstotakeadvan- tageoftheprocessingandvisualizationpowerofcomputers.Toillustrate,Fig.2.1shows thegraphicaluserinterfaceforaprograminBOXERthatismeanttogenerateastaranda trianglefortwodi˙erentobjects.Theunderlyingalgorithmsarelaidoutinsequentialsteps thatrepeataspeci˝ednumberoftimes. 11 Figure2.1:GraphicaluserinterfaceforBOXERshowingthegraphicsdata(e.g.,thestep instructions)andtheresultinggraphicforaspritenamedkey AmorerecentimplementationofcomputationtakesthenameVPython:thePython programminglanguagewiththeVisualmodule.Historically,thegoalofdevelopingVPython wastoeitfeasiblefornoviceprogrammersinaphysicscoursetodocomputermodeling with3-dimensionalvisualizationsThecurrentversionofVPythondoesjustthat. AlthoughVPythonwasidealfornoviceprogrammers,italsocateredtomoreadvancedusers. ItsunderlyingalgorithmisanEuler-Cromerstyleintegrationtocalculatetheconstantly updatingpositionandmomentum(orvelocity)ofanobjectwithinawhileloopthatdepends ontime.Forexample,Fig.2.2showsthebasicstructureofaverysimplebutpowerful program.ThisEuler-Cromeralgorithmcanbeusedtoanalyzerudimentarysituations(e.g., free-fallmotion)aswellasmorecomplicatedandrealistic(e.g.,themotionofsatellitesand rockets). AlongwiththedevelopmentofVPython,asoftwarecalledEasyJavaSimulations(EJS) wasincreasinginuse[28].Thesesimulationsweremeanttogivestudentsalittlemore controlbehindthescenes,similartoVPython,whilestilllimitingthegeneralizabilitylike PhETsimulations(describedbelow).Forexample,asimulationofapendulumcouldbe constructedinEJSbydraggingaparticularobject(e.g,apendulumbob)intothemodel 12 Figure2.2:Aprogramillustratingthatthebasiccontrolstructureandintegrationalgorithm arepre-writtensothatstudentscanfocusonthecomputationalforcemodelthatmustbe constructedinline11. andusingtheirbuilt-ineditortosolvetheassociateddi˙erentialequation(seeFig.2.3). Onlyasmallamountofmodi˝cationisneeded,reducingtheloadonnoviceprogrammers somethingsharedwiththeVPythonprogramsofPER[29]. Anotherimplementationofcomputation,frequentlyusedtoday,arethePhysicsEdu- cationTechnology(PhET)simulations[30].Thesesimulationshaverealisticgraphicsthat displaybuttons,sliders,andknobsthatcanbegraphicallytweakedtochangeparameters inasystem.Thistypeoftestingsearchingforthee˙ectonaphysicalsystemwiththe variationinaparameterismeanttobemoreengagingandconducivetolearning.For example,thePhETsimulationshowninFig.2.4ismeanttodemonstratethedependenceof apendulum'smotion(e.g.,itsperiodoramplitudeofoscillation)onthevariousparameters ofthesystem(e.g.,thelengthofthependulumorthemagnitudeoffriction).Beingableto 13 Figure2.3:GraphicaluserinterfaceforanEJSillustratingthenatureof thesoftware.Elements(e.g.,apendulumbob)canbeaddedorremovedfromthedi˙erent panels(e.g.,thedrawingpanel)inthesimulationview. holdoneparameterconstantwhilevaryingtheotherhelpsstudentstocon˝dentlyidentify itsqualitativee˙ect. Finally,oneofthemostrecentimplementationsofcomputationattheintroductorylevel iscalledGlowscript[9].Glowscriptisanon-lineIntegratedDevelopmentEnvironment(IDE) usingVPythonwhichisdesigned,inpart,toeasilygeneratethree-dimensionalvisualizations. Forexample,therathercomplicatedGlowscriptprogramshowninFig.2.5usesaninverse- squareelectric˝eldmodelwithandloopstogenerateavisualrepresentationofthe electricvector˝eldatanypointinspacesurroundingadiscretechargedistribution. Thismorerealisticanddescriptivethree-dimensionalvisualizationleveragedbyGlowscript andVPythonisthoughttoencouragestudentstoformadeeperunderstandingoftheun- derlyingphysicsconcepts.Althoughmanydi˙erentimplementationsofcomputationexist [19,20,30,9],researchfocusingonimprovingthoseimplementationsinPERisstilllacking. 14 Figure2.4:APhETsimulationillustratingthedependenceofpendulummotiononthelength ofthependulum,themassofthependulumbob,themagnitudeofthelocalacceleration duetothegravity,andanyfrictionalforces. Figure2.5:Glowscriptoutputdemonstratingitsabilitytogeneratethree-dimensionalvisu- alizationsofobjects,vectors,andgraphs.Theabilitytoquicklyandaccuratelygenerate three-dimensionalvectorsallowsformore˛exibilityandadeeperunderstandingofelectric ˝elds. 15 Someofthecriticalresults,though,aredescribedbelow. 2.2.2Prior˝ndings Intheearly2000s,Chabaybegantoresearchtheintegrationofcomputationintothein- troductorycalculus-basedphysicscourseusingVPython[9].Thiscourseincludedacom- putationalcurriculumfollowingthatpresentedby MatterandInteractions .Primarily,the coursesstudiedbyChabayfocusedontheapplicationoftheintegralequationgoverning thelinearmotionofobjects(i.e., d~p = ~ F net dt and d~r = ~p=mdt ).Theseequationswere appliediterativelythroughanEuler-Cromerstyleintegrationalgorithm,andallowedamore thoroughanalysisofposition-dependentforces(e.g.,thespringforce). Chabayfoundthatoneofthepositiveaspectsofincludingcomputationattheintroduc- torylevelwastostimulatecreativityinstudents[9].Thiscreativityinapproachingproblem solvingisthoughttoleadstudentstotheconstructionofmorerealisticcomputationalmod- els.Inotherwords,computationallowsstudentstoeasilyverifyand/ormodifyamodel, encouragingcreativityandanguessandchecapproachtoproblemsolving. Shealsofoundthatrequiringstudentstoprogramattheintroductoryphysicslevelwas adi˚cultbarriertoovercome.Giventhatthereissomuchcontenttobecoveredinsolittle timeinmostintroductoryphysicscourses,˝ndingtheroom/timetodiscussthebasicsof programmingisdi˚cult.Oneofthewaysinwhichthisdi˚cultyisovercomeisbyproviding MinimallyWorkingPrograms(MWPs)tostudents.TheMWPforaparticularproblem usuallyrunswithouterrorfromthestart,andrequiressmall(oratleastlocalized)changes totheunderlyingcomputationalmodels.Forexample,seetheMWPinFig.2.7andits di˙erentcomponents. Aroundthatsametime,Kohlmyerdugdeeperintostudentperformance[10].Hefound 16 that,amongotherthings,computationalmodelingstudentsstruggledtorecognizethatcom- puterscouldevenbeusedtosolvephysicsproblems.Furthermore,oncetheydiddecideto useacomputer,theystruggledwiththeconceptsandcomponentsofcreatingacomputa- tionalmodel.Theseresultsweregeneratedfromtwoexperiments:lookingathowstudents approachnovelproblemswithcomputationandlookingatthedi˙erencesinthefundamental principlesusedascomparedtotraditional(i.e.,anon-computationalcurriculum)students. Interestingly,hefoundthatstudentsdecidedtotakeadvantageoftheEuler-Cromerstyle integrationindiscreteformevenwhentheyweren'tusingacomputationalmodel.Thatis, studentsmadeuseofthekeyconceptualtoolthattheyweretaughtevenifjustonpaper. Healsofoundthatthecomplexprocedureneededtomodelattractiveposition-dependent forceswasadi˚cultchallengeforstudents.Reducingthisandotherdi˚cultiescanbe achievedthroughincreasingthefrequencyofcomputationthroughoutthecourseorrequiring computationalhomeworkproblems.However,Kohlmyermadeexplicitthewidevarietyof unansweredquestionsthatcouldbepursuedinfurtherresearch,hintingthattheprocess ofassumptionandincorporatingthemintoacomputationalmodelwouldbeof particularinterest. In2011,Weatherfordbegantolookatintegratingcomputationintothephysicslab curriculumandthesense-makingthatstudentsengagein[29].Hisstudywasanin-depth qualitativeanalysisofgroupproblemsolving,focusingonthreedi˙erentcontexts:ascat- teringproblem,aspring-massproblem,andaspacecraft-Earthproblem.Acodingscheme wasdevelopedtohelpcategorizedi˙erentportionsoftranscript,asshowninFig.2.6. Hefound,amongotherthings,thatcomputationalphysicsstudentswereabletoreason- ablyinterpretphysicalquantitiesaccordingtotheirvariablename.Forexample,themass ofasatellitemightbede˝nedas m.satellite=1 ,orthenetforceactingonanobjectmay 17 Figure2.6:Asampleofthein-depthanalysisWeatherfordperformed.Eachparticularline ofcodethatthegroupisfocusingonistrackedintimeandcodedaccordingtoaresearcher- developedscheme. 18 bede˝nedas Fnet=vector(0,-m*g,0) .Thesepre-writtenvariablesarenamedsoasto suggesttothestudentswhatphysicalquantitytheyrepresent.However,themorecompli- catedthede˝nitionsget(e.g.,afunctionofmultiplevariableslike Fnet=-k*(ball.pos -origin.pos)/mag(L) ,themorestudentsstruggledatrecognizingit. Additionally,Weatherfordwasabletoencouragestudentstobegintoincorporateacom- putationalmodelinaMWPbyprovidingaminimumlevelofsupport.Thatis,onlyomitting thefundamentalphysicscalculationsthatstudentsaremeanttoengagewith(e.g.,various computationalforcemodels)helpstokeepstudentsfocusedonthephysics.Othertasks thatarenotphysicalinnaturehaveatendencytoderailthephysicsdiscussionandthe problemsolvingprocessingeneral.Forexample,ensuringthattheendofaspringiscon- nectedtotheendofamassinacomputationalspring-massanalysisbeginstoovershadow themorefundamentaltaskofincorporating/constructingaposition-dependentlinearspring force.Similarly,˝guringouthowtousethe mag() functioninPythoncansidetrackthe ultimategoalofconstructingapositiondependentgravitationalforce. WeatherfordclearlypointedoutthattheMWPactivitiesintheirstudyhadmuchroom forimprovement,andthatmoreresearchwasneededonfosteringstudentpro˝ciencyin computationalphysics.ThesequenceofMWPsinhisstudydidn'tquiteraisestudents' programcomprehensionandprograminterpretationskillstoacertainpro˝ciency,buthe believesthatmoreresearchwillshedlightonthesubject. In2011,Caballerowasabletoidentifyanumberoffrequentstudentmistakeswitha satellite-EarthMWP,showninFig.2.7,thatweregroupedintothreedi˙erentcategories: initialconditionmistakes,forcecalculationmistakes,andsecondlawmistakes[31].Aninitial conditionmistakemighttaketheformofanincorrectinitialvelocityormomentumofthe satellite.Aforcecalculationmistakemightmanifestinaconstantspringforceratherthan 19 Figure2.7:Anexpectedsolutiontoacomputationalsatellite-EarthproblemwheretheNew- toniangravitationalforcehasbeenconstructedfromaseparationvectoranditsmagnitude. TheforcecalculationhasbeenincorporatedintothemomentumthroughNewton'ssecond law,andthemomentumisincorporatedintothepositionthroughapositionupdate. apositiondependentspringforce.Asecondlawmistakemightinvolvemissingthedivision ofthemassfromthenetforceonanobjectsothatthevelocityiscorrectlyupdatedaccord- ingtotheacceleration.Thesefrequentmistakesresultinbothunexpectedandphysically inaccuratevisualizations. Basedonhisanalysisofthesatellite-Earthproblem,Caballeroconcludedthatthemajor- ityofstudents( ˘ 60% )wereabletocorrectlycomputationallymodelnovelphysicsproblems andthat,amongotherthings,thepracticeofdebuggingwouldservestudentswell.Partic- ularly,theactoftroubleshootingsyntaxerrorsaswellastheactoftroubleshootingphysics errors. 2.2.3Remainingquestions Althoughmanyaspectsofcomputationandcomputationalthinkingattheintroductory levelhavebeenstudied,therearestillmanyunansweredquestionswithinphysicseduca- tion.Particularly,astothetypesofpracticesstudentsareengaginginthatareindicativeof 20 computationalthinking.Moreresearchisneededtonotonlymoreclearlyde˝nethecom- putationalpracticesobservedinintroductoryphysics,butalsotomoreclearlyunderstand thehabitsofmindandtypesofthinkingthatstudentsareengagingin.Thisthesisattempts toprovideclearandprecisede˝nitionsofthevariouspractices,indicativeofcomputational thinking,thatstudentsengageinwithinintroductoryphysics. 2.3Framework Recently,aframeworkforidentifyingthecomputationalpracticesthatareindicativeofcom- putationalthinkinghasbeenproposedbyWeintropet.al[7].Thisframeworkwasdeveloped usingexistingliteratureoncomputationalthinking,interviewswithmathematiciansandsci- entists,andcomputationalactivitiesfromgeneralscienceandmathematicsclassrooms. Inordertodeveloptheirframework,aliteraturereviewwasperformedtogeneratean initialsetof 10 mathandsciencepractices.Theseinitialpracticesarerepeatedlycited byWeintropet.alasbeingcentraltocomputationalthinking.Forexample,thebroad andrepeatedlycitedpracticeofgeneratingalgorithmicsolutionsmightrequireastudentto engagewithadi˙erentialequationalgorithm.Thesebroadinitialpracticeswereusedto guidethesubsequentqualitativeanalysis. Usingtheinitialpracticesresultingfromtheliteraturereview,tworeviewersindepen- dentlycodedforthevariousofcomputationalthinkingthatwererequiredbythe curricularmaterials.Theyanalyzed 32 di˙erentcomputationalactivitiesfromchemistryto programming,resultingin 208 facetswhichweregroupedinto 45 di˙erentpractices. Next,areviewprocessincorporatingfeedbackfrommultiplesources(e.g.,teachers,con- tentexperts,andcurriculumdesigners)wasusedtoreducethe 45 practicesinto 27 ,which 21 werefurtherorganizedinto 5 di˙erentcategories.Further,externalinterviewswerecon- ductedwith 16 K-12scienceandmathematicsteachers,helpingtoreducethe 27 practices into 22 ˝tting 4 di˙erentcategories,summarizedinTab.2.1. DataModelingSolvingSystems CreatingConceptsPreparingInvestigating CollectingTestingProgrammingUnderstanding ManipulatingAssessingChoosingThinking CreatingCommunicating DebuggingDe˝ning Table2.1:TheframeworkdevelopedbyWeintropet.altodescribethecomputational practicesobservedinscienceandmathematicsclassrooms.Eachcategorycontainsbetween ˝veandsevenindividualpractices,andeachpracticehasbetweentwoandsevenfundamental characteristics. Finally, 15 interviewswithSTEMprofessionalswereconductedtoratetheirframework accordingtoitsapplicabilitytoauthenticprofessionalpracticesandtogivedirectionfor futureimprovement.Forexample,interviewsshowedthatthepracticeoftestinganddebug- gingwasacrucialpractice(seeSec.2.7)thatwasnotadequatelycapturedbytheframework animprovementthatshouldbemadeonfutureiterationsoftheframework. Thefourdi˙erentcategoriesofpracticesarelabeledasdata,modelingandsimulation, computationalproblemsolving,andsystemsthinkingpractices.Thedatapracticesfocus mostlyonthecreationandvisualizationofdata.Themodelingandsimulationpractices focusmostlyonthedesign,construction,andassessmentofacomputationalmodel.The problemsolvingpracticesfocusmostlyonprogramminganddebugging,whilethesystems thinkingpracticesaremoreabstractandfocusmostlyonthestructureoftheprogramitself. Asamoreconcreteexample,thecomputationalpracticeofcreatingdata,oneofthedata practices,hasthreefundamentalcharacteristics:thecreationofasetofdata,anarticulation oftheunderlyingalgorithm,andauseofthedatatoadvanceunderstandingofaconcept. 22 Themoreofthecharacteristicsthatweobserveinaparticularsituation,themorecon˝dent wearethatthatsituationcanbeclassi˝edasthatpractice. Althougheachpracticeisde˝nedlikethis,accordingtoWeintropet.al,thecharacteris- ticsthemselvesarerathervaguesimilartotheoperationalde˝nitionsfromtheNGSS.For example,thecomputationalpracticeofassessingcomputationalmodelsrequirestheiden- ti˝cationofaphenomenon,acomputationalmodel,andacomparisonmadebetweenthe two.Althoughitisclearwhatacomparisonwouldlooklikeinanysituation,thephenomena studiedandthemodelsusedwilldependgreatlyonthecontext(seeCh.3).Forthisrea- son,moreworkmustbedonetoclearlyde˝necomputationalthinkingwithinintroductory physicsclassrooms. Ultimately,Weintropfoundthreemainbene˝tstoincludingcomputation:itbuildson thereciprocalrelationshipbetweencomputationalthinkingandSTEMdomains,itengages learnersaswellasinstructors,anditintroducesanauthenticandmodernelementofdoing science.However,heiscleartoindicatethatmoreresearchisneededtobetteraddress thechallengeofeducatingatechnologicallyandscienti˝callysavvypopulation.Thisthesis attemptstoimprovethateducationprocessbyprovidingclearandprecisede˝nitionswith examplesofthecomputationalpracticesthatareindicativeofcomputationalthinkingatthe introductoryphysicslevel.Accordingly,wehaveusedbothataskandthematicanalysis, describedinthesectionsthatfollow,tofacilitatethoseclearandprecisede˝nitions. 2.4Taskanalysis Ataskanalysisisaprocedurethatcanbeusedtobetterunderstandtherequirementsofa particulartaskandthewayane(orgroupofoperators)mightworktosatisfythose 23 requirements[32].Thistypeoftaskanalysisisusuallyfocusedontheobservableactions thatanoperatormightengageinwhileworkingtowardaparticulargoal(e.g.,producinga graphordiagram),butthereisalsoastrongcognitivelinkbetweentheobservedactionsand therequirementsofthetask[33].Thisindispensabletypeofprocedurehelpedustofocus onthemostimportantstepsthatstudentsweretakingwhilesolvingproblems. Beforebeginningataskanalysis,datamust˝rstbecollected.Often,themethodfor datacollectionisobservationbased(e.g.,observingtheactionsofagroupofoperatorsas theycarryoutatask),althoughdatacanalsobesubjectbased(e.g.,askinganexpertwhat theidealactionswouldbetocarryoutatask).Eitherway,thetaskitselfgenerallyguides thecollectionofdata. Oncethedatahasbeencollected,therearedi˙erenttypesofdescriptionsthatcanbe attachedtoitanddi˙erenttechniquesthatcanbeusedtogeneratethem.Forexample, oneofthetechniquesfrequentlyusedisto chartandnetwork thedata.Thesedescriptions canbewritten,butaremostoftenpresentedvisuallythroughinformation˛owchartsor Murphydiagrams.Thisthesisleveragesatechniqueforgeneratingan organizedhierarchy ofdescriptionofthedata:complextasksarebrokendownintomultiplesmallerbutmore manageabletasks. Thistypeoftaskanalysisisfrequentlyusedinthe˝eldsofmathematicsandcomputer science[34,35,36,37].Thesmallerbutmoremanageablesub-tasksaretheof thatcanthenbesearchedforwithindata.Forexample,anexpertgroupmightproceedin predictingthemotionofanobjectby˝rstconstructinganEuler-Cromerstylealgorithm, constructingthevariousforces,andthenconstructingtheinitialparametersofthesystem. Thesestepscanbedoneinanyorder,butareallnecessarytotheoverarchingtask. ThistypeofprocesswasusedbyCatrambonetoshowthatbreakingaproblemdown 24 intosmallerbutmoremanageablesub-taskshelpsstudentstotransferknowledgetonewand novelproblems[34].Hebelievesthatitisahierarchicalstructureoftasksratherthanalinear structureoftasksthatstudentsneedtotransferknowledgetonewandnovelsituations.The ˛exibilityofahierarchicalstructureisthoughttosupportamoreexpertapproachtosolving problems. Step(Sub-Task)AssociatedCode Constructseparationvector s ep=obj2.pos betweeninteractingobjects - obj1.pos Constructtheunitvector u sep=sep/mag(sep) Constructthenetforce F net=-G*m1*m2*usep vector / mag(sep)**2 Integratethenetforceover o bj.p=obj.p+Fnet*dt timeintomomentum Table2.2:SomeofthenecessarystepsthatmustbetakenwhenconstructingaNewtonian gravitationalforceincode.Eachstepisassociatedwiththeconstruction/modi˝cationofa lineofcode. Catramboneperformedthreeexperiments,eachfocusingonhowstudentstransferknowl- edgetonewandnovelproblems.The˝rstexperimentwasacomparisonbetweenthemean- ingfulnessofalabel'sname.Hefoundthatthemoremeaningfulthelabelwas,thebetter preparedstudentsweretosolvenewandnovelproblems.Thesecondwasadeeperstudy oftheconnectionsbetweenlabelsandsub-tasks.Hefound,toareasonabledegree,that therewasafundamentalconnectionbetweenlabels,sub-tasks,andhowtheyweregrouped. Thethirdwasatalk-aloudstudythatlookedatself-explanationwhilesolvingproblems.He foundthataptlynamedlabelscouldbeusedtocuestudentstogroupsub-tasksandexplain theirpurposethroughself-explanation. Althoughwehaveusedtheconceptofataskanalysistohelpfocusonspeci˝caspects ofourdata,wehavenotuseditinthesamewayasCatrambone.Thereareamyriadof expectedandunexpectedtasksthatstudentsengageinwhilesolvingaparticularproblem 25 inanytypeofclassroom.Forexample,takingthetimetonameavariablewithmeaning, workingtoconstructamultiple-variablefunction,orchangingthecolorofanobjectwithin aprogram.Giventhealmostlimitlessnumberoftasksthatmightdrawstudents'(and our)attention,thetaskanalysiswasusedtoreducetheinitialsetoftasksthatwefocused ourattentionon.Thisinitialsetoftaskswasmodi˝edandexpandedduringsubsequent qualitativeanalysis(seeSec.2.5). Thetaskanalysisoftheproblemthatthisthesisfocusesonwasinitiallyconstructed byasinglecontentexpert.Afterthe˝rstiterationitwaspresentedtoadditionalexperts. Throughthediscussionssurroundingtheseiterations,itbecameclearthattheconstruction ofthepositiondependentNewtoniangravitationalforceincodeisamulti-stepprocedure involvinganumberofdi˙erentsub-tasks.Thetaskanalysiswasiterativelyre˝nedthrough thisprocessuntilallexpertsagreedthatthesub-tasksshowninTab.2.2weresu˚ciently described/de˝nedtobeusefulinvideoanalysis. Ontopofthisexpertgeneratedsolution,therearemanyother(bothexpectedand unexpected)studentgeneratedsolutionsthatweobserveinthedata.However,theexpert generatedsolutionisanidealpathtofollowandsotheinstructorstrytokeepgroupsmoving inthisdirection.Forexample,asu˚cientforcemodelcanbeconstructedintermsofthe polarandazimuthalangleofthesatellite,althoughitrequiresasubstantialamountofwork tocode.Boththeexpertandstudentgeneratedsolutionsareagoodplacetolookfor evidenceofcomputationalthinkinganditsaccompanyingpractices. 26 2.5Thematicanalysis Thematicanalysisisacommonlyusedtypeofqualitativeanalysisthatiscommonlyused withinpsychology.However,Braunmakesthewell-supportedcasethatthematicanalysis cane˙ectivelybeusedinmanyother˝elds(e.g.,nursingorphysicseducation)andclearly de˝nesthesu˚cientstepsthatcanbetakeninordertocompleteareasonablyreliableand validthematicanalysis[38,39,40,41,42,43,44]. WithinPER,thematicanalysisisusuallyusedforanalyzingintervieworwork-aloud dataofstudentssolvingproblems.Forexample,Irvingfoundthatthereweremanydi˙erent themesthatcamefromthevariousperceptionsstudentshaveaboutwhatitmeanstoea ph[45].Thesethemeswerethenbrokendowninto 12 sub-categories(e.g.,highorlow interestinresearch),highlightingthedi˙erentperceptionsstudentshadaboutwhatitmeans tophThistypeofanalysis,asdemonstratedbyIrving,canbeusedtogenerate robustthemesthatcanbeusedtoinforminstructionalchanges/improveinstruction. However,thematicanalysisisjustoneofmanyqualitativetechniquesthatcanbeusedto analyzequalitativedata.Thevariousqualitativemethodologiescanbebrokenintoroughly twomaintypes:thosestronglytiedtoatheory/epistemologyandthosethataredeveloped independentofaguidingtheory/epistemology.Thematicanalysis,accordingtoBraun,is ofthesecondtype.Soastoguardagainsttheoftencitedcritiqueofthematicanalysisas beingill-de˝ned[44],Braunpresentsamethodtoconductingareliableandvalidthematic analysis. Thismethodconsistsof 6 di˙erentphases,usuallyfollowedlinearly,to˝nallyproduce areport(e.g.,athematicmap)ofthevariousthemesandtheirrelationshipswithinaset ofqualitativedata.However,beforeenteringthe˝rstofthe 6 phases,thereareafew 27 fundamentaldecisionsthatmustbemadeandexplicitlystated.Ideally,thesedecisionswill bemadeinrelationtotheresearchquestionandthegoalofthestudy. First,itiscrucialthatresearchersexplicitlystatethemetricbywhichtheyplanto identifythemes.Forexample,athemethatshowsupmorefrequentlyisnotnecessarily moreimportant.Additionally,athemethatshowsuplessfrequentlyisnotnecessarilyless important.Rather,itisimportanttobeconsistentthroughoutanalysis.Thisthesismostly focusesonthemorefrequentthemes,butconsiderationisalsogiventothemesthatare particularlyillustrativeyetinfrequent. Second,researchersmustdecidebetweenarichdescriptionoftheentiredatasetora moredetailedaccountofaparticularsub-set.Forexample,withinphysicseducation,you mightbeinterestedinaroughdescriptionoftheentireprocessthatagroupfollowedto successfullysolveacomplicatedproblem.Alternatively,youmightwanttofocusinona particularsub-taskanditsnuance.Thisthesisfocusesonamoredetailedaccountofa particularsub-setofthethemes(i.e.,thoseinvolvingcomputationalthinking). Third,researchersmustdecidebetweenaninductiveandamoretheoreticalapproachto thegenerationofthemeswithintheirdata.Aninductiveapproachoftenleadstothemesthat arenotrelatedtotheoriginalresearchquestions,butratherareemergentduringanalysisand aremorestronglytiedtothedataitself.Atheoreticalapproach,ontheotherhand,often leadstoasetofthemesthatarelessdescriptivebutarebettersuitedtoansweraparticular researchquestion.Thisthesisfollowsamoretheoreticalapproach,usingthetheoretical frameworkpresentedinSec.2.1asafoundationforthegenerationofourthemes. Fourth,researchersmustdecidewhethertheywillbelookingforsemanticorlatent themeswithintheirdata.Semanticthemesarethosethatareclearlyindicatedwithin thedata,whereaslatentthemesoftengobeyondwhatisactuallybeingobserved.For 28 example,withinphysicseducation,agroupofstudentsmightbestrugglingwithaparticular problem.Thereasonforthisstrugglemightotherwisegounnoticedwithoutlookingbeyond theimmediateandrecognizingthateachstudenthadalateandmentallytaxingchemistry examthepreviousnight.Usuallyathematicanalysisfocusesononelevelthisthesis primarilyfocusesonthesemanticthemesthataredirectlytiedtotheactionsobserved duringtheproblemsolvingprocess. Fifth,researchersmustchoosebetweenanessentialistandasocialconstructionistthe- maticanalysis.Anessentialistthematicanalysisallowsresearcherstotheorizestudentun- derstandingandmeaninginastraightforwardway[43,46].Asocialconstructionistapproach focusesmoreontheoverarchingsocioculturalandstructuralenvironmentthateachstudent liveswithin.Thisthesisfocusesonamoreessentialistapproach,payingspecialattentionto thecomputationalthinkingandhabitsofmindthatstudentsareengagingin. Oncethesedecisionshavebeenmade,thequalitativeanalysiscanproceedthroughthe 6 phaseslaidoutbyBraun.The˝rstphasefocuseson(1)transcribingandfamiliarizing yourselfwiththedata.Readingthroughthetranscriptsmultipletimeshelpstogenerate preliminaryideasthatcanbe(2)codedforfurtherinvestigation.Next,eachcodemustbe(3) collatedwiththecorrespondingtranscriptsoastoprovideacontext.Afterthecodeshave beencollatedwiththecorrespondingtranscript,(4)themesbegintoemerge.Reviewing anythemesthatemerge,particularlyagainstthecodedextractsandthetranscriptasa whole,leadstothenextphaseof(5)de˝ning,validating,andnaminganythemes.These themescan˝nallybepresentedina(6)scholarlyreportwithstep-by-steptranscriptanalysis and/orathematicmap.Athematicmap,liketheoneshowninFig.2.8,showsnotonlythe componentsofatheme,butalsothe relationships betweenthosecomponents. Brauniscleartopointoutthattherearemanypitfallsassociatewiththematicanalysis, 29 Figure2.8:A˝nalthematicmapshowingthecomponentsofathemenamed asThemaincomponentsofthisthemeareowandymen andthatresearchersmustbecognizantofthemthrougheveryphaseoftheprocess.For example,oneofthepitfallsshehighlightsisapossiblemismatchbetweenthedataandthe analyticclaimsthatarebeingmade.Inotherwords,itisimportanttoalwayscloselytie yourclaimstotheactualdata.Thisclosenessoftheclaimstothedatacanbeensured throughfrequentinter-raterreliabilitychecks. Aswehaveshown,thematicanalysisisapowerfuland˛exiblequalitativemethodology. Accordingly,thisthesisleveragesthematicanalysistoguideourstudyofgroupproblemsolv- inginintroductorycomputationalphysicswiththehopesofhighlightingthevariouspractices studentsengageinthatareindicativeofcomputationalthinking.Adetailedaccountofthis processisdescribedinSec.5. 30 Chapter3 Context Itisimportanttounderstandthecoursefromwhichwehavecollectedourdatatobetter understandtheresultsofourstudy.ThatcoursecalledProjectsandPracticesinPhysics ( P 3 )isbasedonasocialconstructivisttheoryoflearninganda˛ipped/problem-based pedagogy[47].Inotherwords,studentsfamiliarizethemselveswithrelevantmaterialbefore comingtoclass,wheretheywillworkinsmallgroupstoactivelyandsociallyconstructknowl- edgewhilesolvingcomplexanalyticalandcomputationalphysicsandengineeringproblems. Thecoursehasintentionallybeendesignedtoencouragecomputationalthinkingwherever possible.Speci˝cally,computationalthinkinghasbeenincorporatedintothenotes,pre- andpost-classhomework,in-classfeedbackandassessments,andaselectionofthein-class problems. 3.1Courseschedule Eachweekin P 3 ,studentsareexpectedtoaccomplishanumberoftasks.Theymustcom- pletethepre-classhomeworkwhichisbasedoninformationthattheyshouldgatherfrom thepre-classnotes.Theymustthenworkinsmallgroups(usuallybetweenthreeandfour members)ontworelatedanalyticalproblemsoramixtureofoneanalyticalandonerelated computational.Theseproblemsaredeliveredduringthetwotwo-hourweeklymeetings(See Fig.3.1).Forthecomputationalproblem,thatmeansreadingandinterpretingpre-written 31 code(i.e.,aminimallyworkingprogram)whiletheydesign,assess,andconstructvarious computationalmodels.Thesmallgroupisfacilitatedbyeitheracourseinstructor,graduate teachingassistant,orundergraduatelearningassistantwhowillaskrelevantandpertinent follow-upquestionstocheckforconceptualunderstanding.Therearealsopost-classhome- workquestionsbasedoninformationgatheredfromthepre-classnotesandthein-class problemsthataredueattheendoftheweek.Thisalloccurswhilestudentssimultaneously prepareforthefollowingweek. 3.2VPython Giventhatthevastmajorityofstudentsenter P 3 withlittletonopriorprogrammingex- perience,weneedtoensurethattheyarepreparedtohandlecomputationalproblemsearly inthesemester.Onewaythatwecanensurethisisbyrequiringstudentstoengagewith thefundamentalprogrammingideas(e.g.,iterationthroughawhileloopcontrolstructureor pre-de˝nedmathematicalfunctions)beforecomingtoclassthroughpre-classhomeworkand notes.Thesenotesandhomeworkquestionshighlightthefundamentalphysicalandpro- grammingideasspeci˝ctoVPythonandthecomputationalproblemsthatwillbedelivered inclass. Forexample,considertheportionofthecoursenotesshowninFig.3.1.Thesenotesare madeavailabletothestudentsatthebeginningofthesemesterandaremeanttoprovide studentswithabasicunderstandingoftheutilityofVPythonalongwithalistofcommon errorsthatnoviceprogrammersmustfrequentlydealwith.Thesenotesprovidenotonlya descriptionoftheerror,butalsoaprocedureforremovingitwhilestudentsaretroubleshoot- ingcode.Troubleshootinganddebuggingaretwooftheproblemsolvingpracticesindicative 32 ofcomputationalthinkingthatwefocusedouranalysison. Figure3.1:Portionofon-linenotesthatismadeavailabletothestudentsduringthe˝rst weekofthecourse.Thesenotesintroducethefundamentalprogrammingideasandalistof commonerrorswithtipsandtricks. 3.3Pre-classwork Thereareotherweeklynotes,madeavailabletothestudentsatthebeginningofeveryweek, focusingmoreonthefundamentalphysicalideasthatwillbeusedduringclass.Forexample, duringthethirdweekthenotesfocusonuniformcircularmotion(mostheavilyusedduring theweek'sanalyticalproblem)andtheNewtoniangravitationalforce(mostheavilyused duringtheweek'scomputationalproblem). Asidefromnotes,materialisalsodeliveredtothestudentsthroughweeklypre-class homeworkquestions.Considerthepre-classhomeworkquestionshowninFig.3.2thatis madeavailableatthebeginningofthethirdweekofthecourse.Thisquestionismeant todemonstratethattherearemultiplecorrectwaysthataunitvectorcanbeconstructed incode.Giventhenatureofthecorrespondingweek'scomputationalproblem,weexpect 33 studentstobeabletodrawonandtakeadvantageofthisknowledgewhenfacedwitha relatedalbeitmorecomplicatedprobleminclass.Thatis,weexpectstudentstobechoosing betweencompetingsolutions.Choosingbetweencompetingsolutionsisaproblemsolving practiceindicativeofcomputationalthinkingthatwefocusedouranalysison. Figure3.2:Pre-classhomeworkquestionfocusingonthedi˙erentwaysthatthemagnitude ofavectorcanbeconstructedinVPythoncode:explicitlycodingthesquarerootofthesum ofthesquaresofthecomponentsandusingthepre-de˝nedPythonagnitudefunction. 3.4In-classwork Thereareanumberofin-classcomputationalproblemsspreadoutthroughoutthesemester, scheduledinTab.3.1.The˝rstfewcomputationalproblemsfocusondi˙erentforcemodels (i.e.,noforce,aconstantforce,anon-constantforce)andtheresultingmotionofobjects. Thelastfewcomputationalproblemsfocusonextendedobjectsandtheirrotation.While 34 M onday T uesday T hursday S unday W1 Pre-H1due A1:constantvelocitymotion C1:constantvelocitymotion Post-H1due W2 Pre-H2due A2:constantaccelerationmotion C2:projectilemotion Post-H2due W3 Pre-H3due A3:Satelliteorbit C3:Newtoniangravitationalforce Post-H3due W4 Pre-H4due A4a:Springforce A4b:Young'smodulus Post-H4due W5 Pre-H5due A5a:Friction A5b:Friction Post-H5due W6 Pre-H6due A6a:Circularmotion A6b:Circularmotion Post-H6due W7 Pre-H7due A7a:Gravitationalpotentialenergy A7b:Springpotentialenergy Post-H7due W8 Pre-H8due A8:Energy C4:Energy Post-H8due W9 Pre-H9due A9a:Heat A9b:Thermalenergy Post-H9due W10 Pre-H10due A10a:Rollingmotion A10b:Rotationalenergy Post-H10due W11 Pre-H11due A11:Elasticcollisions C5:Inelasticcollisions Post-H11due W12 Pre-H12due A12a:Statics A12b:gears Post-H12due W13 Pre-H13due A13:Angularmomentum C6:Angularmomentum Post-H13due W14 Pre-H14due A14:Angularcollisions C7:Angularcollisions Post-H14due W15 Pre-H15due A15a:Chooseyourownadventure A15a:Chooseyourownadventure Post-H15due Table3.1:Ascheduleforthesemesterfocusingontopicscovered,homework/readingdead- lines,andin-classproblems. solvingtheseproblems,groupsareexpectedtoengageinanumberofpracticesthatthe problemshavebeendesignedaround[5]: P1. developingandusingmodels, P2. planningandcarryingoutinvestigations, P3. analyzingandinterpretingdata, P4. usingmathematicsandcomputationalthinking, P5. constructingexplanations, P6. engaginginargumentfromevidence. P7. andobtaining,evaluating,andcommunicatinginformation. Oneofthescienti˝cpracticesusedheavilyonbothanalyticandcomputationdaysisthat of(P1)developingandusingmodels.Whetherthosemodelsbemathematicalorcomputa- tional,weexpectstudentstonotonlyworktogetheringroupstodevelopthemodel,but alsotoutilizethatmodelinfurtherinvestigations.Thistypeofscienti˝cpractice(P1)and 35 itsassociatedlearninggoals[47]werefurtherusedtogeneratethein-classprojectthatthis thesisfocuseson. 3.4.1Analyticproblem Inthethirdweekofthecourse,studentsareaskedtoanalyzethemotionofasatellite orbitingEarthbothanalyticallyandcomputationally.Fortheanalyticday,thegroupswere askedtosolveforthemagnitudeofthevelocityandradiusneededbyasatellitetobeheldin ageostationaryorbit.Thisinvolvesidentifyingtworelevantequationsintwounknownsand combiningthemtosolveforthedesiredradiusandmagnitudeofvelocity.Theinformation gatheredduringthisproblemcanbeusedinthefollowingcomputationalproblem,andthe groupfacilitatorsareoftenobservedreferencingthisinformation. 3.4.2Computationalproblem Thisthesisfocusesonthethirdandmostcomplicatedcomputationalproblemdelivered tothestudents,showninbothFigs.3.1and3.3.Givenitscomplexity,wedevelopeda frameworktohelpguideandgroundouranalysis.Thisframeworkwasconstructedwith thehelpofataskanalysis(seeSec.2.4)oftheproblem.Ultimately,studentsmustdesign, construct,andassesacomputationalmodelfortheNewtoniangravitationalforceactingon asatelliteingeostationaryandothermoregeneralorbits. Oncethecorrectforcehasbeencorrectlycoded,thegroupmustalsograpplewithadding inavisualizationofavectorrepresentingtheforcethattheyhavejustadded.Thistype ofmotiondiagramismeanttoshowthatthegravitationalforcevectorresultinginthe orbitalwayspointsradiallyinward(towardtheEarth).Thistaskrequiresstudentsto 36 Figure3.3:TheNewtoniangravitationalforceproblemstatementdeliveredtothestudents inthethirdweekofclass. programaswellasallowsthemtomoreeasilychecktheirconceptualunderstanding.Using computationalmodelstounderstandaconceptisacomputationalmodelingandsimulation practicethatisindicativeofcomputationthinking,andonethatwefocusedouranalysison. Additionally,inordertocheckthattheirmodelcanproduceageostationaryorbit,groups areaskedtogenerateagraphshowingthemagnitudeoftheseparationbetweenthesatellite andthecenteroftheEarthvs.time.Thisallowsthemtocheckforaconstantdistance whichimpliesacircularorbit.Thistaskofproducingagraphismeant,amongotherthings, toencouragestudentstovisualizedata,yetanothercomputationalpracticeindicativeof computationalthinking. 3.4.2.1Minimallyworkingprograms Whilebeginningtheproblem,thegroupusesaMinimallyWorkingProgram(MWP)similar tothoseseeninthetwopreviouscomputationalproblemslistedinTab.3.1.ThisMWP hasallofthestructureofthecodecorrect(thewhile/calculationloopandtheEuler-Cromer integration)butismissingthecomputationalforceactingonthesatellite(alongwithsome inaccuratenumericalvalues).TheinitialMWPcodewithitsinitialvisualizationareshown inFig.3.4. Thus,themaintaskofthegroupistoconstructaphysicallycorrectforcemodelincode. 37 Figure3.4:TheinitialcodeandvisualizationoftheMWPthatisgiventothestudentsin thethirdweekofthecourse. Secondarily,theymustmodifynumericalvaluestore˛ectthephenomenonbeingmodeled. Ideally,thisforcemodelwillbeofaNewtoniangravitationalform(i.e., F G ˘ 1 =r 2 )with adirectioncodedintermsofaseparationvector(i.e., ^ F G ˘ ^ r ˘ ~r=r ).However,thereare manyotherwaystogoaboutthis,andwedofrequentlyobservegroupsworkingwithother models(e.g.,acentripetalforce). 3.4.2.2Tutorquestions Thereareanumberofpre-writtentutorquestionsaswellasmanyon-the-˛yquestions generatedbythetutorswhileinclass.Thesequestionsaremeanttocheckthestudents forconceptualunderstandingaswellastodirectstudentstowardthecorrectsolution.For example,thetutorquestionsshowninFig.3.5aremeanttoensurethatthemodelthegroup hasconstructedisactuallygeneralenoughtogeneratealltypesofellipticalorbitsgiven variousinitialconditions. Ontheotherhand,atutorinteractionliketheoneshownbelowthathappenson-the-˛y 38 Figure3.5:Aselectionoftutorquestionsthatfocusonthecomputationalmodeleachgroup hasconstructed. mightencouragestudentstouseamoregeneralforceratherthanamorerestrictedone: TA: Youguyswannatalkaboutwhatyourstrategyisatthemoment? SB: Idon'tthinkweknow. SA: Wejust,weneedto˝gureouthowtogetthevelocityofthespacecraftcorrect,aswell astheforcenetcorrect,andthenitshouldbe˝ne... TA: Yeah,myrequest,canIpointinyourprogramthat'swhatyouhaveforFnetnow constantcomponents. TA: Myrequestistouseacompletelydi˙erentstrategywherethatformula[pointsto 39 GmM=r 2 ontheboard]isinforFnet. SC: Yeah,wetriedtomakethat,yeah... SA: Canwejustputthenumberin? TA: Umm,inprincipleyoucould,butI'dreallyratheryounot...Iwouldliketheprogram tobeabletorespondifthesatelliteisfatheraway,sotheforcewouldbeless,ifthe satelliteisclosertheforcewouldbemore... TA: SoIwouldlikeittobeadynamicprogramandnotonethatalwayshavea˝xedforce. Inthison-the-˛yinteraction,thequestionofweatherornottheircomputationalmodelwill beabletohandlealltypesoforbitsisenoughtoindicatethatthegroupneedstoswitch theirmodelup.Inthisway,thetutorisabletomakesurethegroupsstayonthedesired pathwithoutdirectlytellingthemexactlywhattodo. 3.4.3Feedback/Assessment Groupsareassessedonmanylevelsin P 3 .Oneofthemostimportantformsofassessmentis givenweekly,intheformofwrittenfeedbackandanumericalscore.Thewrittenfeedback isbasedontheobservedin-classperformanceandisdesignedtopointoutde˝cienciesand suggestswaystoimprove.Thenumericalscoringisbasedonperformanceinthreecategories: groupunderstanding,groupfocus,andindividualunderstanding. Oftenthewrittenfeedbackpertainstogroupactivitywiththecomputer.Forexample, theportionofwrittenfeedbackshowninFig.3.6isencouragingastudenttoallowother groupmemberstodosomeofthetyping.Thiscouldberequestedforanynumberofreasons 40 mostlikely,though,becausethestudentswithlesspriorprogrammingexperiencearenot beinggivenachancetoparticipate. Figure3.6:Asnippetofwrittenfeedbackgiventoastudentafterthethirdweek. Inthisway,instructorscanencouragetheirgroupstosharetheprogrammingload.While doingthetyping,itisverydi˚culttofollowalongwithoutknowingexactlywhatisgoing on.Thishelpstoengageallofthestudentswiththematerial. 3.5Post-classwork Thereareanumberofpost-classhomeworkquestionsthataremeanttoreinforcethephysics andcomputationalconceptsseeninclass.Duringthethirdweekofthecourse,theseques- tionsfocusmostlyontheNewtoniangravitationalforce.However,thepost-classhomework questionshowninFig.3.7thatisdeliveredinthethirdweekfocusesonthepreviousweek's computationalproblem(i.e.,itinvolvesalocalgravitationalforceasopposedtoaNewtonian gravitationalforce).Nevertheless,thispost-classquestioninvolvesthesameEuler-Cromer styleofnumericalintegrationasseeninallcomputationalproblems.Thestudentsareex- pectedtousetheerrormessageinordertoidentifyanerrorinthecode. 41 Figure3.7:Aportionofapost-classhomeworkquestiondeliveredinthethirdweekofthe course.Thisquestionrequiresstudentstotroubleshootanddebugthecode. Thistypeofproblemhelpstoencouragestudentstoidentify,isolate,reproduce,and correctunexpectedproblemsthatarisewhileconstructingcomputationalmodels.Ideally,it requiresstudentstointerpretthenamesgiventothevariablesbeingusedandverifythat theyarede˝nedinacorrectform. 42 Chapter4 Motivation Asidefromageneralinterestinintroductorycomputationalphysics,itisimportanttoun- derstandtheunderlyingmotivation(s)forthisthesis.Sectionsfromthefollowingchapter, detailingsomeofthosemotivations,werepublishedintheproceedingsofthe2015Physics EducationResearchConference[4],andarepresentedherewithminormodi˝cationsfrom theirappearanceinpublication.ItwaspublishedwithsecondandthirdauthorsPaulW. IrvingandMarcosD.Caballero,respectively. Theprocessofidentifyinganinterestingcomputationalpractice,describedinSec.4.1, wastheearliestmotivationforthisstudy.Wefoundthatitwasextremelydi˚culttode˝ne andidentifytheparticularpracticeofwhatwenamedysicsNotonlydidthe practiceneedtobeclearlyde˝ned,italsoneededtobeclearlyidenti˝edinthedata.This requiredalotofin-depthqualitativeanalysisandinter-raterreliability,motivatingouruse oftheWeintropframeworkandthequalitativemethodsofClarkeet.al. Additionally,wefoundthatitwasverydi˚culttounderstandthequalitativelydi˙er- entwaysinwhichstudentsexperiencedcomputationalintroductoryphysics.Thisdi˚culty motivatedataskanalysiswithafocusonidentifyingpracticesthatthestudentswereen- gaginginthroughin-classobservation,asopposedtotheirexperiencesthroughout-of-class interviews. 43 Figure4.1:Interactionsbetweenindividualsformagroup,andthegroupinteractswiththe computer. 4.1Debugging Inthissection,wepresentacasestudyofagroupofstudentsimmersedinthis P 3 environ- mentsolvingacomputationalproblem.Thisproblemrequiresthetranslationofanumberof fundamentalphysicsprinciplesintocomputercode.Ouranalysisconsistsofqualitativeob- servationsinanattempttodescribe,ratherthangeneralize,thecomputationalinteractions, debuggingstrategies,andlearningopportunitiesuniquetothisnovelenvironment. Wefocusthiscasestudyontheinteractionsbetweengroupandcomputer,illustratedin Fig.4.1,tobegintounderstandthewaysinwhichcomputationcanin˛uencelearning.Par- ticularly,weareinterestedintheinteractionsoccurringsimultaneouslywithsocialexchanges offundamentalphysicsprinciplesspeci˝ctothepresenttask(e.g.,discussing d r = v dt on amotiontask)andthedisplayofdesirableproblemsolvingstrategies(e.g.,divide-and- conquer).Thesegroup-computerinteractionsvaryinform,fromthemoreactiveprocessof siftingthroughlinesofcode,tothemorepassiveprocessofobservingathree-dimensional visualdisplay. Onepreviouslyde˝nedcomputationalinteractionthatreinforcesdesirablestrategies, borrowingfromcomputerscienceeducationresearch,istheprocessofdebugging[36].Com- putersciencede˝nesdebuggingasaprocessthatcomesaftertesting syntactically correct codewhereprogrammersoutexactlywheretheerrorisandhowto˝xit.Given 44 thegenericnatureoftheapplicationofcomputationincomputerscienceenvironments(e.g., datasorting,pokerstatistics,orWtasks),weexpecttoseeuniquestrategies speci˝ctoacomputational physics environment.Thus,weextendthisnotionofcomputer sciencedebuggingintoaphysicscontexttohelpuncoverthestrategiesemployedwhilegroups ofstudentsdebug fundamentally correctcodethatproducesunexpectedphysicalresults. 4.1.1Analysis InFall2014, P 3 wasrunatMichiganStateUniversityinthePhysicsDepartment.Itwas this˝rstsemesterwherewecollected insitu datausingthreesetsofvideocamera,micro- phone,andlaptopwithscreencastingsoftwaretodocumentthreedi˙erentgroupseachweek. Fromthesubsetofthisdatacontainingcomputationalproblems,we purposefullysampled a particularlyinterestinggroupintermsoftheircomputationalinteractions,asidenti˝edby theirinstructor.Thatis,wechoseourcasestudynotbasedongeneralizability,butrather onthegroup'sreceptiveandengagingnaturewiththeprojectasan extremecase [49]. Theprojectthattheselectedgroupworkedonforthisstudyconsistsofcreatingacom- putationalmodeltosimulatethegeosynchronousorbitofasatellitearoundEarth.Inorder togenerateasimulationthatproducedthedesiredoutput,thegrouphadtoincorporatea positiondependentNewtoniangravitationalforceandtheupdateofmomentum,usingreal- isticnumericalvalues.TheappropriatenumericalvaluesareGoogleable,thoughinstructors encouragedgroupstosolveforthemanalytically. Thisstudyfocusesononegroupinthefourthweekofclass(thefourthcomputational problemseen)consistingoffourindividuals:StudentsA,B,C,andD.Thegrouphad primaryinteractionwithoneassignedinstructor.Broadly,weseea50/50splitongender, withoneESLinternationalstudent.StudentAhadthemostprogrammingexperienceout 45 Figure4.2:Thedebuggingprocessnecessarilycorrespondstoaphasebesetoneithersideby thephasesofrecognitionandresolution.Notetheabsenceofaverticalscale,asthevertical separationmerelyactstodistinguishphases. ofthegroup.Itisthroughtheaudiovisualandscreencastdocumentationofthisgroup's interactionwitheachotherandwiththetechnologyavailablethatwebeganouranalysis. Tofocusinonthegroup'ssuccessfulphysicsdebuggingoccurringoverthe 2h classperiod, weneededtoidentifyphasesintimewhenthegrouphadrecognizedandresolvedaphysics bug.Thesetwophasesintime, bugrecognition and bugresolution arethenecessarylimits oneithersideoftheprocessof physicsdebugging ,asrepresentedinFig.4.2.Weidenti˝ed thesetwoboundingphasesataround60minutesintotheproblem,andfurtherexaminedthe processofdebuggingin-between.Thatis,wefocusedonthecrucialmomentssurrounding the˝nalmodi˝cationsthattookthecodefromproducingunexpectedoutputtoexpected output. 4.1.1.1Recognition Ataround55minintotheproblem,followinganinterventionfromtheirinstructor,thegroup begantoindicatethattheywereatanimpasse: SB: We'restuck. SD: Yeah... ThesimulationclearlydisplayedthetrajectoryofthesatellitefallingintotheEarthnotthe 46 geostationaryorbittheyexpectedasobservedonthescreencast.Thisimpassewasmatched withanindicationthattheybelievedthefundamentalphysicsprinciplesnecessarytomodel thisrealworldphenomenonwereincorporatedsuccessfullyintothecode: SB: Andit'sgonnabesomethingreallydumbtoo. SA: That'sthethinglike,Idon'tthinkit'saproblemwithourunderstandingofphysics, it'saproblemwithourunderstandingofPython. Insteadofattributingtheunexpectedoutputwithamistakeintheirunderstandingoren- codingofthefundamentalphysicsprinciples,theyinsteadseemedtoplaceblameonthe computationalaspectofthetask. Duringthisinitialphase,weseeaclearindicationthatthegrouphasrecognizedabug thereisanunidenti˝ederrorinthecode,whichmustbefoundand˝xed: SA: Idon'tknowwhatneedstochangehere... SD: Imean,thaterrormeanswecouldhavelikeanythingwrongreally. Althoughtheyhaveidenti˝edtheexistenceofthebug,theystillarenotsurehowto˝xit thisnecessitatestheprocessofdebugging. 4.1.1.2Physicsdebugging Withinthepreviouslyidenti˝edphaseofbugrecognition,thegroupdevelopedaclearand primarytask:˝gureoutexactlyhowtoremovethebug.Eventually,followingalittleo˙- topicdiscussion,thegroupacceptedthatinordertoproduceasimulationthatgenerates thecorrectoutput,theymustonceagaindelveintothecodetocheckeveryline: SA: I'mjusttryingtobreakitdownasmuchaspossiblesothatwecan˝ndanymistakes. 47 Inthisway,thegroupbegantonotonlydeterminethecorrectnessoflinesofcodethathave beenadded/modi˝ed,butalsobegantoexaminetherelationshipsbetweenthoselinesof code. Forexample,thegroupbeganbycon˝rmingthecorrectnessoftheformofonesuchline ofcode: SA: Finalmomentumequalsinitialmomentumplusnetforcetimesdeltat.True? SC: Yeah... SB: Yes. SA: O.K.That'sexactlywhatwehavehere.Sothisisnottheproblem.Thisisright. SD: Yeah. Thatis,StudentA(1)readaloudandwrotedownthelineofcode ~p f = ~p i + ~ F net dt whiletheentiregroupcon˝rmedonitscorrectform.Thiswrittenlinewasthenboxed,and wasshortlyfollowedupwith(2)asimilarcon˝rmationoftheline ~r f = ~r i + ~v dt that immediatelyprompted(3)thecon˝rmationof ~v = ~p=m .Thus,notonlydoweseethegroup determiningthecorrectnessofadded/modi˝edlinesofcodeasin1,2and3,wefurthersee con˝rmationwiththelinksbetweenthoselines.Thecon˝rmationofthelinkbetweenthe linesofcode1and2,representingtheincrementalupdateofpositionandmomentumin time,respectively,wasevidencednotthroughthemereadditionofthelinkingequation(3) tothelistoflinesadded,butfurtherthroughthegesturesexhibitedbystudentA.Pointing at(3),the ~v in(2),andthe ~p f in(1),demonstratedthatthegroupunderstoodthatwithout thislinkingequation(3),thevelocityusedin(2)wouldnotre˛ectthetimeupdatedvelocity bymeansof(1). 48 Thegroupranthroughthesetypesofcon˝rmationswithfundamentalphysicsprinciples rapidlyoverthespanofafewminutes.Oncethegrouphadcon˝rmedalltheadded/modi˝ed linesofcodetotheirsatisfaction,thediscussionquieteddown.Thefundamentalphysics principleswerewinnowedfromthediscussion,andafteralittlemoreo˙-topicdiscussionwe ˝ndthemseekinghelpfromtheinstructor: SD: Maybeweshouldjuststareathimuntilhecomeshelpus... Suddenly,ahaphazardchangetothecode: SA: Youknowwhat,I'mgonnatrysomething... whereStudentAchangedtheorderofmagnitudeoftheinitialmomentumafewtimes.This modi˝cationeventuallyresultedinasimulationthatproducedthecorrectoutput. 4.1.1.3Resolution Atabout65minintotheproblem,StudentAchangedtheorderofmagnitudeofthemo- mentumone˝naltime,whichproducedsomethingclosertotheoutputthattheyexpected: SA: Ohwait...Ohgod... SD: Isitworking? ThesatellitenowellipticallyorbitedtheEarth.Thismarkstheendofthedebuggingphase andthebeginningoftheresolutionphasegiventhatthebughadsuccessfullybeenfound andremedied.Giventhattheonlylineofcodemodi˝edtoproducethischangewasthe initialmomentum,theybegantorethinktheproblem: SD: Ithinkthatistheissueisthatwedon'thavetheinitialmomentum... 49 SA: Momentumcorrect? Thatistosay,thegrouppursuedtheissueofdeterminingthecorrectinitialmomentum withtheaddedinsightgainedthroughdebuggingfundamentallycorrectVPythoncode. 4.1.2Discussion Tosummarize,inanalyzingthisparticulargroup,we˝rstidenti˝edthetwophasesintime whenthegrouphadrecognizedandresolvedaphysicsbug.Wethennecessarilyidenti˝ed thephasein-betweenastheprocessofphysicsdebuggingin P 3 ,wherethefundamentally correctcodewastakenfromproducingunexpectedoutputtoproducingexpectedoutput. Givenourassumptionthattheprocessofcomputersciencedebuggingencouragesdesirable strategies,wethenbegantoanalyzethisprocessofphysicsdebuggingfurtherforstrategies uniqueto P 3 . Giventheactionsexhibitedduringthedebuggingphase,wecanseparatethemintotwo distinctparts:amorestrategicpartandalessstrategicpart,asshowninFig.4.2.Thegroup initiallygaveindicationthattheywereworkinginaconsiderate,thorough,andconsistent manner,whichweclassifyasmorestrategic.Thisiscontrastedbythelaterindications ofmorehaphazardactions,whichweclassifyaslessstrategic.Thesearethetwophysics debuggingstrategiesthat,together,ledtotheresolutionofthebuginthiscontext. Themorestrategicstrategywasexhibitedthroughthecon˝rmationofindividualFPPs aswellastheirrelationtoothers.Notonlydidthegroupcon˝rmthroughdiscussion,they simultaneouslywrote,boxed,andreferencedequationsinthecodethishelpedtoreducethe numberoffundamentalphysicsprinciplestheyneededtocognitivelyjuggleatanygiventime [4].Thiscon˝rmationofFPPsthroughdiscussionpresentedagreatlearningopportunity 50 fortheentiregroup,wherecreativeandconceptualdi˙erencescouldbejointlyironed-out. Accordingly,wetentativelyrefertothisstrategyasself-consistency. Althoughtheresolutionofthebugmightnotbetieddirectlytothisself-consistency, thatdoesnotnegatethelearningopportunitiesa˙ordedtothegroupalongtheway.Specif- ically,wesawthegroupdouble-checkingeveryfundamentalideausedand,possiblymore importantly,thelinksbetweenthoseideas.Beingphysicallyself-consistentinthismanner isadesiredstrategyin P 3 . Thelessstrategicstrategywasexhibitedduringthehaphazardchangestotheinitial momentum.Thesechangestothecodethateventuallyresolvedthebug,thoughoneofthe bene˝tsofcomputation(i.e.,theimmediacyoffeedbackcoupledwiththeundofunction), couldhavebeenmorethoughtful.Adeeperunderstandingofthephysicsorcomputation couldhavetippedthegroupo˙tothefactthattheinitialmomentumwastoosmall. Again,thisdoesnotnegatethelearningopportunitiesa˙ordedtothegroupthroughthis lessstrategicstrategy,whichresemblesthatofductivemessingab[4]Accordingly, wetentativelyrefertothisstrategyasplay. Bothofthesestrategiesidenti˝edhere,self-consistencyandplay,providedlearningop- portunitiestothegroupwhicharebolsteredbythecomputationalnatureofthetask.In otherwords,thenecessityoftranslatingacollectionofphysicalideasintolinesofcodewhich mustlogically˛owandthebene˝tofimmediatevisualdisplayresultedinlearningopportu- nitieswhichmightotherwisehavebeenmissedinananalytictask.Moreresearchisneeded todissecttheselearningopportunitiesandtodeepenourunderstandingofthestrategies themselves. 51 4.1.3Conclusion Thiscasestudyhasdescribedtwostrategies(onemoreandonelessstrategic)employed byagroupofstudentsinaphysicscoursewherestudentsdevelopcomputationalmodels usingVPythonwhilenegotiatingthemeaningoffundamentalphysicsprinciples.These strategiesarosethroughthegroup'sprocessofdebuggingafundamentallycorrectprogram thatmodeledageostationaryorbit.Theadditionaldatawehavecollectedaroundstudents' useofcomputationisrich,andfurtherresearchisneededtoadvancethedepthandbreadth ofourunderstandingofthemyriadofwaysinwhichstudentsmightdebugcomputational modelsinphysicscourses. 52 Chapter5 Observations Throughoutouranalysisinthisthesis,wehavemademanydi˙erenttypesofobservations, andhaveusedthoseobservationstohelpanswerourresearchquestion(i.e.,whatarethe computationalpracticesindicativeofcomputationalthinkingthatarecommonto P 3 ?). Accordingly,itisimportantthatwetakesometimetoelaborateontheprocessofand resultsfromthoseobservations.Morespeci˝cally,inthischapter,wedetailthemethodof ouranalysis(i.e.,thedatareduction,thecodingprocess,andtheinter-raterreliability)and illustratetheidenti˝cationofsomeofthemostinterestingpractices(e.g.,troubleshooting anddebugging,assessingcomputationalmodels,andcreatingcomputationalabstractions). TheremainingpracticesarepresentedinCh.6andApps. 5.1Analysis Ourfullanalysisinvolvesdi˙erentstages:˝rst,theinitialdatawascollectedandsubse- quentlyreducedinordertoprovideamanageablesetofdata;next,acodingschemewas generatedusingtheWeintropframeworkfromSec.2.1tohelpidentifycomputational practices;and˝nally,inter-raterswereusedtoensurethereliabilityoftheanalysis.Eachof thesethreestagesaredetailedbelow. 53 5.1.1Datareduction Ourtotalsetorcorpusofdataconsistsofin-classvideoofninegroupsoffourindividuals working.Eachgroupworksonthreecomputationalproblems(twenty-sevenvideosintotal) thatincreaseindi˚culty/complexityasthesemesterprogresses.Thesecomputationalprob- lems,presentedinSec.3.1,requirestudentstoconstructvariouscomputationalforcemodels incode.Eachweek,theappropriateforcemodelincreasesincomplexityandgenerality. Speci˝cally,the˝rstprobleminvolvesaconstantzeroforce,thesecondprobleminvolvesa constantnon-zeroforce,andthethirdprobleminvolvesanon-constantforce. Inorderto˝rstreducethecorpusofourdatatoamorefocusedandmanageableset,we followedthesuggestionsoftaskanalysis(seeSec.2.4).Thatis,wepaidattentiontowhen studentsweremakingthemostprogresstowardasolution.Thefrequencyofindependent progressbeingmadeincreasedasthecomplexityoftheproblemincreased(i.e.,studentsmade themostindependentprogressonconstructingtheNewtoniangravitationalforcemodel). Here,wearede˝ningendentasprogressthatisultimatelymadebythegroup withoutanyinstructorintervention.Webelievethisrapidincreaseofindependentprogress isdue,inpart,totheirlackofpriorprogrammingexperiencecomingintothecourse.For example,onthe˝rstproblem,manygroupsstruggledwithabasiccalculationloop.Bythe timetheyseethethirdproblem,theyhavealreadygainedalittleexperienceandknowwhat toexpectinthecourse. Ourinitiallyreducedsetofdataconsistsoftranscriptsfromin-classvideo(bothside- viewandoverhead-view)ofninegroupsworkingonthegeostationarysatelliteproblemfrom Sec.3.3.Wealsocollectedcomputerscreencaststocaptureexactlywhatstudentsaredoing whentheytype/clickontheirgrouplaptop.Followingthesuggestionsofthematicanalysis 54 Figure5.1:Aportionoftranscriptmeanttohighlighttheindicationofunspokenandinferred actions.Forexample,line367showsthisgrouplookingintheirnotesforanequation.The equationthatthey˝ndiswrittendowninline370. (seeSec.2.5),webeganwithafulltranscriptionofthein-classvideo.Anyinaudiblesections areindicated,withlongpausesbeingindicatedbyellipses( ::: ).Todistinguishbetweenun- spokenactions(e.g.,pointingtoanequation)andinferencesmadebytheprimaryresearcher (e.g.,agroupreferringtoapreviouslyusedequation),wefollowtheconventionofsquare brackets( [] )andcurlybrackets( fg ),respectively.Forexample,Fig.5.1showsaportionof transcripthighlightingthesevariousindications. Oncewehadreducedourdatacorpustoamoremanageableandfocusedsetofnine transcripts,wecontinuedourinvestigationintothecomputationalpracticesstudentswere engagingin.Eachtranscriptwasreadmultipletimesinordertogeneratealow-resolution butcoherentpictureofwhateachgroupwasdoing.Thistypeofwiththe dataisacrucialstepasoutlinebyBraunet.al.Ultimately,thislow-resolutionpicture helpedustoidentifytheo˙-topicandotherwiseirrelevantdiscussioninordertoremove thoseportionsofthedatafromouranalysis. 55 Morespeci˝cally,eachtranscriptwasinitiallyanalyzedwithaneyetowardsidentifying discussionwherestudentsweresolvingthesatelliteproblem.Allotherdiscussionthencould beconsideredo˙-topicandsafelydiscarded.Forexample,groupsareoftenseendiscussing homeworkforotherclassesthatinnowayrelatestotheNewtonianproblem.Similarly, groupscanoftenbeseendiscussingrecentsocialevents(e.g.,aconcert).Thistypeofo˙- topicandotherwiseirrelevantdiscussion,althoughimportantforthesocialcohesionofthe group,cansafelybediscarded.Inthisway,wefurtherreduceourdatasetbyaboutone quarter.Witheachofninetranscriptbeingabout˝fteen-hundredlinesofspeech/action, thistranslatestoabout˝fteen-thousandlinesofon-topicdiscussionforfurtheranalysis. Acloseranalysisofthison-topicdiscussioniswherewebegintomoreclearlyde˝newhat computationalpracticeslooklikewithinourdata.Thiscloseranalysisstartedwiththesearch foranumberofcharacteristics(asdescribedinSec.2.1),withintheon-topicdiscussion.For example,thekeycharacteristicsforthepracticeoftroubleshootinganddebuggingare:i) toidentifyandisolateanunexpectederror,ii)articulatehowtoreproducetheerror,and iii)worktosystematicallycorrectit.Thesecharacteristics,onceidenti˝ed,canbeusedto justifytheclassi˝cationofanexcerptasthecomputationalpracticeoftroubleshootingand debugging.Recallthateachcomputationalpracticemaybeindicativeofthecomputational thinkingasdescribedinSec.2.1.Thisjusti˝cationallowsustode˝nethecomputational practicesweseeinourdata.Adetailedaccountofthisprocessofjusti˝cationisdescribed below,withapplicationstospeci˝cexamplesfollowinginSec.5.2. 5.1.2Codingprocess Inordertojustifytheclassi˝cationofanexcerptasaparticularcomputationalpractice, westartedbysystematicallycodingourdata.Thissystematiccodingprocesswasapplied 56 tothreestreamsofdata:theside-viewvideo,over-headvideo,andcomputerscreencasts. Thesethreestreamswerethenusedtogeneratethreetypesofrationale:rationaleaccording totheframework,rationalewithinanindividualexcerpt,andrationalebeyondanindividual excerpt.Thesethreetypesofrationalearedescribedindetailbelow. Intermsoftheframework,weidenti˝edthevariouscharacteristicsthatmanifestedthem- selvesintheactionsandspeechofeachgroupandcomparedthemtotheWeintropframework. Eachpractice,accordingtotheframework,hasanynumber(betweenoneandseven)ofre- latedcharacteristics.Themorerelatedcharacteristicsthatweseeinanexcerpt,themore con˝dentweareinclassifyingthatexcerptasaparticularpractice.Forexample,ti- fyinganunexpectederrorincoisoneoftherequiredcharacteristicoftroubleshooting anddebugging.Similarly,orkingtosystematicallyrectifytheunexpectedisclearly arelatedbutdistinctcharacteristic.Theidenti˝cationofeitherofthesecharacteristicsin- dividuallywouldbehintingatthepracticeoftroubleshootinganddebugging,butbothof themsimultaneouslymakesastrongerclaim.ThistypeofrationalecanbefoundinColumn GofFig.5.2. Withinanindividualexcerpt,weareabletofocusinonwhateachmemberofthegroup saysanddoesastheyworktowardaclearandfocusedgoal.Anyrationaleofthistypeusually referenceslinenumberspertainingtospeci˝clinesofspeech/actionwithintheexcerptthat embodiesthecharacteristicinquestion.Inthisway,wecloselytieourrationaleandthe frameworktothedata.Forexample,agroupmightidentifyanunexpectederrorintheir programandsay: SC: (756) Ohthereitis{theerrormessage}. SB: (757) Where? 57 SC: (758) Inthething{shell}onthescreen... Inthisexchange,StudentChasfoundtheerrormessagefromtheshellburiedunderafew otherwindows.Thiserrormessageisultimatelyusedbythegrouptotrackdownthecauseof theunexpectederror.Inthisway,weclearlyseeagroupworkingtoidentifyanunexpected errorinourdata.ThistypeofrationalecanbefoundinColumnHofFig.5.2. Beyondeachindividualexcerpt(i.e.,lookingateachtranscriptasawhole),weareable togeneratealow-resolutionpicturethatcapturestheoverarchinggoalsthateachgroupis workingtoward.Thislow-resolutionpicturehelpsustocontextualizeeachindividualexcerpt withinthebroadertranscript.Therearedi˙erentwaystocontextualizeaparticularexcerpt ofdata(e.g.,inthecontextofthegroup,theclassroom,theuniversity,thestate,etc.),and relatingittootherexcerptsisoneofthemostimportant.Forexample,withinanindividual excerpt,agroupmightreferencewithoutde˝ninganequation: SA: (894) Shouldwetrythatoneequation? SB: (895) Yeah,Ithinkweshoulddothat... SA: (896) Okay. SC: (897) Yeahthat'sagoodidea,let'susethatone. Usingourlow-resolutionpictureofthetranscriptasawhole,wecantrackbackthrough time(oftenminutes,sometimeslonger)to˝ndoutexactlywhatvagueequationtheyare referencing: SC: (120) Howaboutweusetheequation... SC: (121) [writes GmM=r 2 ]. 58 Figure5.2:Thetemplateusedforthecodingprocess.Eachexcerptisnumbered,eachline ofspeech/actionisnumberedandattributedtoanindividualmemberofthegroup,andthe threetypesofrationaleareusedtojustifytheclassi˝cationofaparticularpractice. SC: (122) Andthenmultipliedbyrhat... SD: (123) Idunno... Anyrationaleprovidedatthislevelusuallyreferencesthenumberofanotherexcerptthat providesthenecessaryadditionalinformation.ThislevelofrationalecanbefoundinColumn IofFig.5.2. Thiscodingprocesswasfollowedforninegroupstogenerateabout˝ve-hundredcandidate excerpts,eachexcerpthavingmultiplepractices,andeachpracticehavingthethreetypes ofrationaledescribedabove.Eachexcerpthasanywherefromonetofourpossiblepractices identi˝edwithsupportingrationale.Thatequatestoroughlythree-thousandindividual justi˝cationsthatmustbefoundwithinourdata.Afterconcludingourinter-raterreliability, describedbelow,wehadareduceddatasetofroughlyonesevenththeinitialset. Thethreetypesofrationaledescribedabove,thoughnotnecessarilypersuasiveindivid- ually,whentakentogethercanprovideareasonablejusti˝cationfortheclassi˝cationofan excerptasbelongingtoaparticularcomputationalpractice:therationalefromtheframe- workprovidesincompletebutguidingde˝nitions,therationalewithinanindividualexcerpt tiesuscloselytothedataandtheimmediateactionsthatagroupistaking,andtherationale 59 Figure5.3:Examplesofthethreelevelsofcon˝denceareshowningreen,yellow,andredto indicatehigh,medium,andlowcon˝dence,respectively.Eachinter-ratersuggestionisused tomodifyorsolidifythelevelofcon˝dencegiventoaparticularpractice. beyondanindividualexcerpthelpstocontextualizethoseimmediateactionsandspeech. 5.1.3Inter-raterreliability Inordertoensurenotonlyreasonable,butalso reliable justi˝cationsfortheclassi˝cation ofthevariouscomputationalpracticeswithinourdata,wefollowedaniterativeprocessof inter-raterreliability.Oneprimaryresearcherwasjoinedbythreeinter-raters,ensuringa robustcodingprocessandstrongerclaimsthroughiterativecritiqueanddiscussion. Initially,thedatawascodedbytheprimaryresearcher,relyingheavilyontheWeintrop frameworkandthequalitativemethodsdescribedinCh.2,togenerateaninitialsetof rationaleforeachcandidateexcerpt.Thisinitialsetofrationaleforaparticularexcerpt, consistingofthethreetypesofrationaledescribedinthesectionabove,wasthentakenasa 60 Figure5.4:Theinitialrationalegeneratedforanexcerptalongwithinter-ratersuggestions andsubsequentmodi˝cationovertime.Withtheadditionofsomerequestedinformation, thestrengthoftherationalewasimprovedandthecon˝dencewaspromotedfrommedium tohigh. wholetoformulateaninitiallevelofcon˝dence:low,medium,orhigh.Lowcon˝dencewas usuallygiventoexcerptscontainingonlyafewofthecharacteristicsneededbyapractice, ortoexcerptswheretheidenti˝cationofanindividualcharacteristicwasinseriousquestion. Mediumcon˝dencewasgiventoexcerptscontainingmostofthecharacteristicsrequiredby apractice,ortoexcerptswheretheidenti˝cationofindividualcharacteristicswasprobable. Highcon˝dencewasgiventoexcerptscontainingalloftherequiredcharacteristicsfora practice,ortoexcerptswheretheidenti˝cationofeachindividualcharacteristicwasself- evident.Examplesofexcerptsbelongingtothesedi˙erentlevelsofcon˝denceareshownin Fig.5.3. Asubsetofthedatacontainingavarietyofcomputationalpracticesandlevelsofcon˝- dencewasthensharedwithinter-raters,rangingfromundergraduatestudentstoprofessors. Eachinter-ratersubsequentlytestedthestrengthofourinitialclaimsthroughdiscussionby 61 askingquestionsandmakingsuggestions.Thesesuggestions,oncemutuallyagreedupon, wereincorporatedintotherationale.Forexample,Fig.5.4showsoneinter-raterasking aclari˝cationquestionastowhattheverbatimoutputoftheshellinaparticularexcerpt was.Theanswertothisclari˝cationquestion,thoughnotobviousgiventheinitialrationale, provestoberelevantandnecessarytothestrengthoftherationale.Thisprocessofgener- atingreliabilitythroughaskingquestionsandmakingsuggestionswasfollowediterativelyto furtherstrengtheneachclaim. 5.2Computationalpractices Byanalyzingallofthedatawiththemethodsdescribedabove,wehaveidenti˝edanumber ofpracticesthatshowupinourdata.Thesepracticesandtheirfrequencieswithinourdata aresummarizedinFig.5.5.Intotal,weidenti˝edroughly300occurrencesofindividual practices,withsomepracticesoccurringfrequentlyandsomeoccurringnever.Themost frequentpractices,thoughfoundwithinourdata,canbeexpectedtoarisejustasfrequently insu˚cientlysimilarclassroomsanddeserveafairamountofattention. Theremainderofthissectionprovidesconcreteexamplesofsomeofthemostfrequent computationalpracticesthatwefoundinourdata.Wearefocusingonthosepracticesthat occurwithhighfrequencywithinonegrouporoccurwithmoderatefrequencyacrossmultiple groups.Thesepracticesare(innoparticularorder):creatingandanalyzingdatawithin thedatapractices;designing,constructing,andassessingcomputationalmodelswithinthe modelingandsimulationpractices;programming,creatingabstractions,andtroubleshooting anddebugginginthecomputationalproblemsolvingpractices;andthinkinginlevelsand communicatinginformationwithinthesystemsthinkingpractices. 62 Figure5.5:Thefrequencyofeachpracticethatwasfoundwithinouruniquedataset. 63 Characteristic Qualities Automating Thedatathatisbeingcreatedshouldbedonesoinanautomaticoral- gorithmicmanner.Forexample,anEuler-Cromerstyleintegrationisfre- quentlyusedtogeneratelargesetsofnumericaldatarepresentingvarious physicalphenomenaintime. Advancing Eachgroupshouldultimatelybeadvancingtowardcompletionofthespec- i˝edtask.Forexample,creatinganalgorithmthatgeneratesthevarious momentaofthesatellitecanultimatelybeusedtohelpgenerateasimu- lationofitstrajectory. Table5.1:Thecharacteristicsandassociatedqualitiespertainingtothecomputationalprac- ticeofcreatingdata:automatingthecreationofdatathathelpstoadvancetowardgoals. Althoughtheexamplesthatfollowaremeanttoclearlyillustratesomeofthecommon computationalpracticesthatwehaveobserved,theydonotcomewithouttheirownlimita- tions.Accordingly,Ch.6providesadiscussionofthoselimitations,aswellaspresentssome ofthelesscommonandunobservedpractices. 5.2.1Creatingdata Thecomputationalpracticeofcreatingdata,asde˝nedbyWeintropet.al,involvesthe generation(asopposedtothecollection)ofcomputerdatawhilevestigatingphenomena thatcannotbeeasilyobservedormeasuredorthataremoretheoreticalinThis typeofdatacreationfrequentlyarisesinphysicsandengineeringgiventhatdatacollection isinfeasibleinmanyrealisticsituations.Forexample,complexcomputermodelscanbeused togeneratedatathatcanbeusedtooptimizelaunchconditionsforsatellitesandmanned rocketswhenreal-worldcollectionofdataistoocostlyordangerous.Thefundamental characteristicsassociatedwiththispractice,assummarizedinTab.5.1,are:i)de˝ninga computationalprocedurethatautomatically/algorithmicallycreatesdataandii)usingthat procedureortheresultingdatatoadvancetheoverallgoalsofthetask. ConsiderExcerpt9fromGroupH.Overthecourseoftwohours,thisgroupcanbeseen 64 ensuringthattheirMWPwilldynamicallyupdatethepositionofthesatellite.Thisentails ensuringthatthemomentumofthesatellitewillalsodynamicallyupdate.Accordingly,the groupworkstoconstructacomputationalalgorithmthatwillautomaticallycreatesetsof datarepresentingthepositionandmomentumofthesatelliteovertime.Thesesetsofdata arethenultimatelyusedtoadvancetowardcompletingthegoalofproducingofarealistic visualizationofthetrajectoryofthesatellite. Earlyon,thegroupcanbeseendiscussingtheirgoalofgeneratingavisualizationofthe satellite'sorbit(lines195-196).Theyconsiderchangingtheinitialpositionofthesatellite (line199)towhattheycalculatedfromthepreviousproblem: SD: (195) So,it'smostlyjusttryingto˝gureouthowtogetit{theprogram}todisplay anorbit... SA: (196) Yeah,itis. SC: (197) Wait,wehavetochangetheposition,don'twe? SB: (198) Ithinktheinitialpositionstaysthere,wehavetoupdatepositionthough... SC: (199) Yeah,wehavetochangetheinitialpositiontowhatwefound...itwasthisfar away,youknow? SA: (200) Yeah. SB: (201) Yeah. SA: (202) Whichwas...fourpointfourtwotimestentotheseven. SB: (203) Fourpointtwo...[codes] 65 Theymakethedistinctionbetweenchangingtheinitialpositionofthesatelliteandchanging thewaythatthepositionupdatesovertime(line198).Thisisanimportantdistinction becauseeachchangeinvolvesvastlydi˙erentamountsworktoaccomplish,andonlyone resultsintheautomatic/algorithmiccreationofdata.Thatis,changingtheinitialposition ofthesatelliteisasimplechangeofanumericalvalue,whereaschangingthewaythatthe positionupdatesovertimeinvolvesde˝ningasetofalgorithmswithmultiplevariablesinside ofthecalculationloop.Ensuringthatthepositionupdatesproperlyisabigadvancement towardtheirgoalofproducingarealisticvisualization. Eventually,theyproposeanEuler-Cromerstylealgorithmtoautomaticallyupdatethe positionofthesatellite(line222)intermsofitsmomentum,mass,andtime: SB: (217) Alright... SB: (218) Okay,sowehavetoadditsnewposition. SA: (219) Butithastoupdateitspositioneverytime... SB: (220) Right. SA: (221) Sowehavetomakeitupdate. SB: (222) Satellitepositionplusmomentumofthesatellite... SA: (223) Overthemass? SB: (224) Timesthechangeintime...yeahsoit's,yeah. SB: (225) Butthemomentumisalwayschanging... Althoughthegrouphasclearlylaidoutthewaythatthepositionofthesatellitewillneed tochange(line222),theyhaveraisedanotherconcernintermsofthemomentumofthe 66 satellite(line225).Inotherwords,theyhavede˝nedaproceduretoautomaticallycalculate thepositionsofthesatellite,butstillneedtode˝neaproceduretoautomaticallycalculate themomenta. Later,asthegroupworkstowardde˝ningaproceduretochangethemomentumofthe satelliteovertime,theyrecalltheconceptofbothiterativeprediction(line684)andNewton's secondlaw(line695)fromthenotes: SB: (681) Sowegotta˝gureouthowtochangethemomentuminthere{thecode}. SB: (682) Whatwastheequationfromlastweek? SA: (683) Umm...Fgrav...No. SD: (684) Whataboutusingiterativepredictionforlikefuturepositions? SD: (686) Right? SC: (687) Thechangeinmomentumwouldbethenetforcetimes... SB: (688) Becausetheforceismasstimesacceleration... SC: (689) Thatwouldbeit,yeah. SB: (690) Sointegratethat. SC: (692) Changingmomentumisforcetimeschangeintime... SB: (693) Oh,therewego,nice. SA: (694) Waitwhatisit? SB: (695) Thechangeinmomentumisthenetforcetimeschangeintime. 67 Withthesetwoalgorithmsde˝ned,theirMWPisreadytoautomaticallyanddynamically updatethepositionandmomentumofthesatellite.Afterward,thegroupspendsafair amountoftimeincorporatingtheappropriateforcemodelintotheircode.Theconstruction ofthesealgorithms,alongwiththecorrectforcemodel,showsaclearadvancementtoward theirgoal(line195)ofgeneratingavisualizationofthesatellite'sorbit. Tosummarize,thegroupcanbeseen automating thegenerationofsetsofdatarepre- sentingthepositionandmomentumofthesatelliteovertime.Further,withthesesetsof data,thegroupisultimately advancing theirprogresstowardproducingavisualizationof orbitalmotion.Giventheidenti˝cationofthesetwocharacteristics,weclassifythisexcerpt asthecomputationalpracticeofcreatingdata. 5.2.2Analyzingdata Thecomputationalpracticeofanalyzingdata,asde˝nedbyWeintropet.al,usuallyinvolves largesetsofdata(thathaveeitherbeencreatedorcollected)wheregroupsareokingfor patternsoranomalies,de˝ningrulestocategorize,andidentifyingtrendsand Thistypeofanalysisshowsupfrequentlywithinthe˝eldofphysics,especiallygiventhe computationalnatureofmany(ifnotmost)moderninvestigations.Forexample,extremely largesetsofdataaregeneratedwhileinvestigatingtheformationandevolutionofgalaxies throughouttheuniverse.Beingabletoe˙ectivelyanalyzealargesetofdataisacrucial skillwithinmanyinterdisciplinary˝elds.Thefundamentalcharacteristicsassociatedwith thiscomputationalpractice,assummarizedinTab.5.2,are:i)ageneralprocessofanalysis (detailedinTab.5.2)andii)aconclusionbeingdrawnbasedonthatanalysis. ConsiderExcerpt35fromGroupH.Overall,thisgroupcanbeseenengaginginthe processofanalysisofasetofdatathatrepresentsthenetforceactingonthesatellite,and 68 Characteristic Qualities Analyzing Thisisabroadtermthatusuallyinvolvesatleastoneofmanytypes ofanalysis.Forexample,sortingasetofdataintodi˙erentcategories, lookingfortrendsorpatternswithinagivenset,lookingforcorrelations betweenmultiplesets,and/oridentifyingoutliersandanomaliesareall consideredtobedi˙erenttypesofanalysis. Concluding Theinformation(e.g.,apatternortrend)gatheredfromtheanalysisofa setofdatashouldultimatelybeusedtomakeordrawsomeconclusion. Thischaracteristic,thoughanimportantone,isnotnecessarilyrequired foragrouptobeanalyzingdata. Table5.2:Thecharacteristicsandassociatedqualitiespertainingtothecomputationalprac- ticeofanalyzingdata:ageneralprocessofanalysisleadingtoconclusionsbasedonevidence. drawingaconclusionbasedontheresultsofthatanalysis.Theparticularprocessofanalysis observedinthisexcerptinvolvesbothcategorizationandpatterning.Thecategoriesthat thedataareplacedinare:a)large-scalenumbersandb)vectorquantities.Thetrendthat thegrouprecognizesisthatthesetofdatarepresentingthenetforceistimedependent. Priortothebeginningofthisexcerpt,thegroupaddsaprintstatement(i.e., print(Fnet) ) intotheircalculationlooptoprinto˙thenumericalvalues( x -, y -,and z -components)of thenetforceactingonthesatelliteovertime.Theydothistocheckthattheirmodelis producingtheexpectedvalues: SD: (1330) Howmanytimesdoesthiscalculationlooprunthrough? SB: (1331) Alot... SD: (1332) Yeah..alot[lookingattheoutput]. SB: (1333) Howevermanysecondsareinaday. SA: (1334) Eightysixthousand. SD: (1335) Wow... 69 SB: (1336) Yeahdoingitlinebylineisnotgonnabeeasy. Withthisprintstatement,theyarecreatingalargesetofdata(line1332)thatissubsequently analyzed. Thegroupcon˝rmsthattheirprintstatementisdisplayingalargesetofdatathat representsthenetforceonthesatellite(line1338).Atthesametime,theybegintocategorize thedataandlookfortrends: SD: (1337) It'snotshowingthesatellitebecauseIthinkthe{window}scaleistoosmall. SD: (1338) Butit'soutputtingalloftheforces,anditis... SD: (1339) It'schangingtooIthink. SB: (1340) Howbigarethey? SB: (1341) I'massumingweretalkingaboutFgrav... SC: (1342) Yeah,itisbig. Onetrendthatthegroupsuggests(line1339)isthatthevaluesinthesethavesomesort oftimedependence.Similarly,onecategorythatthegroupplacesthedatain(line1342)is thatofhavingalargeorderofmagnitudewhichisexpectedgiventhetypeofforcethat theyareanalyzing. Mistakenly,thegroupbelievesthatthetrendoftimedependencethattheyhaveidenti˝ed intheirdataisnottheexpectedordesiredone.Inotherwords,theysuggestthatthesetof datashouldbeconstantintime(line1343): SB: (1343) Uhh...Idon'tthinkit'ssupposedtobechanging. SB: (1344) Notagoodsign. 70 SB: (1345) Dowehaveitasavectororascalarrightnow? SD: (1346) Rightnowwehaveitasavector. Additionally,thegroupfurthercategorizesthesetofdataasbeingacollectionofvectorsas opposedtoacollectionofscalars(line1346).Thisfocusonthevectorialnatureofthenet forceultimatelyhelpsthemtodrawaconclusionabouthowitshouldbehaveasthesatellite changesposition. Afteralittleo˙-topicdiscussion,thegroupbeginstoconsiderhowthevariouscomponents ofthenetforceshouldnotonlychangeintime(line1425),butshouldalsoremainaparticular size(line1434): SB: (1420) WeneedFgravtobeavector. SD: (1421) Wehaveitasavector...itisavectorrightnow. SB: (1422) How? SA: (1423) Howdoyouhaveitasavector? SD: (1424) Iinitiateditasavector. SB: (1425) Right,butitneedstomove. SD: (1426) Oh,doesithavetobenegative? SB: (1427) Eitherway,ithastobeinthexandtheydirection... SD: (1428) Ohwellthenyoujustdothis[addstheforceforthex-component]... SB: (1429) Because...Butit'sthecomponentsthatwouldmakeFgravbiggerthanwe needittobe? 71 SD: (1430) Why? SB: (1431) Becauseacomponentvector...Ifwehaveonelikethat[drawsavectortoward thefourthquadrant]thenit'sgonnabeouttothere... SC: (1432) No,itwouldbedouble. SB: (1433) Rightitwouldbethatlong. SB: (1434) Andwejustneedittobethatlong. SD: (1435) Sojustdivideitbytwothen? SB: (1436) Exceptitchangesintime... SB: (1437) Becausewhenit'srighthere,it'sonlygoingdown,andwhenit'srighthereit's onlygoingacross... SB: (1438) Butwhenit'srighthere,it'sgoingdownandacross... SC: (1439) Yeah. Inotherwords,althoughthenetforcehasbeeninitiatedasavector,ithasbeeninitiatedas aconstantvector(pointingonlyinthe y -direction).Thegroupreachestheconclusion(line 1437)thattheforcemustbemodi˝edsothatitcanchangedirectionsdependingonwhere thesatelliteislocatedrelativetotheEarth.Furthermore,theyconcludethatitisimportant thatmagnitudeofthenetforceremainaconstant(line1434).Theseconclusionsultimately leadthemtorethinktheirforcemodel. Tosummarize,thisgroupcanbeseen analyzing asetofdatarepresentingthenetforce actingonthesatelliteovertime.Theyhaveidenti˝edthe trend thatthedatachangesover 72 time,andthedatawereplacedinthe categories ofbeinglarge-scalenumbersandbeing vectorsquantities.The conclusion thatthegroupmakesisthatthenetforceshouldnot onlybeavector,butthatitscomponentsshouldbeabletooscillatebetweenthe x -and y -componentsdependingonwherethesatelliteis.Giventhisprocessofanalysisandthe conclusionsbeingdrawn,thisexcerptisthoughttoillustratethecomputationalpracticeof analyzingdata. 5.2.3Designingmodels Thecomputationalpracticeofdesigningcomputationalmodels,asde˝nedbyWeintropet. al,involvestheprocessofmakinghnological,methodological,andconceptual ThesetypesofdecisionsarefrequentlydealtwithintheSTEMdisciplinegiventhecom- plexityofmodernscienti˝cendeavors.Scienti˝crigorandsoundmethodologymustbe maintainedwhileusingtoolsattheforefrontoftechnology(i.e.,computation)toinvestigate modernphenomena.Atthesametime,developingadeepconceptualunderstandingofthe modelsandthephenomenathattheyrepresentisplayinganincreasinglyimportantrole inthesharingandcommunicationofscienti˝cinformation.Accordingly,thefundamental characteristicsassociatedwiththiscomputationalpractice,assummarizedinTab.5.3,are: i)de˝ningthecomponentsofamodel,ii)describinghowthecomponentsofthemodelin- teract,andiii)articulatingwhatpredictionscanbemadewiththemodel.Inkeepingwith therecentliteratureonmodelingineducationresearch,welimitourinvestigationtomodels pertainingtotheforceactingonthesatellite(e.g.,alocalgravitationalforcemodelora Newtoniangravitationalforcemodel). ConsiderExcerpt11fromGroupB.Throughoutthisexcerpt,thegroupcanbeseen workingtoincorporateacentripetalforcemodel(i.e., ~ F cent = mv 2 R h cos ; sin ; 0 i )into 73 Characteristic Qualities De˝ning Eachindividualcomponentofamodelmustbeseparatelyde˝nedincode. Forexample,themassofanobjectandthelocalaccelerationduetoa planetcanbeseparatelyde˝nedandusedtoconstructthecorresponding localgravitationalforce. Relating Thegroupmustdescribethewaythattheindividualcomponentsofthe modelrelatetothephenomenonthatisbeingstudied.Thisrelationship usuallymirrorsanequationoranexpectedtypeofbehavior.Forexam- ple,theNewtoniangravitationalforcefollowsaninversesquareposition- dependence. Predicting Thegroupmustarticulatewhatinformationtheirmodelwillprovidethem, andusethatinformationtomakepredictionsaboutthetimeevolutionof aphenomenongiveninitialconditions.Forexample,aforcemodelcan generatethevariousvaluesoftheforceactingonanobjectatdi˙erent positionsintime.Thissetofdatacanthenbeusedtomakepredictions aboutthemotionoftheobject. Table5.3:Thecharacteristicsandassociatedqualitiespertainingtothecomputationalprac- ticeofdesigningacomputationalmodel:de˝ningcomponents,relatingthemtooneanother, andusingthemtomakepredictions. theircode.Ultimately,thegroupisdissuadedfromusingthisparticularmodelthrough discussionwiththeTA.Nevertheless,thisexcerptisaclearillustrationofthepracticeof designingacomputationalmodel. Afewminutesintobeginningtheproblem,thegrouphasrecognizedthattheyneedto useaforcemodel(line118)tocalculatethetrajectoryofthesatellite,asopposedtojust plottingitusingtheexpectedradius(line116): SA: (111) Nowweresayingthatit's{theradius}avariable... SA: (112) Sowhatdowewanttodowiththisothernumber? SB: (113) Well,yousaidtheradiusfromheretohereisnotgonnabethesameasfrom heretohere? SA: (114) Yeah. 74 SB: (115) Well,shouldwe...CouldweGooglehow,likehowmuchfartherorshorteritis fromheretohere? SD: (116) Okay,Ithinkactuallywhatit'stryingtogetustosayisthatwecan'tjustplot itspatharoundbyusingtheradiusoftheorbit... SA: (117) Right. SD: (118) Wehavetoactuallyusetheforcethatisactingonitto˝ndit'spath. SA: (119) Wehavetousetheforce. SD: (120) Wehavetousetheforce. Thegrouphasbeguntoarticulatetheinformationthattheirmodelwillprovidethem,even iftheyhavenotyetdecidedontheparticularmodel.Inotherwords,theirforcemodelwill allowthemtomakepredictionsaboutthepositionandtrajectoryofthesatellite. Afteralittleo˙-topicdiscussion,thegroupdecidesonaparticularforcemodeltouse: SD: (152) Theforceislikevsquared...Theforceisuhhvsquaredtimesmovertheradius oforbit. SD: (153) CorrectmeifI'mwrong... SA: (154) Sorry? SD: (155) Theforceisequaltomasstimesvsquaredovertheradiusoftheorbit. SB: (156) Somaybewecouldjust˝ndit{theforce}atthatdistance? SD: (157) Well,wehaveaccesstoavariablethatrepresentsourradiusoforbit... 75 SB: (158) Andwehavemass. SD: (159) Andwehavemass. SC: (160) Wefoundthevelocitylasttime... SB: (161) Andweknowtheradiusandknowthevelocity. SB: (162) Sowecanjust˝ndthenetforce. Here,thegroupisclearlyidentifyingtheindividualcomponentsofthecentripetalforcemodel (lines157-161)andmakingsurethattheyareseparatelyde˝nedincode.Additionally,they haveidenti˝edaclearmathematicalrelationshipbetweenthem(line152)thattheyrecall frommemory. Beforejumpingintotheconstructionofthenewlyproposedmodel,theyspendalittle timediscussingitsbehaviorandhowitrelatestothephenomenon: SD: (182) Ifwecouldgetit{theforce}tooscillatebetweenmaximumswecouldgeta rotation... SD: (183) Buthowdowerepresentthatasaforce...Becauseit'sobviousthattheywant ustodothat. SA: (184) Sineandcosine? SD: (185) Sineandcosine? SA: (186) Ifwedosineandcosine,ifwehavebothofthem,oneinthex,oneinthey,like this[pointstonotes]issaying... SA: (187) Thenevenifonegoestozero,likeyouweresaying,thentheotheroneisgonna beclosetoone. 76 SA: (188) Andso... SD: (189) Wehavetouseourangles? SC: (190) Ohhh... SD: (191) Andwehaveaccesstoanglesthatarede˝nedbelow. SD: (192) Ohmygod,that'ssogreat,that'sperfect,you'retotallyright. Speci˝cally,theyarticulatethewaythatthecomponentsoftheirforcemodelwillneedto oscillatetocausearotation(line182).Thisoscillatorybehaviorhasadirectrelationtothe mathematicalsineandcosinefunctionsthattheyplantouse(line184)asonecomponent approachesavalueofzero,theothercomponentwillapproachavalueofone(line187). Theyalsoidentifyyetanotherindividualcomponentoftheirmodel(line191)withtheangle ofthesatellite. Tosummarize,thegroupbeginsbyrecognizingthatusingaforcemodelwillallowthem to predict thetrajectoryofthesatelliteinamoregeneralway(line118).Afterdecidingona centripetalforcemodel,theythenseparately de˝ne theindividualcomponentsofthemass, velocity,radius,andangleofthesatellite(lines157-161and191).Finally,they relate the sinusoidalnatureofthemodeltotheexpectedsinusoidalbehaviorofthesatellite'strajectory (lines182and186).Giventhesethreecharacteristics,thisexcerptisaclearillustrationof thecomputationalpracticeofdesigningamodel. 5.2.4Assessingmodels Thecomputationalpracticeofassessingacomputationalmodel,asde˝nedbyWeintropet. al,involveshowthemodelrelatestothephenomenonbeingrepresen 77 Characteristic Qualities Assuming Indesigningacomputationalmodel,certainassumptionsareinvariably takenintoaccount.Theseassumptionsregardlessofhowappropriate orvalidshouldbeidenti˝edandclearlyarticulatedbythegroup.For example,theassumptionthatthesatellitewillalwaysbetravelingina perfectlycircularorbit,althoughapoorone,isstillanassumption. Validating Asmoreassumptionsarebuiltintoamodel,itsvalidityshouldcontinually becheckedtoensureitspredictiveaccuracy.Forexample,assumingthat anorbitingsatelliteisactedonbyaconstantnetforceisnotvalidfor longperiodsoftime. Table5.4:Thecharacteristicsandassociatedqualitiespertainingtothecomputationalprac- ticeofassessingacomputationalmodel:identifyingassumptionsandvalidatingthem. Thisisacrucialstepintheprocessofmodelingwithoutanassessmentofthevalidity andmeaningoftheresults(i.e.,withoutadeepunderstanding),themodelisalmostcer- tainlyuseless.Thefundamentalcharacteristicsassociatedwiththiscrucialcomputational practice,assummarizedinTab.5.4,are:i)identifyingassumptionsbuiltintothemodel andii)validatingthemodel.Thesetwocharacteristics,ifcon˝dentlyobservedwithinan excerpt,wouldservetoclassifythatexcerptasthecomputationalpracticeofassessinga computationalmodel. ConsiderExcerpt9ofGroupC.Generallyspeaking,thegroupcanbeseenworkingto incorporateagravitationalforceintotheircode.Earlyon,theyrecognizethattheircodeis missingthenetforceonthesatellite,andsubsequentlyspendaboutthirtyminutesdeciding ifandhowtheyshouldshouldincorporateone.Eventually,theyreachaconclusiontoadd agravitationalforcebasedontheirassessmentofacoupleofdi˙erentmodels(i.e.,alocal gravitationalforceandaNewtoniangravitationalforce). Afewminutesintotheproblem,thegroupconsiderswhathappenstotheinitialmomen- tumofthesatelliteastheirprogramruns(line256): SA: (253) Umm... 78 SB: (254) Okay. SA: (255) Sothat'sourinitialmomentum. SA: (256) Andthenwhathappens{tothemomentum}? SD: (257) Andthen... SA: (258) Weneed,wehaveit... SA: (259) Thenetforceequationiswhat'swrong... SD: (260) Yeahandthenetforceequalstolikegravity,right? Obviouslythegroupisconcernedwiththestateofthenetforceequation(line259),anda proposalismadetosetthenetforceequalsomesortofgravitationalforce(line260).This isthebeginningoftheassessmentoftheirnetforcemodel. Theycontinuetodiscussandvalidatethetypeofgravitationalforcethattheyplanto incorporateintotheircode.Speci˝callytheywonderwhatnumericalvaluetheyshouldbe using(line262),andtheysuggestusingthelocalgravitationalconstant( g =9 : 81m = s 2 ): SD: (261) Sowejustneedtolikepluginthevalueofgravityright? SA: (262) Yeah...butwhat'sthevaluethatweneed? SA: (263) Becausewehaveum...wehaveum...wehave... SA: (264) Massinkilogramsandwehavetheradiusoforbitinkilometers,obviouslywe allknowlikeninepointeightnumber... SC: (265) That'sonlyclosetothesurfaceoftheEarth... 79 SA: (266) Ninepointismeterspersecondthough... However,theyrecognizesthattheirsatelliteisnotparticularlyclosetothesurfaceofthe Earth(line265),andthatthelocalgravitationalconstantisnotparticularlyvalidatthe actualdistance.Inotherwords,thegroupcanbeseenvalidatingtheircomputationalmodel basedontheparticularsituation. Eventually,thegroupdoesdecideonaparticulargravitationalforcetouse(line270): SC: (267) [looksinnotes] SD: (268) Yeahit'sthat[pointstoequation]. SD: (269) Gravityequalstolikegravity,ofgravityequalsFnet... SD: (270) Equalstogravityequalsto[writesNewtonianforceonboard]... SB: (271) What'sthat? SD: (272) That'stheconstantof... SB: (273) OhtheGyeah. SB: (274) Sixpointsix... Thisforceinvolvestheuniversalgravitationalconstant( G =6 : 61 10 11 Nm 2 = kg 2 )as opposedtothelocalgravitationalconstant,whichtheyclearlystate(line273).Again,the grouphasensuredthevalidityoftheirnetforcemodelbyassessingthelocationofthe satelliteandsubsequentlyusingtheappropriategravitationalconstant. Beforegettingtofar,thegrouptakessometimetoclearlyarticulateanassumption(line 275)builtintotheirmodel: 80 SA: (275) Sorryijustwantedtowriteherethatwe'remakinganassumption[writeson WB]. SD: (276) Yes. SD: (277) Fnetequalstogravity. SD: (278) Yes. SD: (279) Equalsto... SA: (280) Ijustdidthat[addinganE]toshowthatthat'softheEarth. SA: (281) Doeseveryoneagreethatthisisanassumption? SC: (282) Yeah. Thefactthattheonlyforceactingonthesatelliteisagravitationalforceisreallyjustan assumption(althoughagoodone)madeatthispoint.Thegroupspeci˝callytakesthetime toarticulateandagreeuponthisimportantassumption. Tosummarize,thisexcerptdemonstratestwofundamentalcharacteristics:thegroupis validating theirmodelwhentheycomparewhichgravitationalforce/constanttheyshouldbe using,andthegroupis assuming thingsabouttheirmodelwhentheysaythatthenetforce iscomprisedofonlyagravitationalforce.Giventhesetwocharacteristics,wefeelcon˝dent incategorizingthisexcerptasastrongillustrationofthecomputationalpracticeofassessing acomputationalmodel. 81 Characteristic Qualities Conceptualizing Thereneedstobesomeconceptthatagroupisfocusingon.Conceptsusu- allyrangefromindividualphysicalquantitiestomorecomplicatedphysical relationships. Representing Aparticularconceptshouldberepresentedmathematically.Thispro- cessofrepresentationusuallyinvolvestranslatingamathematicalequation fromthenotesintoamoregeneralcomputerfunction. Table5.5:Thecharacteristicsandassociatedqualitiespertainingtothecomputationalprac- ticeofcreatingcomputationalabstractions:representingphysicalconcepts. 5.2.5Creatingabstractions Thecomputationalpracticeofcreatingabstractions,asde˝nedbyWeintropet.al,requires abilitytoconceptualizeandthenrepresentanideaoraprocessinmoregeneral ThisabilityshowupfrequentlyintheSTEMdomainsespeciallywithinintroductorycom- putationalphysics.Thetwofundamentalcharacteristicsofthiscomputationalpractice,as summarizedinTab.5.5,are:i)conceptualizinganideaandii)representingitinmoregeneral terms.Thesetwocharacteristics,ifcon˝dentlyobservedwithinanexcerpt,wouldserveto classifythatexcerptasthecomputationalpracticeofcreatingcomputationalabstractions. ConsiderExcerpt13fromGroupDinthefollowinganalysis.Overall,thegroupcan beseengivingtheirnetforceadirectionthroughtheuseofaunitvector( ^ r ).They˝rst recognizethattheirforceneedstobeavector,andproposeanequationtousethatspeci˝cally involvesadirection( ~ F / ^ r=r 2 ).Oncetheyhavetheirequationtoworkwith,theybeginto discusshowtheycande˝neitasageneralfunction.Inotherwords,thegroupcanbeseen conceptualizing and representing anideaingeneralterms. Theystartbylookingforanequationthattheycanusetotrytocalculatethenetforce onthesatellite: SA: (108) [calculatingthemagnitudeoftheforceonhiscalculator] 82 SC: (109) Yeahjusttrythatoneequation˝rst. SC: (110) Ifthat'snotgonnawork,then{I}think{the}other... SD: (111) ButthedirectionofFis{avector}... SD: (112) Soweneedtoturntherintoavector. SC: (113) Ithinkweshould... SD: (114) [writesforceequationwith ^ r ] Herethegroupcanbeseendeciding(oratleastsuggestinginline109)thatthecomputational forcemodelthattheyareusingwillneedtotakeadirectionintoaccount(i.e.,itneedsto beavector).Thisequation, ~ F / ^ r=r 2 (retrievedfromtheirnotes),iswrittendownonthe WB.Noticethatitinvolvesusing ^ r togivetheforceadirection.Thisunitvectoristhe computationalabstractionthatthegroupidenti˝esandultimatelybeginstoconstructin theirprogram.Thisabstractionhelpsthemtoworktowardtheirgoalofconstructingthe non-constantNewtoniangravitationalforceonthesatellite. Oncetheunitvector( ^ r )hasbeenidenti˝edasacomputationalabstraction,theybegin itscreationincode: SD: (115) Sojustputthervalue,vectorvalue... SD: (116) Justputthis[pointstorhat]uhhfunction... SB: (117) Asaparameter? SD: (118) Justgivethecomputerafunctionsowedon'thavetocalculateFlikeSAis doing. 83 SB: (119) That'sagoodidea. AlthoughtheyareclearlyfocusingontheconceptofthedirectionoftheNewtoniangrav- itationalforce,theyarealittlestuckonhowtoactuallygoaboutcreatingit.However, theyatleastknowthattheywantittobeafunction(line118)ratherthanjustaconstant numericalvalue.Presumably,thisisbecausetheyknowthatthenumericalvalueswillneed tochangeintime(line271): SD: (269) NoImeanthisisthedistance...andithasadirection... SB: (270) Soit'savector. SD: (271) Yeahthisthepositionofthesatelliteisavector. SD: (272) Changewithtime... SC: (273) YeahI'mtalkabouttheverybeginningwiththeD...here[pointstoWB]. SB: (274) SotheDistheradius... Tosummarizethisexcerpt,thecomputationalabstractionthatthegrouphascreated isafunctionfortheunitvectorofthepositionofthesatellite(line116).Theydecideto createafunction(asopposedtoahard-codedvalue)sothatitwillbeabletochangeover time(line272).Thatis,thegrouphas conceptualized thedirectionoftheforcewithaunit vector( ~ F / ^ r )andhave represented thatideaaspositiondependentandthereforemore generalizablefunction( rhat=satellite.pos/R ).Giventhesecharacteristics,thisexcerpt illustratesthecomputationalpracticeofcreatingcomputationalabstractions. 84 Characteristic Qualities Isolating Thecauseofanunexpectederrorthatarisesinaprogrammustbetracked down.Thissometimesinvolvesretracingsteps(orkeystrokes)through theundocommand,butusuallyinvolvestestingtheprogramthrougha processofguessingandchecking. Correcting Theunexpectederrormustultimatelybecorrectedinalong-termand generalizablemanner. Systematizing Whenisolatingorcorrectingtheunexpectederror,itshouldbedoneis asystematicande˚cientway.Thischaracteristicisnotnecessarilyre- quired. Table5.6:Thecharacteristicsandassociatedqualitiespertainingtothecomputationalprac- ticeoftroubleshootinganddebugging:isolatinganunexpectederrorandcorrectingitina systematicmanner. 5.2.6Troubleshootinganddebugging Thecomputationalpracticeoftroubleshootinganddebugging,asbroadlyde˝nedbyWein- tropet.al,referstoprocessof˝guringoutwhysomethingisnotworkingorbehavingas expThisprocessisfrequentlyundertakenbystudentsinall˝eldsofstudyespecially withinintroductorycomputationalphysics,giventheirrelianceonincomplete/approximate computationalandphysicalmodels.Thethreefundamentalcharacteristicsofthiscompu- tationalpracticethatwehaveidenti˝ed,assummarizedinTab.5.6,are:i)isolatingan unexpectederror,ii)correctingthatunexpectederror,andiii)doingsoinasystematic/e˚- cientway.Thesethreecharacteristics,ifcon˝dentlyobservedwithinanexcerpt,wouldserve toclassifythatexcerptastroubleshootinganddebugging. Forexample,considerExcerpt2fromGroupIinthefollowinganalysis.Broadly,the groupcanbeseenworkingtoincorporaterealisticvaluesandgeneralizablefunctionsinto theirMWP.Acoupleofminutesintostartingtheproblem(Sec.3.3),theymodifythepre- writtennumericalvalueforthemassofthesatellitefrom 1 to 1E4 .Thisleads,overthecourse ofaboutthirtyminutes,tothegroupde˝ningthemomentumofthesatelliteasafunction. 85 Thatis,thegroupcanbeseen isolating thecauseofanunrealisticsatellitetrajectoryand ultimately correcting itina systematicway byrede˝ningthemomentumofthesatellite fromahard-codedvaluetocomputerfunction. ThegroupbeginsbyreadingthroughtheEuler-Cromerupdateofthepositionofthe satelliteinthecalculationloop(line6).Thisupdateinvolvesthepositionofthesatellite, themomentumofthesatellite,themassofthesatellite,andthediscretetimestep(i.e., satellite.pos=satellite.pos+satellite.p/msatellite*dt ): SC: (6) It{theMWP}doesthesatellitespositionplus,vector,zero,˝vethousand,zero, thatsthemomentumofthesatellite... SC: (7) Dividedbythemass,so,satellitesposition... Theyalsobegintoconsiderthenumericalvaluesthathavebeenassignedtothephysical quantitiesbeingused(i.e.,theinitialpositionandmomentumofthesatelliteandthemass ofthesatellite).Notably,thegrouppointsout(line8)thatthemassofthesatelliteshould bechangedtore˛ecttherealisticvaluegivenintheproblemstatement: SD: (8) This[pointstothescreen]isthemass?shouldwechangethatthen? SC: (9) Yeahweknowthatthisis...theygaveittousdidn'tthey? SD: (10) Fifteentimestentothethird[readingfromtheproblemstatement]. SA: (11) Ihaveallofthenumbersuphere[pointsto4Q]. SC: (12) [changesthemassofthesatellitefrom1to1.5E4] Bychangingthemassofthesatellitefrom 1 to 1.5E4 (line12),theyhavecorrectlymodi˝ed theprogramtore˛ecttherealisticsituationpresentedtothem.However,bychangingthe 86 massofthesatellitetheyhavealsointroducedanunexpectederrortheirsatellitelooksas ifitis˛oatingmotionlessinspace. Aftermakingtheirchangetotheprogram(line12),thegroupbeginstowonder(line 15)whatthenewvisualizationwilllooklike.Aftersomebackandforthaboutwhatthe visualizationusedtolooklike(line18),theydecidetoruntheprogramandobservethenew visualization.Thegroupdiscovers(line20)thatthesatellite,althoughitusedtotravelina straightlinetrajectory,nowremainsstationaryrelativetotherotatingEarth: SA: (15) WellIwonderwhatit{thevisualization}lookslikenow... SD: (16) Itjustlikeshootsstraight. SA: (17) Areyou{sure},didyoualreadytryit? SC: (18) Yeah{previously},butitmightbedi˙erent... SD: (19) Wejustchangedthemass. SC: (12) [runstheprogram] SA: (20) Uhhitsnotmoving,maybeweshould... Giventhisunexpectederror,thegroupbeginstoisolatethecauseoftheunexpectederror. Theyconsiderthattheymayhaveintroducedasyntaxerrorsincetheylastrantheprogram (e.g.,inusingEasopposedto**),resultinginitcrashingtheprogram(line22).Theyalso considerthatchangingthemassmighthaveleadtotheunexpectederror,andworktoat leasttemporarilyrectifyit(line25): SC: (21) Weprobablywroteitwrong... SC: (22) Maybeitmighthavecrashedthe... 87 SA: (23) Welljustexitoutthen. SD: (24) Yeah. SD: (25) Shouldwechangeitbackandseeifitrunsagain? SC: (26) Wellifwechangeitbacktooneit'llprobablyrunagainbecausewedidn'tchange anythingelse. SA: (27) WellcanIseewhatitlookslikewhenitrunswithone? SC: (28) Yeah. Changingthemassofthesatellitebacktoitsinitialdummyvalueisindeedatemporary ˝xtotheirunexpectederror.However,amorelong-termcorrectionisneededtoensurethe generalizabilityoftheirprogram.Ultimately,thegroupdoesworktocorrecttheerrorina moresystematicandlong-termmanner: SB: (745) So,okayso,we'reallinunderstandingofwhywearedoingitlikethis{de˝ning themomentumofthesatelliteasthemasstimesvelocity}insteadofdeclaringthis{a hard-codednumericalvalue}? SB: (746) Italsolikeitmakesitreallyexplicittoo,likewhenwegodownhereanddothis thingwhereyoutakepdividedbymyouareliterallyjustleftwithvelocity... SB: (747) Sothat'sgood. SD: (748) Yeah. Here,thegrouprecognizesthatthemomentumofthesatelliteshouldbede˝nedasafunction utilizingthevelocityandmassofthesatelliteseparately(line745).Thatway,whenthe 88 momentumisusedintheEuler-Cromerupdate,itwillcorrectlydivideoutthemassno matterwhatvaluetheyuse(line746). Thetypeofsystematiccorrectionofanunexpectederrorseeninthisexcerptcanbe contrastedwithourmotivatingcasestudy(Sec.4).Thatis,thechangesthatthegroupmade inthecasestudycouldbecharacterizedasamorehaphazardapproach,asopposedtothe presentexcerptwherethegroupshowsacertainlevelofreasoningbehindtheiractions(line 746).Accordingly,thisexcerptseemstoillustrateagroupworkinginasystematic/e˚cient wayastheytroubleshootanddebugtheirprogram. Tosummarize,theunexpectederrorthatthegrouprunsintoisthatinchangingthe massofthesatellitetore˛ecttherealisticsituation,thesatelliteremainsmotionlessrelative totherotatingEarth(line20).Thisintroducesconcerntothegroup,presumablybecause astraightlinetrajectoryisclosertoageostationaryorbitascomparedtonotrajectoryat all.Thegroupworksto isolate theerrorbychangingthemassofthesatellitebacktoits initialdummyvalueand˝ndingthatthisdoesindeedrectifytheunexpectederror(line25). Ultimately,thegroupworksto correct thiserror˝rsttemporarilybychangingthemassof thesatellite,andthenmore systematically andpermanentlybyrede˝ningthemomentumof thesatelliteasafunction(line745).Giventhesecharacteristics,thisexcerptillustratesthe computationalpracticeandprocessoftroubleshootinganddebugging. 5.2.7Thinkinginlevels Thecomputationalpracticeofthinkinginlevels,asde˝nedbyWeintropet.al,involvesthe analysisofasystemthatrangesamicro-levelviewthatconsidersthesmallestelements ofthesystemtoamacro-levelviewthatconsidersthesystemasaThistypeof high-andlow-resolutionanalysisofasystemisaskillthatshowsupfrequentlyinscienti˝c 89 Characteristic Qualities Leveling Agroupshouldeitherimplicitlyorexplicitlyde˝nethedi˙erentlevelsof asystem.Forexample,everyMWPcanbebrokendownintoaninitial conditionlevelandacalculationlooplevel. Featuring Theuniquefeaturesofeachlevelshouldbearticulatedbythegroup.For example,agroupmightarticulatethatphysicalquantitiesthatneedto changeintimemustbeplacedinthecalculationloop. Table5.7:Thecharacteristicsandassociatedqualitiespertainingtothecomputationalprac- ticeofthinkinginlevels:breakingaprogramintodi˙erentlevelsandattributingfeatures tothem. disciplinesandespeciallywithinthedomainofcomputerscience.Thevariouscontrol structurescommontocomputerprogramming(e.g.,awhileoraforloop)mustnotonlywork independently(i.e.,atthemicro-level)butmustalsoworktogether(i.e.,atthemacro-level) withothercontrolstructurestoproducethedesiredresultsoftheprogram.Accordingly,the twofundamentalcharacteristicsthatwehaveidenti˝edforthiscomputationalpractice,as summarizedinTab.5.7,are:i)identifyingthedi˙erentlevelsofasystemandii)correctly attributingfeaturesofthatsystemtotheappropriatelevel. Forexample,considerExcerpt6fromGroupAinthefollowinganalysis.Broadlyspeak- ing,thisexcerptfocusesonthegroupmakingdecisionsaboutwhatneedstobeaddedto theircodeand,moreimportantly,wherethosethingsneedstobeadded.Morespeci˝cally, theyworktoconstructafunctionforthemomentumofthesatellite(whichdependsonits velocity)aswellthenetforceactingonit. Earlyon,thegroupdecidesthattheyshouldconstructafunctionforthemomentumof thesatelliteintheirprogram(line76): SC: (74) Umm,sowehavelikeit'sde˝ningpofthesatellite,andthat'slikepismomentum youknow?Likepequalsmv. SC: (75) Butthere'snothinginherethatactuallyde˝nesthepofthesatelliteasbeingm 90 v. SC: (76) SoIfeellikeweneedtoputinav,andthenthevelocityofthesatelliteisa variable. SC: (77) Andthenmakethemomentumofthesatelliteasacombinationofthemassand velocity... SB: (78) Umm,myquestionforyou,fromtheperspectiveof... SB: (79) We'redoingcircularmotion,andasyougoaroundfrompointatob,yourvelocity ischangingcauseit'schangingdirection SB: (80) Maybe,Iguesswecande˝nespeed,butuhhthetrickwithvelocity...sinceit's goingtobechanging. SB: (81) Likeyouwantthevariabletocontinuechanging... SB: (82) Andforthevariabletocontinueupdatingyouhavetoputitinthecalculation loop... SC: (83) Umm,okay. However,thisraisestheissueofwheretoactuallyplacethefunctioninthecode(line78).The groupdecidesthattheymustde˝nethevelocityinsidethecalculationloop(line82)given thatitmustcontinueupasitsdirectioncontinuestochange.Thecrucialfeature thatthegroupisarticulatinghereisthatthecalculationloopiswheretime-dependentor changingquantitiesmustbeplaced. AfterashortTAinteractionfocusingonthegeneralizabilityoftheirprogram,thegroup returnstotopicofwherecertainthingsare/shouldbeplacedintheircode: 91 SB: (107) MayIumm,mayIuhh... SB: (108) Okayso,thereislike,there'stwosectionsinthecode... SB: (109) Sointhecode,youhaveyourcalculationloopandyourparametersandinitial conditions. SB: (110) Sofromwhatwehave,we'rede˝ningourinitialconditionsasthismodelright here,whichisjustEarthandthesatellite,likeit'sde˝nedthesetwobodiesandithas setthemomentumofthis... SB: (111) AndthenIwasthinking,inthecodehereinthecalculationlooptheforceisset tozerozerozero,sowereneverde˝ningFnetatanypoint. SB: (112) IthinkwhatweneedtodoisdescribeFnet.Theonlyotherthingwehaveto declareistheradius. SC: (113) Yeah,sure,wecoulddothat. SD: (114) Okay. Here,thegroupclearlyarticulates(line108)thetwodi˙erentsections/levelsoftheprogram (i.e.,theinitialconditionsandthecalculationloop)anddetailssomeofthecomponents belongingtoeachlevel.Thatis,theobjectsoftheEarthandthesatellitebelongtothe initialconditions(line110)andthenetforceactingonthesatellitebelongstothecalculation loop(line111). Tosummarize,thegrouphasbrokentheirprogramintothetwodi˙erent levels ofinitial conditionsandcalculationloop(line108).Similarly,theyhaveattributedtheparticular 92 Characteristic Qualities Communicating Theactofcommunicationcanrangefrompuredialoguebetweentwoor moreindividualstodetailedvisualizationsthatcapturetherelevantinfor- mationtobeshared.Forexample,creatingagraphofaphysicalquantity vs.timecanbeusedtosuccinctlyshareinformationaboutthetimede- pendenceofthatphysicalquantity.Alternatively,thistimedependence couldbearticulatedverballythroughdialogue. Understanding Theinformationbeingcommunicatedshoulddemonstrateanunderstand- ingthatthegrouphasoftheunderlyingmechanics.Forexample,agroup mightcommunicatethewaythattheposition,force,andmomentumof thesatelliteareinterrelatedassimulatedtimeprogresses. Table5.8:Thecharacteristicsandassociatedqualitiespertainingtothecomputationalprac- ticeofcommunicatinginformation:ageneralprocessofcommunicationthatdemonstrates anunderstanding. feature oftime-dependencetothecalculationloop(line82).Giventhesetwocharacteristics, thisexcerptiscanbeusedtoillustratethecomputationalpracticeofthinkinginlevels. 5.2.8Communicatinginformation Thecomputationalpracticeofcommunicatinginformation,accordingtoWeintropet.al, usuallyinvolvesavisualizationorrepresentation(e.g.,agraph)thatcanbeusedtoh- lightthemostimportantaspectsofwhathasbeenlearnedaboutthesysteminsucha waythatitcanbeunderstoodbysomeonewhodoesnotknowalltheunderlying Thiscommunicationskillisespeciallyimportantin˝eldsinvolvingcomplexandinterrelated systems,suchasthoseobservedinphysicsandengineering.Theabilitytoshareusefulinfor- mationwithcolleagueswithoutgoingthroughalloftheunderlyingdetailsandmechanisms iscrucial.Accordingly,thetwofundamentalcharacteristicsassociatedwiththisparticular practice,assummarizedinTab.5.8,are:i)ageneralprocessofcommunication(detailedin Tab.5.8)andii)thedemonstrationofanunderstandingthathasbeenreachedaboutthe system. 93 Forexample,considerExcerpt30fromGroupE.Atthisstage,thegrouphasbegunto constructaNewtoniangravitationalforcemodel,butisstrugglingwithitsimplementation. AbriefinteractionwiththeTAshowsthemcommunicatinginformationabouttheirunder- standingoftheunderlyingconceptofcircularmotion,aswellasanunderstandingofthe powerandgeneralizabilityoftheprogram.Afterthisinteraction,thegroupcontinueswith theconstructionoftheNewtoniangravitationalforce,andmorespeci˝cally,itsdirection. Abouthalfwayintotheprogram,thegroupisstruggling(line231)toconstructtheir Newtonianforcemodel.TheTArecognizesthattheyneedalittlehelp,andasksforthem toexplaintheirprocess(line232): SB: (231) Physicsman...thisisamess[pointstoscratchworkonWB]. TA: (232) Nono,goaheadandexplain... SB: (233) Okay,so... SB: (234) Withthatbeautifullittleformularighthere[pointstoNewtonianforceequa- tion]... SB: (235) Wedecided...thisforcehastobenegative. SB: (236) Becauseourinitialmomentumis˝vethousandinthepositivey-direction. TA: (237) Okay,Icandigthat. SB: (238) Andthenourunitvector{forposition}rightnow,isonezerozero[inaudible]. SB: (239) Soifwehavetheforcemultipliedbythat,negative,soithastobenegative. SB: (240) Thenthis{themomentum/velocity}willslowlystartapproachingnegative˝ve thousandhereinthex-direction. 94 SB: (241) Andthenoncethatreaches{negative}˝vethousand,thenourpositionishere atzeroonezero... SB: (242) Andthensinceit's{theforce}negative,it'llmoveitdownward. SB: (243) Andthatwillhappenateverystep[drawsfourpointsonaunitcircle]. TA: (244) Okaygood. Thus,amemberofthegroupcanbeseencommunicatinginformation(lines234-243)about thewaythattheforce,momentum,andpositionarerelatedatvariouspointsonthe x -and y -axesforacirculartrajectory.Althoughjustonememberofthegroupisdoingamajorityof thecommunication,theyareactingasaspokespersonorarepresentativeforthegroup(line 235).Thisinformationshowsaclearunderstandingoftherathercomplicatedinterrelation (i.e.,sinusoidalandout-of-phase)ofthesephysicalquantities. However,theTAcontinuestopressthemontheirunderstanding(lines245,248,and 250)byaskingthemtoconsiderpositionsotherthanthoseonthe x -and y -axes: TA: (245) Whatabouthere[drawsadotinthe˝rstquadrant]? SD: (246) Point˝ve... SB: (247) Thenit'llbe...thesquarerootoftwo,squarerootoftwo,zero. TA: (248) Okay,whatabouthere[drawsadotinthesecondquadrant]? SB: (249) Squarerootoftwo,negativesquarerootoftwo,zero. TA: (250) Whatifit'snotatforty˝vedegrees? TA: (251) Whatifit'sjustatsomearbitraryangle? 95 SB: (252) Well,thereasonthatweredoingthis... SD: (253) Therhatisgonnaupdateasitgoes. TA: (254) Okay... SD: (255) Soyoudon'tneedtoknowthat. SB: (256) Yeahyoudon'tneedtoknowthat... TA: (257) Okay,that's˝ne. Inresponse,thegroupdemonstratesastrongunderstandingofnotjusttheinterrelation betweenphysicalquantities(lines234-243),butalsoofthecomputationalpoweroftheir program.Thatis,anothermemberofthegrouparticulates(line253)thattheirde˝nition ofthedirectionoftheforceincode(i.e., ^ r )willautomaticallyupdatetoaccountforthese various/arbitrarypositions. Tosummarize,thisexcerptshowsaTAinteractionfocusingontheconstructionofthe Newtoniangravitationalforceactingonthesatellite.Thegroupcanbeseen communicating informationabouttheinterrelationoftheposition,force,andmomentumofthesatellite. Throughthisdialogue,theyaredemonstratingaclear understanding oftheseinterrelations, aswellasaclearunderstandingofthepowerandgeneralizabiltyoftheirprogram.Given thiscommunicationanddemonstrationofunderstanding,thisexcerptcanbeclassi˝edas anillustrationofthecomputationalpracticeofcommunicatinginformation. 96 Chapter6 Discussion Thischapterprovidesadiscussionofthereasonswhywemightobservecertainpractices, thelimitationsoftheunderlyingframework,andtheconstraintsofthecourseandactivity. 6.1Findings Inthesectionsthatfollow,wepresentour˝ndingsofthecommon,lesscommon,andthe unobservedpracticeswithinourdataset.Thecommonpracticesarethosethatwereidenti- ˝edatleastthreetimesinamajorityofthegroups(individualpracticesoccurredbetween zeroandseventimespergroup,withanaverageofthreeoccurrences).Thelesscommon practicesaretheremainingoftheobservedpractices.Theunobservedpracticeswerenot identi˝edatall.Alongwithsomeofthestatisticsofthefrequenciesofthesepractices(i.e., rawnumbersandpercentages),weprovidetheirde˝nitionsandreferencedetailedexamples intheappendices.Additionally,andperhapsmostimportantly,wediscussthereasonsasto whyaparticularpracticemighthaveagivenfrequency. 6.1.1Commonpractices Elevenofthepracticeslaidoutbytheframeworkhavebeenidenti˝edmultipletimesin eightofthegroupsthatwereanalyzed.Accordingly,thesepractices(listedinTab.6.1)are 97 deemedcommonandarediscussedbelowandinApp.A.Itisimportanttopayattentionto thesepracticesbecause,asinstructors,wenotonlywantstudentstobeabletoaccomplish tasks,butwealsowanttomakesurethattheyareengaginginthingslikecriticaland computationalthinkingwhiledoingso.Beingabletoidentifyandencouragethesepractices astheyoccur(ordon'toccur)inaclassroom,therefore,iscrucialtoe˙ectivepedagogyand coursedesign.Accordingly,wemustdevelopclearandreliablede˝nitionsforeachofthe commonpractices. CategoryPracticeNumber%ofcategory%ofall DataCreatingdata27319 Analyzingdata23278 Visualizingdata364213 ModelingDesigningmodels323311 Constructingmodels18196 Assessingmodels27289 ProblemsolvingCreatingabstractions22278 Programming21267 Troubleshootinganddebugging24298 SystemsThinkinginlevels18336 Communicatinginformation23438 Table6.1:Thecomputationalpracticesthathavebeendeemedcommonareshownwiththe numberoftimeseachpracticewasidenti˝ed,thepercentageofitscategorythatitoccupies (i.e.,thenumberoftimesapracticewasobserveddividedbythetotalnumberofpractices fromthatcategory),andthepercentageofallthepracticesthatitoccupies(i.e.,thenumber oftimesapracticewasobserveddividedbythetotalnumberofpracticesfromallcategories). Horizontaldividersseparatethedi˙erentcategories(i.e.,data,modeling,problemsolving, andsystemsthinking). Overall,thereareanumberofreasonsthatwehavecommonlyobservedthesepractices intermsofthelearninggoalsandthecoursedesign.Thelearninggoalofbeingableto usemathematicalandcomputationalthinking(P4)comesintoplayanytimestudentsare dealingwiththeabstractionsthattheyhavede˝nedinthecalculationloop,whichareneeded tobeabletoaccuratelypredictmotionusingtheEuler-Cromeralgorithms.Thelearning 98 goalofbeingabletoanalyzeandinterpretdata(P3)showsupanytimestudentsprintor visualizethedatarepresentingaphysicalquantity,acommontroubleshootingtechniqueand somethingrequiredbytheproblemstatement.Thelearninggoalofbeingabletodevelop andusemodels(P1)showsupwheneverstudentsareworkingtoconstructthevariousforce modelscoveredinthecourse,whicharenecessaryforanaccuratetrajectoryofthesatellite. Thelearninggoalofbeingabletoobtain,evaluate,andcommunicateinformation(P7) showsupcontinuallyasgroupsareengagingindiscussionwitheachotherandwiththe tutors,somethingthatwestronglyencouragethroughtutorquestions.Alloftheselearning goalsseemtostronglyin˛uencethepracticesthatweobserve. Forexample,thepracticeof visualizingdata involvesthe production ofavisualization thatclearly conveys someinformation.Thecomputationalproductionofadynamically updatinggraphofthedistancebetweenthesatelliteandtheEarthvs.simulatedtimecan beproducedandusedtoclearlyconveyinformationaboutthenatureoftheorbit(i.e.,how closetheorbitistoperfectlycircular).Thistypeofpracticewasobserved 36 timesacross thedataset,accountingfor 42% ofthedatapractices,and 13% ofallpractices. Visualizingdataisexpectedtoshowupcommonlyinourdatagiventhelearninggoal ofanalyzingandinterpretingdata(P3).Oneofthewaysthatdatacanbeanalyzedand interpretediswithavisualization.Speci˝cally,thevisualizationsweseestudentsmaking arethatofthetrajectory,theforce,themomentum,andthegraphofdistancevs.time. Thesevisualizationse˚cientlyconveyinformationbothtothestudentsandtotheTA(e.g., thevisualizationoftheforceconveysinformationaboutitscentralnature). Additionally,studentshavebeenworkingwithMWPstoproducedynamicvisualizations ofmotionsincethe˝rstweekofclass.The˝rstandsecondcomputationalproblems,focusing onboatsandhovercrafts,respectively,werevisualizedinanumberofways(e.g.,producing 99 visualizationsoftheirtrajectories).Inotherwords,studentsarefamiliarwiththevisualiza- tionofdatacomingintothethirdproblem.Nottomention,theproblemstatement,shown inFig.3.3,explicitlyasksstudentstoproduceasimulation/visualizationofanelliptical orbit. Furthermore,afteragrouphascorrectlyconstructedtheNewtoniangravitationalforce, manyofthetutorinteractionsfocusonthegenerationofagraphtoclearlyshowthatthe satelliteisn'ttravelinginaperfectlycircularorbit.Forexample,considerExcerpt38from GroupAwheretheTAispromptingthegrouptoaddinagraphtotheircode: TA: I'dlikeyoutographtheorbital...themagnitudeoftheradiusoftheorbitvs.a functionoftime... SC: Okay... TA: Andhavethatgraphedaswellanditupdates SC: Okay. Thetutorispresentingtheadditionalgoalofproducingagraphtothegroup.Afterthis interaction,thegroupaddsagraphtotheirprogramandusingtheresultstoconcludethat thesatelliteisnottravelinginaperfectlycircularorbit.Thisgraphcanbeusedtoe˚ciently conveyinformationabouthowclosethesatelliteistoperfectlygeostationary. Amongallthings,wewant P 3 studentstounderstandthatcomputerscanbeusedto quicklygeneratevisualizationsthatcaneasilybetweaked,andthatthosevisualizationscan beusefulwhenitcomestounderstandingandcommunicatingthephysicsoftherealistic phenomenonbeingmodeled.Accordingly,wearelikelytocommonlyobservethisandother (seeApp.A)practicesinourdata. 100 6.1.2Lesscommonpractices Fiveofthepracticeslaidoutbytheframeworkhavebeenidenti˝edrelativelyfewertimesand inonly˝veofthegroupsthatweanalyzed.Accordingly,thesepractices(listedinTab.6.2) aredeemedlesscommonandaregivenareasonableamountofattentionhereandinApp.B. Thedatapracticeswereeitherunobservedorcommonlyobserved,andsothistableshows onlythelesscommonlyobservedmodeling,problemsolving,andsystemspractices. CategoryPracticeNumber%ofcategory%ofall ModelingUnderstandingconcepts772 Findingandtestingsolutions13134 ProblemSolvingAssessingsolutions9113 SystemsthinkingInvestigatingsystems13134 Understandingrelationships13134 Table6.2:Thecomputationalpracticesthathavebeendeemedlesscommonareshown withthenumberoftimeseachpracticewasidenti˝ed,thepercentageofitscategorythatit occupies,andthepercentageofallthepracticesthatitoccupies. Overall,thereareanumberofreasonsthatweobservedthesepracticesgiventhelearning goals,thedesignofthecourse,andtheactualproblemstudentsaresolving.Thelearning goalofbeingabletoengageinargumentfromevidence(P6)manifestswhenstudentsare defendingtheuseofaparticularmodel,whichisnecessarytochoosingthemostappropriate forcemodel(i.e.,theNewtoniangravitationalforce).Thelearninggoalofbeingableto constructexplanations(P5)happenswhenstudentsarecomparingdi˙erentforcemodels, whichsimilarlyisneededtochoosethemostappropriateforcemodel.Thelearninggoalof beingabletoobtain,evaluate,andcommunicateinformation(P7)showsupcontinuallyas groupsareengagingindiscussionwitheachotherandwiththetutors,somethingthatwe stronglyencouragethroughtutorquestions.Thelearninggoalofbeingabletoplanand carryoutinvestigations(P2)happensfrequentlyasgroupsaredealingwiththecomplicated 101 systemthatwehaveprovidedtothem,whichisacrucialpartofprogrammingandcomputer engineering.Thelearninggoalofbeingabletodevelopandusemodels(P1)showsup wheneverstudentsareworkingtoconstructthevariousforcemodelscoveredinthecourse, whichmustbecorrectlytranslatedintocodefortheprogramtoruncorrectly.However,one ofthereasonsastowhyweonlyobservethesepracticeslesscommonlyrelativetotheother practicesasidefromthelimitationsoftheframeworkandthecoursedesignmaybedue tothevariationinstudentpreparationcomingintothecourse. Forexample,thepracticeof assessingsolutions involvesthe comparison oftwoormore di˙erentsolutions.Thisisdi˙erentthanjusttestingsolutions,giventhatitfocusesona comparisonbetweentwoormoresolutions.Groupsoftencomparetheexpectedbehavior usingalocalgravitationalforcetousingaNewtoniangravitationalforce.Inthisway, multiplesolutionsareassessedintermsoftheirvalidity.Thistypeofpracticewasobserved 9 timesacrossthedataaccountingfor 11% ofthesystemsthinkingpractices,and 3% ofall practices. Weexpecttoseethispracticeinourdatagiventhelearninggoalofbeingabletoconstruct explanations(P5).Thatis,wenotonlywantedstudentstomakecomparisonsbetween models,butwealsowantedthemtoclearlyexplainthosedi˙erences.Theselongandoften complicatedexplanations,then,areoftenaclearindicationofagroupassessingasolution. Additionally,manyofthetutorinteractionscanencouragethispractice.Forexample, considerExcerpt22fromGroupBwheretheTAisaskingthemaboutthetwomodelsthey havewrittenontheirboard: TA: Sothesetwoforcesthatyouhaveonyourboard... TA: FgandFcent... 102 TA: Whichofthosedoyouwanttouse? SB: WearethinkingFcent... TA: Why? SB: Becausewehaveeverythingweneedforit... Herethemodelsthatthegroupiscomparingarethegravitationalforceandthecentripetal force.Speci˝cally,thegroupexplainsthedi˙erencebetweenthetwoasbeingintermsof therequirementsofthemodel.Afterthisinteraction,thegrouprunsthroughtheindividual elementsofthemodelthattheyhavedecidedon,andbegintoconstructitincode.Given interactionslikethese,wefrequentlyobservethistypeofpracticeinourdata. However,onereasonthatweobserveassessingsolutionslesscommonlywithinourdata mightbethatsomegroupsarebetterorworseatmakingcomparisonsakeycharacteristic ofthispractice.Mostgroupstakeaguessandcheckapproachratherthanacontrastand compareapproach[50],andsoweonlyseethispracticeslesscommonlyrelativetotherest. Forexample,Excerpt7fromGroupGshowsacommonexchange: SC: ButIfeellikeweshouldjusttrythisoneFcentequation... SC: Andseeifitworks SC: Becauseifitdoesn't,thenwecanworryaboutitlater SB: Yeahokay,letsjustdothat. Here,thegrouptentativelydecidesonamodelwithoutcomparingittoanyotherpossible models(e.g.,aNewtoniangravitationalforce).Agroupwithatleastonememberthatis 103 highlymotivedtocomparetheprosandconsofdi˙erentsolutionswilllikelyengageinthis andother(seeApp.B)practicesmoreoften. 6.1.3Unobservedpractices Sixofthepracticeslaidoutbytheframeworkhavenotbeenobservedatall.Accordingly, thesepractices(i.e.,collectingdata,manipulatingdata,choosingcomputationaltools,devel- opingmodularsolutions,preparingproblems,de˝ningsystems)aregiventheirdueattention hereandintheappendices.Althoughthesepracticesareunobserved,itisstillimportantto discusswhytheyareunobserved. Thereareanumberofreasonsthatwedidnotobservethesepractices.Speci˝cally, thereisalackoflearninggoalsrelatedtodatacollection,datamanipulation,choosingtools, developingmodularsolutions,preparingproblemsforsolutions,andde˝ningsystemsand managingcomplexity.Further,giventhelackoffocusontheselearninggoals,manytutor interactionsworkedtointentionallydissuadestudentsfromengagingintheseunobserved practices. Forexample,thepracticeof collectingdata isnotexpectedtoshowupinourdata giventhattherearenosensorsormetersthatareprovidedtostudents,astheymightbeina labsetting.TheonlytoolstudentsarerequiredtouseisthecomputeralongwithVPython. Thistool,althoughitcanbeusedtohandlethecollectionofdata,isprimarilymeanttobe usedtocreatethedataalgorithmically.Thisalignswiththelearninggoalsofdevelopingand usingmodels(P1)tocreatedata,ratherthancollectingit.Giventhelackofthesetypesof learninggoals,wedonotexpectstudentstoengageinthisandother(seeApp.C)practices. 104 6.2Limitations Althoughtheframeworkgivesasolidfoundationandagoodstartingplace,itdoesnotcome withoutitslimitationsthatin˛uencethepracticeswewereabletoidentify.Additionally,the courseanditsdesigninvariablyconstrainsthepracticesthatwehaveandhavenotobserved. Furthermore,theactivityitselfplacesadditionalconstraintsonthepracticesavailableto thestudents.Finally,thequalitativelensthatwehaveusedin˛uencestheanalysiswehave conducted.Inthissection,wedescribethelimitationsoftheframework,theconstraintsof thecourseandactivity,andtheprimarylenswehaveusedtoguideouranalysis.Finally,we discusshowthoselimitationsrelatetothepracticeswehavebeenabletocon˝dentlyidentify. Itisimportanttonotethatforeverylimitation,wealsosawpossibleresearchopportunities, whicharedescribedinCh.7. Itisalsoimportanttorecognizethatouranalysishasfocusedononespeci˝cproblem (i.e.,theNewtoniangravitationalforceproblemdetailedinSec.3.3)intheearlystagesof thedevelopmentof P 3 .Eventhoughourscopeofanalysisisfairlyrestricted,ourresearch isstillusefulinaddingtetothebroadde˝nitionspresentedbyWeintrop et.al,aswellastofutureresearchonthecomputationalpracticeswithinphysics.Ex- pandingouranalysistootherproblems(e.g,springforces)infutureiterationsof P 3 orto otherimplementationsoftheclassroom(e.g.,theelectricityandmagnetismversion,aptly named EMP 3 )willinvariablyintroduceneworotherwisepreviouslyunobservedpractices. Inaddition,MSU'snewStudioPhysicscoursesarespeci˝callydesignedtoengagestudents withcollectingandanalyzingdata. 105 6.2.1Framework Althoughtheframeworkthatweusedhasmanybene˝ts,therearealsosomelimitations thatcomealongwithit.Forthemostpart,theselimitationsarecenteredonthebroad de˝nitionsthatareprovidedbyWeintropet.al.Althoughtheirbroadde˝nitionsarewidely applicabletomanydi˙erenttypesofscienceandmathematicsclassroomsgenerally,theycan berathervague/ambiguouswhenapplyingthemtoaparticularclassroom.Accordingly,the followingsectionsdescribethevague/ambiguousde˝nitionswithin P 3 intermsofthefour di˙erentcategoriesofpractices:thedata,modeling,problemsolving,andsystemsthinking practices. Additionally,theframeworkitselfhasarathernarrowfocus(i.e.,onpracticesindicativeof computationalthinking),whichcouldbeexpandedtocapturepracticesthatareindicative ofothertypesofthinkingorcognitivedevelopment.Forexample, P 3 doesanexcellent joboffacilitatingthedevelopmentofphysicsidentitywithinitsstudentssomethingthe frameworkisnotintendedtocapture.Despitetheselimitations,theframeworkdoescapture manypracticeswell(seeSec.5.2). 6.2.1.1Data Withinthedatapractices,itisoftendi˚culttoidentifypreciselywhenstudentsare advanc- ing towardthegoalsoftheproblemthroughthecreationofdata.Althoughtheconstruction ofthecomputationalalgorithmthatcreatesthedatacanbeeasilyidenti˝edincode,the advancementthattheyundergoismoresubjective.Thegoalsoftheproblem(e.g.,sim- ulatinganorbitorproducingagraph)oftentaketheentiretwohoursoftheclasstobe accomplished,andthedi˙erentpathstogettherearewindingandoftennon-linear.For example,afterconstructingaworkingforcemodelincode,agroupisreadytoautomatically 106 andalgorithmicallycreateasetofdatarepresentingtheforceonthesatellite.However, thispossiblyincorrect/inaccuratesetofdatadoesnotimmediatelyorcompletelyadvance themtowardtheirgoalofcreatinganellipticalorbit.Still,thisincorrectsetofdatacan eventuallypromptthemtocorrectlymodifytheirforcemodelultimatelyadvancingthem towardtheirgoalofcreatinganellipticalorbit.Inotherwords,sometimesyouneedtotake astepbackbeforeyoucantaketwostepsforward.Giventhisdi˚cultyinde˝ningwhat constitutesadvancingtowardagoal,identifyingthispracticewasoftendi˚cult. 6.2.1.2Modeling Withinthemodelingandsimulationpractices,itisoftendi˚culttoidentifywhenagroupis progressing intheirunderstandingofaconceptastheyinteractwiththemodelsofthecourse. Althoughitiseasytoidentifythedi˙erentforcemodels,andtoidentifywhenagroupis interactingwiththem(e.g.,designingonthewhiteboardorconstructingincode),itismore di˚culttosaywhenagroupisusingthatmodeltomakeprogressintheirunderstanding oftheunderlyingconcepts.Nottomention,itisdi˚culttoclearlyde˝newhatitmeansto truly understand something.Forexample,groupsarefrequentlyseeninteractingwiththe computationalmodeloftheNewtoniangravitationalforce.Astheynecessarilyconstruct thedirectionoftheforceintermsoftheseparationvectoranditsmagnitude,theyshould beusingittodevelopanunderstandingofthewaythattheforcechangesdirectionover time.Thisunderstanding,althoughdi˚culttodirectlyobserve,canbeteasedoutthrough tutorinteractions(seeSec.5.8).Itoftentakesalonglineoftutorquestioningtocon˝dently checkagroup'sunderstanding.Giventhisdi˚cultyinde˝ningwhatconstitutesprogressing intheunderstandingofaconcept,identifyingthispracticewasdi˚cult. 107 6.2.1.3Problemsolving Withintheproblemsolvingpractices,itcanbedi˚culttoclearlyde˝newhatitmeansfor agrouptobe systematic whiletroubleshootinganddebuggingtheircode.Althoughitis relativelyeasytoidentifywhenagrouphasisolatedandcorrectedanerror,itisnotsoeasy toidentifywhenagroupisbeingsystematicinthatprocess.Althoughmanygroupsdevise planstomethodicallyisolateandcorrecterrorsuntiltheyhavesuccessfullytroubleshootedor debuggedtheircode(seeCh.4),thoseplansarenotalwayssuccessful.Additionally,many groupsstumblehaphazardlyonunexpectederrors,andcorrecttheminasimilarfashion. Giventhisdi˚cultyinde˝ningwhatitmeanstosystematicallytroubleshootanddebug, identifyingthispracticewasdi˚cult. 6.2.1.4Systems Withinthesystemsthinkingpractices,understandingrelationshipsinasystemisambigu- ouslyde˝neditcanhaveasigni˝cantamountofoverlapwiththepracticeofdesigninga computationalmodel(seeSec.5.2.3).Thisoverlapultimatelydependsontheambiguous de˝nitionofasystemgivenbyWeintropet.al.Forexample,ifacomputationalforcemodel canbeconsideredasystem,thenanytimestudentsaredesigningacomputationalmodelthey arealsounderstandingtherelationshipsinasystem.Rather,ifasystemreferstosomething morelikeacollectionof˝lesthatarerelatedtooneothertocreateaprogram(e.g.,aPython scriptthatloadsdi˙erentmodules/libraries),thenunderstandingtherelationshipsinasys- temwouldmostlikelynotoccuralongsidedesigningacomputationalmodel.Thistypeof ambiguityhasatendencytolowercon˝denceduringinter-raterreliability.Accordingly,this practiceiscon˝dentlyobservedlessoften. 108 6.2.2Course Althoughthecoursethatwecollectedourdatafromwaswelldesignedandimplemented(see Ch.3),itwasnotconducivetosomeofthepracticeslaidoutbytheframework,stemming fromthemanydesignchoicesthatweremadeearlyon.Thesecrucialdesignchoicesand theirrami˝cationsonouranalysis/˝ndingsaredescribedinthesectionsthatfollow. 6.2.2.1Groupvs.individual Giventhatthecoursefollowedagroup-basedapproach,itwasoftendi˚culttosaywhich individualstudentswereactivelyengaginginthepractice.Forexample,theprocessof agreeingontheassumptionsoftheforcemodelofteninvolvesmultipleviewpointsthatmust betakenintoaccountsimultaneously.Accordingly,itisdi˚culttoascribethepracticetoany individualfromthegroup.Additionally,justbecauseindividualsarenotphysicalengaged (e.g.,talkingorwriting)doesnotmeanthattheyarenotmentallyengaged.However,follow- upinterviewdatacouldbeusedasadditionalevidenceforascribingaparticularpracticeto aspeci˝cindividual. 6.2.2.2Sca˙oldingvs.discovery Atthebeginningofthecourse,adaywasheldasameanstoprovidestudentswith anoverviewofthecoursestructureandareasonastowhyitwasbeingrunthatway.An importantcomponentofthatdaywasonthemodelingcomponentofthecourse,illustrated inFig.6.1.Giventhisfocusonmodelingatthebeginningofthecourse,itisnosurprisethat wefrequentlyobservedstudentsengaginginthevariouscomputationalmodelingpractices de˝nedbyWeintropet.al. Additionally,thecoursewasreasonablysca˙oldedwiththepre-classreadingandpre- 109 Figure6.1:Theiterativeprocessofmodelingphysicalsystemsthatwasdescribedtothe classonthe˝rstday. classhomework.Thepre-classreadingismeanttointroducethefundamentalconcepts whilethepre-classhomeworkismeanttocheckforcorrectapplicationofthoseconcepts. Thispreparationhelpsthestudentstoframetheprobleminawaythatusesmanyofthe practices.Forexample,someofthehomeworkproblems(seeSec.3.7)focusonVPython errorsthatmustbeidenti˝ed.Giventhistypeofpreparation,manystudentsengagedinthe practiceofthingsliketroubleshootinganddebuggingandprogramming. Moreover,thefrequenttutorinteractionsthroughoutthecoursearemeanttocheckfor understandingoftheconceptsandtheirapplicationwhileinclass.However,thesefrequent tutorinteractionsmakeitdi˚culttosaywhetherornotthepracticesobservedaregenerated bythestudentsorbythetutorsasocialobservere˙ect.Manytimes,TAsdissuade studentsfromengagingincertainpractices,andintentionallyencouragethemtoengagein others.Forexample,anygroupthatgetslostinthePhysUtilsystem˝lewillinvariablybe encouragedtostopthatandstartfocusingontheMWP. 110 6.2.2.3Introvs.advanced Thecoursewasdesignedaroundintroductoryphysicsconcepts,whichlimitedthetypes offorcesandmotionmodelsthatcouldbeanalyzed.Theanalysisofamoreadvanced classroom(e.g.,computationalNewtonianmechanicsusinghigherorderalgorithms)may provideadditionalpractices.Forexample,inmoreadvancedphysicsclassroomswemay expectgroupsto optimize theirmodelsbyaddingmorethanjustoneforcesomethingthat islackingintheWeintropframework.Inotherwords,wecanbegintosearchfornewand uniquepracticesinamoreadvanced P 3 -styleclassroom. Animportantlimitationofthestudyisthatwedidnotfocusonclassifyingthelevelsof sophisticationofthepractices.Rather,wejustfocusedonidentifyingthem.Ascomputation continuestogrowinindustryandacademia[3],andasnewandmoreadvancedcomputational techniquesarediscovered,itisimportantthatwebegintoclassifytheWeintroppracticesin termsoftheirlevelsofsophistication.Althoughwedonotexpecttoseeextremelyunique orsophisticatedpracticesinourdata,itissomethingthatshouldbefocusedoninfuture research. 6.2.3Activity Attheleveloftheactivityitself,itsboundarieslimitthepracticesthatweobserve.For example,theproblemstatementcontainsmanydirecttasks,aswellasafewthatareim- plied.Additionally,thefocusoftheactivityisonrelativelyintuitivephysicalconcepts(i.e., thegravitationalforceandNewton'ssecondlaw).Thesetypesofrequirementsandtheir implicationsonour˝ndingsaredescribedinthesectionsthatfollow. Giventhattheproblemstatementcontainsbothexplicitandimpliedtasks,ithasalarge 111 in˛uenceonthepracticesthatwedoanddonotobserve.Speci˝cally,thedirecttasksinthe problemstatementaretoi)produceasimulationofanellipticalorbit,ii)produceadiagram showingthemomentumofthesatellite,andiii)produceagraphoftheradiusoftheorbit overtime.Giventhedirecttaskof(i),weexpecttoseegroupsdesigning,constructing,and assessingmodelsincodeastheyworktoproducetheirphysicallyaccuratesimulation(see Sec.5.3).Additionally,given(ii)weexpecttoseegroupsanalyzingandvisualizedataas theycommunicateinformationaboutthewaythatthemomentumofthesatellitechanges overtime(seeSec.5.2).Similarly,given(iii)weexpecttoseestudentsengaginginthe practiceofvisualizingdataastheycommunicateinformationabouttheradiusoftheorbit overtime(seeSec.5.8).Thedirecttasksleadtothemorecommonlyobservedpractices. Thedirecttasksintheproblemstatementareexplicitlylaidoutforgroupsandsotake precedence,whereastheimpliedtasksintheproblemstatementareoftenadi˚cultthingfor studentstoinferontheirown.Forexample,oneimpliedtaskistodevelopapro˝ciencyin dealingwithrelativelysmallprogrammingsystems(i.e.,theself-containedMWP).Another impliedtaskistodevelopanunderstandingoftheconceptofadirectionalunitvectoras itshowsupintheNewtonianforce(seeSec.5.2.8).Theserathersubtleimpliedtaskmay bereasonsthatweobservegroupsengaginginsomeofthepracticeslesscommonly(see Sec.6.2). 6.2.4Analysis Itisimportanttonotethatthisstudywasconductedthroughthelensofaninstructor, lookingtoe˙ectivelyincreasetheamountofcomputationtaughtattheintroductoryphysics level.Accordingly,therewasaheavyfocusonthewaythatthetutorsinteractedwith thestudentstoeitherencourageordiscouragecertainpractices.Althoughthesetypesof 112 student-instructorinteractionsareunavoidableinmostclassroomsespeciallyin P 3 type classroomsourfocusonthemmayhavemadeitmoredi˚culttoidentifythepractices comingsolelyfromstudent-studentinteractions. Additionally,giventheheavyfocusoncomputationandcomputationalthinkinginour analysis(seeCh.2),otherimportanttypesofthinking(e.g.,creativethinking)mayhave beenoverlooked.Adetailedanalysisoftherelatedanalyticproblem(seeTab.3.1),which doesnothaveacomputationalelement,mayprovideadditionalinsightintothepractices indicativeofcreativeandothertypesofthinking. Furthermore,thetypeofdatathatwehadtoworkwith(i.e.,in-classvideoofgroupwork) oftenmadeitdi˚culttoascertainexactlywhatstudentswerethinkingastheyworked,and mayhavein˛uencedthepracticesthatweobserved.Conductingpost-classmighthave helpedtovalidateresearcherinferencessomethingthatshouldbelookedintoinfuture research. 113 Chapter7 Conclusion 7.1Summary Thisthesisattemptstomoreclearlyandpreciselyde˝nethecomputationalpracticesob- servedwithinintroductorycomputationalmechanicsthatareindicativeofcomputational thinking.Althoughasetofpreliminarypracticesde˝nedinaframeworkdevelopedby Weintropet.alprovideastartingpoint(seeSec.2.1),theyweresu˚cientlyvagueand/or ambiguousastowarrantfurtherde˝nition.Thisisespeciallytruewithinthedisciplineof introductoryphysics.Accordingly,thisthesisi)describestheoverallprocessofde˝ningthe computationalpracticescommontointroductoryphysicsthatareindicativeofcomputa- tionalthinkingandii)presentsthosede˝nitionswithconcreteexamples(seeChs.5and6 andApps. Webeganbycollectingdatafromaclassroomthatwasdesignedaccordingtomultiple learninggoalsandtheoriesoflearning.Speci˝cally,aproblem-basedlearningenvironment, calledProjectsandPracticesinPhysics,focusingontheprinciplesofconstructivealignment andusingthetheoreticalframeworkofcommunitiesofpractice[47].Thistypeofclassroom wasarichenvironmenttoconductqualitativeeducationresearchwithin,andsowecollected multiplestreamsofdata(seeSec.3)forpossibleanalysis[51,52].Althoughthisclassroom wasanidealplacetosearchforinstancesofgroupsengagingincomputationalpractices 114 helpingustomoreclearlyandpreciselyde˝nethemitisimportantnottogeneralizeour ˝ndingstoclassroomsthataresu˚cientlydi˙erent. Earlyresearchincomputationalphysicseducationsuggestedcontinuingtoinvestigate aphenomenoncalledysicsdebuggingAccordingly,apilotstudywasconducted intheFallof2016tobetterunderstandthisphenomenonwhichultimatelyraisedmore questionsthanitanswered(seeCh.4).Theseadditionalquestionsmotivatedtheneedfora morerigorousandin-depthanalysisofthedatasothatwecouldmakeandsupportstronger claims. Ourcorpusofdata,consistingofin-classvideoofninegroupsworkingonthreedi˙erent computationalphysicsproblems,wastranscribedverbatimwithgesturesandactionsindi- cated.Similarly,overheadvideoandcomputerscreencastswerecollectedtocross-reference withthetranscripts.Giventhesethreestreamsofdata,weperformedbothataskanda thematicanalysis(seeSecs.2.4and2.5)tohelpfacilitateamorerigorousandin-depthanal- ysissoastogenerateclearandprecisede˝nitionsofthecommoncomputationalpractices thatwereobserved. Thecommonpracticesthatweobservedwere:creating,analyzing,andvisualizingdatain thedatapractices;designing,constructing,andassessingmodelsinthemodelingpractices; creatingabstractions,programming,andtroubleshootinganddebuggingintheproblemsolv- ingpractices;andthinkinginlevelsandcommunicatinginformationinthesystemsthinking practices.Thelesscommonpracticesthatweobservedwere:understandingconceptsand ˝ndingandtestingsolutionsinthemodelingpractices;assessingsolutionsintheproblem solvingpractices;andinvestigatingsystemsandunderstandingrelationshipsinthesystems thinkingpractices.Theunobservedpracticeswere:collectingandmanipulatingdatainthe datapractices;choosingcomputationaltools,developingmodularsolutions,andpreparing 115 problemsforsolutionsintheproblemsolvingpractices;andde˝ningsystemsinthesystems thinkingpractices.Alongwiththesede˝nitions(seeSecs.5.2and6.1,andApps.we provideadetailedaccountofthedatareduction,codingprocess,andinter-raterreliability (seeSec.5.1). 7.2Futureresearch Althoughwehaveattemptedtoprecisesomeofthebroadde˝nitionsofthecomputational practicesthatareindicativeofcomputationalthinkingasprovidedbyWeintropet.al,more researchisneededtofullyunderstandthemwithinandbeyondthedisciplineofintroductory physics.The˝ndingsofthisthesis,thoughuseful,haveraisedadditionalquestionsand presentmanyopportunitiesforfutureresearch. Tostart,adeeperanalysisoftheNewtoniangravitationalforceproblemaspresentedin P 3 ,andfocusedoninthisthesis,couldbepursued.Additionaltypesofdata(e.g.,post-class interviews)couldbecollectedtoprovidemoreinformationonthewaystudentsperceiveand experiencethedi˙erentpracticesthattheyareengagingin. Additionally,abroaderanalysisofallofthemechanicsproblemspresentedin P 3 could beconducted.Althoughwehavefocusedouranalysisononeparticularproblemnearthe beginningofthecourse,thereareothercomputationalproblemsfocusingonothermechanical concepts(e.g.,collisionsorrotation)occurringlaterinthecoursethatmyprovideadditional insightintotheassociatedpractices.Itmayalsobeofinteresttoinvestigatethewaythat thesepracticesevolveovertimeasthecourseprogresses. Further,ouranalysiscanbeextendedbeyondanintroductorymechanicscourse(e.g.,ad- vancedmechanicsorintroductoryelectricityandmagnetism).Therearemanyotherphysical 116 conceptsthatcanbe,andsometimesmustbe,usedwhilesolvingengineeringproblems(e.g., Lagrangianmechanicsorcyclotronmotion).Similarly,theEuler-Cromeralgorithmhigh- lightedinthisthesisisnottheonlyone,andisnotalwaysthemostprecise.Moreadvanced classesfocusingonmorecomplicatedyetmoreprecisealgorithmsmightbeoffutureresearch interest. Although P 3 waswellsuitedtotheanalysisthatweconducted,notallclassroomssub- scribetoitsformat.Accordingly,extendingthistypeofresearchtootherphysicsclassrooms, thatatleastutilizecomputersinsomecapacity,wouldbeofvalue. 7.3Concludingremarks Abetterunderstandingofmodernscienti˝cpracticescanonlyhelptoinformthemanyde- cisionsthatmustbemadewhiledesigningacoursesoastofosterthelearningofknowledge, skills,andcomputationalthinking.Asstudent-centeredlearningenvironmentslike P 3 be- comeincreasinglypopular,andascomputationcontinuestopermeatetheSTEMdisciplines, our˝ndingscontributetothatunderstandingandpresentmanyopportunitiesforcontinuing research. 117 APPENDICES 118 AppendixA Commonpractices Thefollowingsectionsdescribethecommonpracticesthatweobserved. Creatingdata Thepracticeof creatingdata involvestheconstructionofan automatic oralgorithmic processthatwillquicklyproducealargesetofdataandusingthatsetofdatato advance towardtheirgoals.Forexample,constructinganEuler-Cromeralgorithmtocreateasetof datarepresentingthepositionofthesatelliteovertimeadvancesthegrouptowardtheirgoal ofsimulatingtheorbitofthesatellite(seeSec.5.2.1).Thistypeofpracticewasobserved 27 timesacrossthedataset,accountingfor 31% ofthedatapractices,and 9% ofallpractices. Creatingdataisexpectedtoshowupcommonlyinourdatagiventhelearninggoalof usingmathematicalandcomputationalthinking(P4).Wewantedstudentstotakeadvantage oftheEuler-Cromeralgorithmstogeneratethesetsofdatarepresentingthepositionand momentumofthesatelliteovertime.Wealsowantedthemtoconstructandusedi˙erent modelstogeneratethesetofdatarepresentingtheforceovertime.Thesealgorithmsand modelsofmotionrequirealotofmathematicalandcomputationalthinking,aligningwell withthatlearninggoal. Additionally,theproblemcannotbesolvedanalyticallywithjustintroductorylevelmath- ematics.However,itcanbesolvednumericallywithintroductorylevelmathematicsand 119 computation.Forexample,considerExcerpt7fromGroupCwheretheTAisprompting thegrouptocreatedata: TA: Butyouneedtheforcetokeepchangingdirectionasitmovesaround SC: Right TA: Soyoucan'tjusthardcodethenumericalvaluethatyoufoundlasttime SC: Oh...becausethispositionofthesatelliteisgoingtochange,whichmeanstheforceis goingtochange... TA: Exactly SB: Oh,gotcha Here,thetutorisfacilitatingthecreationofdatabyfocusingonthewaythattheforceneeds tocontinuallychangeasthesatellitemoves.Afterthisinteraction,thegroupgoesonto codetheirnetforceasaposition-dependentfunctionratherthanahardcodedvalue.These typesofinteractionsusuallyinitiatetheprocessofdesigning,constructing,andassessing computationalmodelsandalgorithmsthatultimatelycreatelargesetsofdata. Overall,wewantstudentsin P 3 tobeabletousesimplecontrolstructureswithforce modelsofvaryingcomplexitytogeneratelargesetsofdataforcomplicatedandrealistic motionproblemstocreatedata. Analyzingdata Thepracticeof analyzingdata involvesabroadprocessofanalysisthatincludessorting datainto categories ,lookingfor trends ,lookingfor correlations ,and/oridentifying outliers 120 thatcanbeusedtoreachsome conclusion .Forexample,whenaprintstatementisusedto verifythattheforceactingonthesatellitehasthetrendofremainingconstantinsimulated time,aconclusioncanbedrawnaboutthecorrectnessoftheunderlyingforcemodel(see Sec.5.2.2).Thistypeofpracticewasobserved 23 timesacrossthedataset,accountingfor 27% ofthedatapractices,and 8% ofallpractices. Analyzingdataisexpectedtoshowupcommonlyinourdatagiventhelearninggoalof analyzingandinterpretingdata(P3).Werecognizethatlargesetsofdataneedtobegener- atedusingcomputationalalgorithmsandmodels,andthatthesesetsneedtobeanalyzedin ordertoassesandvalidatetheunderlyingalgorithmsandmodels.Therearemanydi˙erent waystoanalyzedata,butitusuallyleadstosomeinterpretationorconclusionthatismade. Giventheutilityofanalyzingdatawhenitcomestodesigning,assessing,andconstructing theunderlyingcomputationalmodels,weexpecttoseethispracticecommonlyinourdata. Onetechniqueofanalysisthatisoftensuggestedistouseaprintstatementinthe calculationloopsothedataitselfcanbeanalyzed.Forexample,considerExcerpt23from GroupIwheretheTAmakesthistypeofsuggestion: TA: Checklike,soIknowyouknowhowtodothis...useaprintstatement. TA: Checkifit'sdoinganything,makesenseofwhereit'snot,orifit'srunningorifit's notrunning... SB: Yeah,okay. TA: Talkeverybodythroughwhatyou'redoingthough... SB: Yeah,Iwill. Here,theTAsuggeststhattheyuseaprintstatementsothattheycananalyzethedata 121 representingtheforceactingonthesatelliteandtomakedecisionsbasedonthatanalysis. Afterthisinteraction,thegroupconstructsaprintstatementintheircalculationloopto printthecontinuallyupdatingnetforceactingonthesatellite,therebycreatingasetof data.Theythenanalyzethissetastheyassesstheunderlyingforcemodel.Thesetypesof TAinteractionsfocusingonprintstatementsusuallyinitiatethepracticeofanalysisofaset ofdata. Ultimately,wewantstudentsin P 3 tobeabletointerpretandattachmeaningtothe patternsthatcanbefoundinlargesetsofdata. Designingcomputationalmodels Thepracticeof designingcomputationalmodels involves de˝ning theindividualcom- ponentsofamodel, relating themodeltothephysicalphenomenonunderinvestigation,and articulatingwhat predictions themodelwillbeabletomake.Forexample,themassofthe satellite,themagnitudeofitsvelocity,theradiusofitsorbit,andthepolaranglethatit makescanallbeseparatelyde˝nedincode.Additionally,theseindividualcomponentscan becombined,followinganequation,toproducetheexpectedoscillatorymotionofthesatel- lite.Finally,theresultingforcemodelcanbeusedtomakepredictionsaboutthemotionof thesatellite(seeSec.5.2.3).Thistypeofpracticewasobserved 32 timesacrossthedataset, accountingfor 11% ofthedatapractices,and 33% ofallpractices. Weexpecttoseethispracticecommonlyinourdatagiventhelearninggoalofdeveloping andusingmodels(P1).Thecoursewasspeci˝callydesignedtofocusondi˙erentforce modelswitharangeofcomplexities.Thatis,the˝rstthreeweeksofthecoursefocuses onaconstantzeroforce,aconstantnon-zeroforce,andanon-constantforcemodel.Given 122 thatthestudentsmustactuallydevelopthesemodelsincode,wefrequentlyobservethem designingcomputationalmodels. Further,thefour-quadrantsaremeanttosca˙oldthedesignprocessbyhighlighting theknowns,unknowns,andassumptionsofthemodel(seeCh.3).Thissca˙oldingoften facilitatesthedesignprocessbyhelpinggroupstode˝netheindividualelementsoftheir model.Forexample,considerExcerpt12fromGroupFwhereonestudentisclearto articulatetheindividualelementstheyarede˝ningbywritingthemonthefour-quadrants: SA: SoI'mjustgonnagoaheadandde˝nethoseovertherethen. SA: [writingon4Q]. SA: Shouldwedothat?. SB: Yeahgoaheadand...Wehavethemass. SB: Andthepositionofthesatellite. SC: Andthevelocityfromlasttime. SA: Okayholdon[writingthemdown]. Here,theindividualelementsofthemass,position,andvelocityofthesatelliteareindivid- uallyde˝ned.Oncethisisdone,theybegintorelatethemtooneanotherandtoconstruct theircentripetalforcemodel.Thatis,weseestudentsusingthefour-quadrantstohelpthem designtheirmodel. Mainly,wewantstudentsin P 3 tobeabletode˝netheindividualelementsofamodel, relatethemtoeachother,andmakepredictionsusingvariouscomputationalforcemodels. 123 Constructingcomputationalmodels Thepracticeof constructingcomputationalmodels involves implementing newbehavior incodebyeither creating anewmodelorby extending apreviouslywrittenmodel.For example,implementinganattractionbetweentwomassiveobjectsincodecanbeachieved throughtheconstructionofaforcemodel.Thisbehaviorcanbeimplementedinoneshot (e.g.,immediatelyconstructingaNewtoniangravitationalforcethatcanhandleelliptical orbits)orcanbeimplementedbysuccessivelyextendinganapproximatemodel(e.g.,moving fromaconstantgravitationalforcethatgeneratesaparabolictrajectory,toacentripetalforce thatgeneratesacircularorbit,toaNewtoniangravitationalforcethatgeneratesanelliptical orbit).Thistypeofpracticewasobserved 18 timesacrossthedataset,accountingfor 19% ofthedatapractices,and 6% ofallpractices. Constructingcomputationalmodelsisexpectedtoshowupfrequentlywithinourdata giventhelearninggoalofdevelopingandusingmodels(P1).Developingamodelincode invariablyrequiresstudentstomapmathematicalequationsontoVPythonsyntax.This involvesusingproperoperations(e.g.,adding,multiplying,calculatingmagnitudes),using properorderofoperations(e.g.,usingparenthesestoclearupanyambiguity),andensuring computationalabstractionsareofthepropertype(e.g.,thatpositionisavector,orthat distanceisascalar).Giventhatthesethingsmustallbeconstructedincode,wefrequently observestudentsconstructingmodels. Additionally,manytutorinteractionsareintendedtofacilitatethispractice.Forexample, considerExcerpt9fromGroupIwherethegrouphasdesignedtheirmodelandisbeginning toconstructitincode: TA: No,whatyouhavetheironthewhiteboardlooksgood... 124 SD: Okaysowejustneedtoliketakethisequationandlike... SD: Putitintheprogram... TA: Right... SD: Right,buthowdowedothat? SC: SojusttakebigG...Andthenlikemultipliedtimes... SC: msat,err,yeahthemassofthesatellite. SA: Okay...[beginstotype]. Here,themodeltheyhavedesignedistheNewtoniangravitationalforceandtheybegin toconstructitincodeintermsoftheuniversalgravitationalconstant,themassofthe satelliteandtheEarth,andthesatellite'spositionrelativetotheEarth.Giventhesetypes ofinteractions,wefrequentlyobservestudentsconstructingmodels. Ultimately,wewantstudentsin P 3 tobeabletoconstructmodelsincode,whetheror notthemodelsarecorrect. Assessingcomputationalmodels Thepracticeof assessingcomputationalmodels involvesidentifyingthe assumptions builtintoamodeland validating thembycomparingtorealitytoensurepredictiveaccu- racy.Forexample,groupsfrequentlyassumethattheorbitofthesatellitewillbeperfect circular.Althoughthisassumptionisagoodstartingpoint,itisinvariablycheckedforva- liditywhenconsideringarbitraryinitialconditionsthatleadtomoregeneralellipticalorbits 125 (seeSec.5.2.4).Thistypeofpracticewasobserved 27 timesacrossthedataset,accounting for 28% ofthedatapractices,and 9% ofallpractices. Assessingcomputationalmodelsisexpectedtoshowupfrequentlywithinourdatagiven giventhelearningofdevelopingandusingmodels(P1).Onceamodelhasbeendesigned andconstructedtoareasonabledegree,itcanbeusedtogenerateinformation(e.g.,a trajectoryofthesatellite).Thisinformationcanultimatelybeusedasevidencetomake anargumentfororagainstthevalidityofthatmodel.Thus,throughouttheprocessof designing,constructing,andmostimportantlyassessingacomputationalmodel,students shouldbeengaginginargumentbasedonevidence. Manytutorinteractionscanhelptofacilitatethispracticeaswell.Forexample,consider Excerpt15fromGroupCwheretheyarticulateanassumptionbuiltintoamodelandvalidate itsusegivenprompting: TA: Yeahbutwhenisthatequationgood? SB: Whenitsinfree... SC: Likewhenitsfalling... TA: Right,closetotheEarth. SB: Yeah. SC: Whichiswhywehavethatwrittenhereunderassumptions{onthe4Q}. TA: Okaygoodbut...isthatwhatyouhaveoverhere? SB: No. SC: Noweneedadi˙erentequation... 126 Here,thepoorassumptionisthatofauniformgravitationalacceleration,whichinvalidates theirmodel.Afterthisinteraction,theyscrapthelocalgravitationalforceandbegintotry acentripetalforcemodel.Giventhesetypesoftutorinteractions,weexpecttofrequently observestudentsassessingmodels. Overall,wewantstudentsin P 3 tobeabletovalidatedi˙erentcomputationalmodels byidentifyingtheirassumptions,whetherornottheydidthedesignand/orconstruction themselves. Creatingcomputationalabstractions Thepracticeof creatingcomputationalabstractions involvestakingaphysical concept and representing thatconceptincode.Forexample,thephysicalconceptoftheunitvector givingaproperdirectiontotheNewtoniangravitationalforceactingonthesatellitecanbe mosteasilyrepresentedincodebycombiningthepositionofthesatelliteanditsmagnitude (seeSec.5.2.5).Thistypeofpracticewasobserved 22 timesacrossthedataset,accounting for 27% ofthedatapractices,and 8% ofallpractices. Creatingcomputationalabstractionsisexpectedtoshowupfrequentlywithinourdata giventhelearninggoalofbeingabletodevelopandusemodels(P1).Allofthemodelsusedin thecourse(i.e.,thevariousforceandmotionmodels)havesomemathematicalformthatcan betranslatedintoVPythonsyntax.Thatis,inordertoconstructacomputationalmodel, youmust˝rstcreatethecomputationalabstractionsthatitdependson.Giventhefocuson modelinginthecourse,weexpecttocommonlyobservesstudentscreatingabstractions. Additionally,someofthetutorinteractionsthatwehaveobservedfacilitatethispractice well.Forexample,considerExcerpt9fromGroupFwherethetutorquestionsthemonthe 127 de˝nitionsthattheyhaveintheircode: TA: SoIseethatyouhavethosethingsde˝nedonyourwhiteboard... TA: Butwheredoyouhavethosede˝nedinthecode? SA: ButthatswhatI'msaying,thatswhatwereworkingon. TA: Okay,sowhatareyouthinkingthen? SA: Wehavethesethings[pointstoboard]de˝ned... SA: Andwe'regonnalikeinputthosevaluesforthosevariables... Here,thede˝nitionsthattheyhaveonthewhiteboardarethemassofthesatellite,its speed,andradiusofcircularorbit.Thisinteractionultimatelypromptsthemtoconstruct thecorrespondingcomputationalabstractionsincode,whethertheyhardcodevaluesorcon- structmorecomplicatedfunctions.Giventhesetypesofinteractions,weexpecttocommonly observethispracticeinourdata. Ultimately,wewantstudentsin P 3 tobeabletomakeabstractionsincodewhendealing withvariousphysicalconcepts. Computerprogramming Thepracticeof computerprogramming involves modifying codewhile arranging that codeinpropersyntax.Forexample,whilemodifyingtheforcemodelinthecalculation loop,alllinesmustbearrangedwiththeproperindentation.Inotherwords,asidefrom thevalidityoftheforcemodel,thesyntaxmustbeinorderforthecomputertobeableto 128 interpretthingscorrectlyandtorunwithouterror.Thistypeofpracticewasobserved 21 timesacrossthedataset,accountingfor 26% ofthedatapractices,and 7% ofallpractices. Computerprogrammingisexpectedtobecommonlyobservedinourdatagiventhe learninggoalofusingmathematicalandcomputationalthinking(P4).Groupsareworking withMWPsinVPython(seeSec.3.4.2.1),whichcomeswithitsownuniquesyntaxthatmust beadheredtostrictly.AlthoughthesyntaxinVPythonisveryintuitive(e.g.,calculating themagnitudeofavectorcanbedonebycallingthe mag() function),smallandsometimes di˚cultto˝ndsyntaxerrors(e.g.,amissingparenthesis)canleadtofrustratingruntime errors.Giventhesedi˚culties,weexpecttoseestudentsengagingfrequentlyinthispractice. Additionally,thispracticeisheavilysca˙oldedthroughtutorinteractions.Giventhat manystudentshavelittletonopriorprogrammingexperience,tutorssometimesguidestu- dentsintheirprogramming.Forexample,considerExcerpt30fromGroupDwherethe groupknowswhattodo,butisunsureofhowtoprogramit: SB: TA,weneedhelp. TA: OkayIcantry... SB: Wedon'tknowhowtoliketakethemagnitudeofthis. TA: Where?... SB: Righthere,inourforce,equationfortheforce. TA: Ahhokay,youneedtoputparentheses. SB: Wherehere? Here,thetutorisremindingthegroupthatthepropersyntaxthatmustbeadheredto requiresparentheses.Afterthisinteraction,theymodifytheircode,andcontinuetodesign, 129 construct,andassesstheassociatedforcemodel.Giventhesetypesofinteractions,we frequentlyobservegroupstobeengaginginthepracticeofcomputerprogramming. Mainly,wewantstudentsin P 3 tohaveexperiencewithprogrammingandthedi˚culties associatedwithit. Troubleshootinganddebugging Thepracticeof troubleshootinganddebugging involves isolating anunexpectederror inthecode, correcting thaterrorinalong-termandgeneralizablemanner,anddoingsoin a systematic fashionwhereapplicable.Forexample,withoutde˝ningtheinitialmomentum ofthesatelliteasafunctionintermsofitspreviouslyde˝nedmassandinitialvelocity, changingthemassofthesatellitewon'tcorrectlypropagatethroughtheprogram,leading tounexpectedandundesirableresults.Systematicallyisolatingthecausesoferrors(e.g., notde˝ningthemomentuminadynamicway)allowsforittonotonlybecorrected,butto becorrectedinalong-termandgeneralizablemanner(seeSec.5.2.6).Thistypeofpractice wasobserved 24 timesacrossthedataset,accountingfor 29% ofthedatapractices,and 8% ofallpractices. Troubleshootinganddebuggingisexpectedtoshowupfrequentlywithinourdatagiven thelearninggoalofbeingabletodevelopandusemodels(P1).Duringtheprocessof developingandusingamodel,unexpectederrorsfrequentlyoccurandmustbecorrected. Theseunexpectederrorscaninvolvethingslikesyntaxerrorsorunexpected/unphysical behavior.Ineithercase,studentsmustidentifythoseerrors,andultimatelycorrectthem inasystematicmanner.Giventhisfocusondevelopingandusingmodels,weexpecttosee thispracticecommonlyinourdata. 130 Further,manytutorsintentionallyguidegroupsastheytroubleshootanddebug.For example,considerExcerpt22fromGroupHwherethetutorpointsoutthattheirforce modelincodedoesnotmatchtheirforcemodelontheboard: TA: Oh,Iseewhatitis... SB: What? TA: Okaysointhedenominatorofyourforce,youhavethemagnitudeofthepositionof thesatellite. SB: Right... TA: Butwhatdoyouhaveonyourboard? SB: Ohhh... SC: Weneeditsquared. Here,theincorrectforcemodelproducesanextremelylargeforcethatrapidlyaccelerates thesatellitetoludicrousspeed.Thissmallerror,althoughsyntacticallycorrect,produces unphysicalresults.Afterthisinteraction,thegroupmodi˝estheircodesothatitaccurately re˛ectstheequation.Giventhesetypesoftutorinteractions,weexpecttoseethispractice commonlyinourdata. Overall,wewantstudentsin P 3 tobeabletohandleunexpectederrorsthatarisewhile programming,whethertheybesyntacticalorphysical. 131 Thinkinginlevels Thepracticeof thinkinginlevels involvesbreakingtheMWPintodi˙erent levels andat- tributingthosedi˙erentlevelswiththeircharacteristic features .Forexample,theprogram asawholecanbrokendownintothetwodi˙erentlevelsoftheinitialconditionsandthe calculationloop(seeSec.5.2.7).Eachlevelhasitsownde˝ningfeatures:theinitialcondi- tionsleveliswheretime-independentcomputationalabstractionscanbede˝ned,whereasthe calculationloopiswheretime-dependentcomputationalabstractionsmustbede˝ned.This typeofpracticewasobserved 18 timesacrossthedataaccountingfor 33% ofthesystems thinkingpractices,and 6% ofallpractices. Thinkinginlevelsisexpectedtoshowupfrequentlywithinourdatagiventhelearning goalofdevelopingandusingmodels(P1).TheNewtoniangravitationalforcemodeland Euler-Cromermotionalgorithmsconstituteamodelofmotionthatmustbedevelopedincode andultimatelyusedforsomepurpose.Whilestudentsaredevelopingthismodelofmotion, theymustmaintaintheoverallstructureoftheMWPwritteninVPython(seeFig.3.4.2.1) withoutproperstructureandsyntax,theprogramasawholerunsintofatalerrors.This structurethatmustbemaintained,isnaturallybrokendownintoseveraldi˙erentlevels: theobjects,initialconditions,timeset-up,andcalculationloop.Theselevelsareindicated intheMWPwithcomments(e.g., #CalculationLoop ),andeachlevelhasitsownunique features.Maintainingthesefeaturesforeachleveliscriticaltoarunnableprogram. Additionally,studentsareintroducedtotheconceptofiterativepredictionofmotion asanalgorithmicchangeindi˙erentphysicalquantitiesovertime.Speci˝cally, ~p new = ~p old + ~ F net dt , ~r new = ~r old + ~vdt ,and t = t + dt ,asdescribedinthecoursenotes(see Sec.3.1).Thesetime-dependentphysicalquantitiescanbecontrastedwithtime-independent 132 (orapproximatelytime-independent)physicalquantities(e.g.,thelocalaccelerationdueto gravity).Identifyingthecorrecttime-dependenceofaphysicalquantitiesisnecessaryto ensuringproperplacementofitsde˝nitiontime-independentquantitiescanbeplaced intheinitialconditionslevel,whereastime-dependentquantitiesmustbeplacedinthe calculationloop.Forexample,considerExcerpt17fromGroupEwheretheyarediscussing theplacementofalineofcode: SA: DoweneedFnettobecalculatedinsidetheloop? SA: ThatisdoweneedtorecalculateFneteverytime?isitchanging? SB: No. SA: Sowecouldjustthrowitoutsideoftheloop. Here,thestudents(incorrectly)articulatethatthenetforcedoesnotneedtoplacedinthe calculationloopbecauseitdoesnotneedtoupdate.Thatis,theyidentifythedi˙erentlevels ofinsideandoutsidetheloop,andcorrectlyattributedthefeaturethatupdatingquantities mustbeplacedinsidetheloop,whereasothercanbeplacedoutside. Aboveall,wewantstudentsin P 3 tounderstandthedi˙erencebetweentime-dependent andtime-independentphysicalquantities,andtobeabletoproperlyde˝neandplacethem incode.Accordingly,weobservethispracticecommonlyinourdata. Communicatinginformation Thepracticeof communicatinginformation involvesthebroadprocessof communication thatrangesfrompure dialogue toself-contained visualizations thatcommunicatesome un- derstanding thatthegrouphasachieved.Forexample,anunderstandingofthecomplicated 133 butpowerfulcomputationalinterrelationbetweentheforce,position,andmomentumofthe satelliteisfrequentlycommunicatedverballywithinandbeyondgroups(seeSec.5.2.8).This typeofpracticewasobserved 23 timesacrossthedataaccountingfor 43% ofthesystems thinkingpractices,and 8% ofallpractices. Communicatinginformationisexpectedtoshowupfrequentlywithinourdatagiventhe learninggoalofbeingabletoobtain,evaluate,andcommunicateinformation(P7).Once informationhasbeenobtainedandevaluated,itiscrucialtoensurethateachmemberof thegroupcancommunicateanunderstandingofit.Accordingly,studentsarerequiredto continuallyexplaintheirthoughtprocessthroughouttheday.Giventhisfocusonencour- agingexplanation,weexpecttofrequentlyobservestudentscommunicatinginformationin ourdata. Further,manytutorinteractionscanhelptofacilitatethispractice.Forexample,consider Excerpt35fromGroupEwheretheTAcontinuespressesthemontheirunderstandingof thedirectionoftheforcebyaskingthemtoconsiderpositionsotherthanthoseonthe x - and y -axes: TA: Whatabouthere[drawsadotinthe˝rstquadrant]? SD: Point˝ve... SB: Thenit'llbe...thesquarerootoftwo,squarerootoftwo,zero. TA: Okay,whatabouthere[drawsadotinthesecondquadrant]? SB: Squarerootoftwo,negativesquarerootoftwo,zero. TA: Whatifit'snotatforty˝vedegrees? TA: Whatifit'sjustatsomearbitraryangle? 134 SB: Well,thereasonthatweredoingthis... SD: Therhatisgonnaupdateasitgoes. TA: Okay... SD: Soyoudon'tneedtoknowthat. SB: Yeahyoudon'tneedtoknowthat... InresponsetotheTAprompting,thegroupdemonstratesastrongunderstandingthattheir de˝nitionofthedirectionoftheforceincode(i.e., ^ r )willautomaticallyupdatetoaccount forthesevarious/arbitrarypositions.Giventhesetypesoftutorinteractions,wefrequently observethispracticeinourdata. Ultimately,wewantstudentsin P 3 tobeabletoclearlycommunicatetheirunderstanding ofphysicalconcepts,boththroughdialogueandbygeneratingvisualrepresentations. 135 AppendixB Lesscommonpractices Thefollowingsectionsdescribethelesscommonpracticesthatweobserved. Understandingconcepts Thepracticeof understandingconcepts involves progressing towardadeeperunderstand- ingofaconceptby interacting withacomputationalmodel.Forexample,whiledesigning, constructing,orassessingaNewtonianforcemodelincode,studentsprogressintheirunder- standingoftheabstractconceptofaunitvectorasprovidingpurelyadirectiontoaphysical quantity.Thistypeofpracticewasobserved 7 timesacrossthedataaccountingfor 7% of thesystemsthinkingpractices,and 2% ofallpractices. Weexpecttoseethispracticeinourdatagiventhelearninggoalofengaginginargument fromevidence(P6).Speci˝cally,individualsmustbeabletodefendtheirunderstandingof variousphysicalconceptswhileusingtheirprogramasevidence.Eachprogramproducesa numberofpiecesofevidence(e.g.,graphs,numericalvalues,visualizations,etc.)thatcanbe usedtosupportclaimsofunderstanding.Accordingly,weexpecttoseestudentsengaging inthispracticefrequently. Further,manytutorinteractionshelptofacilitatetheunderstandingofmanyconcepts. Forexample,considerExcerpt25fromGroupCwheretheTAisaskingthemtoillustrate apointtheyaretryingtomakewiththeirprogram: 136 SA: Theforcehastopointinthesamedirectionasthemomentumofthesatellite... TA: Yeahbutwhyareyousayingthat? TA: Canyouuseyourprogramtosortofprovethattome? SA: Yeah,so,ifyoulookatthearrowsonthesatellite. SA: TheyalwayslikemovetowardtheEarth. SA: SoweknowthattheforcehastobepointingtowardtheEarth. SA: Soourdirectionhastobecorrect.... TA: Okay...okaythatmakessense. Here,astudentclearlyusesthevisualizationofthemomentumofthesatellitetodemonstrate herunderstandingoftherelationshipbetweentheforceandthechangeinmomentum.Given thesetypesofinteractions,weexpecttoseegroupsunderstandingconceptsinourdata. However,onereasonthatweobserveunderstandingconceptslesscommonlywithinour datamightbethatcertaingroupsaremoreorlessfocusedontrulyunderstandingthe underlyingmaterialandframetheproblemassuch[53].Inotherwords,groupsareoften seenastakingonananswer-makingmoderatherthanasense-makingmode[52].Astrong focusontheunderstandingoftheunderlyingmaterialisarelativelyrareoccurrence,andso weonlyseethispracticelesscommonlyrelativetotherest.Forexample,Excerpt13from GroupEshowsamoretypicalexchangethatdoesnothaveastrongfocusonunderstanding: SD: Socanyoujust...whyareweusingthat{GmMoverrsquaredequation}now? SD: Likewhydon'twehavetousethat{mg}one? 137 SB: Whycan'twejustusethis{mg}one? SC: Wellthisone[pointstoNewtonianforce]...let'sjusttryitandseewhathappens... SD: Okay. Here,thegroupdoesnotfocusonunderstandingwhytheyareusingtheNewtoniangravita- tionalforce,rathertheyjustguesstouseitandeventuallycheckitlater.Giventhesetypes oftypicalexchanges,weexpecttoseeunderstandingconceptslesscommonlyrelativetothe otherpractices. Findingandtestingsolutions Thepracticeof ˝ndingandtestingsolutions involves justifying theuseofaparticular solution.Often,theparticularsolutionsthatweseearethedi˙erentforcemodelscovered inthecoursenotes(e.g.,localgravitationalforce).Asstudentsprogressfromincorrector approximateforcemodels(i.e.,thelocalgravitationalorcentripetal)tothecorrectmodel (i.e.,Newtoniangravitationalforce),weseethemcontinuallytestingalongtheway.For example,agroupmightrecognizethatthelocalgravitationalforcemodeldoesnotallowfor thesatellitetotravelinaboundorbit,andmoveontosearchingforanewmodel.This typeofpracticewasobserved 13 timesacrossthedataaccountingfor 13% ofthesystems thinkingpractices,and 4% ofallpractices. Weexpecttoseethispracticeinourdatagiventhelearninggoalofobtaining,evaluating, andcommunicatinginformation(P7).Mostimportantly,groupsarerequiredtoevaluate informationbyusingtheirprogramtomakepredictionsandtoevaluateitintermsofits predictivevalidity.Ifthemodelisnotjusti˝ed,anewmodelmustbesoughtout,andthe 138 processoftestingbeginsagain.Giventhisfocusonevaluatinginformationwhenitcomes tothejusti˝cationofasolution,weexpecttoseethispracticeinourdata. Additionally,manytutorinteractionscanfacilitatethetestingprocess.Forexample, considerExcerpt20fromGroupAwheretheyareusingalocalgravitationalforcemodel withadeceptivesatellitetrajectory: TA: Sotheproblemis... TA: Ifyoulookatyourforce,youhavealocalgravitationalforce... TA: Butthatisonlygoodwhen? SB: Whenit'sclosetoEarth. TA: Andisthatwhatwehavehere? SA: Butitlookslikeitsorbiting[pointstoscreen] TA: Itsactuallyparabolic,Iknowitlookslikeitsgonnaorbitbutitsnot SB: Ohcausetheforcehereisonlyinthexdirection TA: Right...soyouneedtochangethat... Here,thetutorispromptingthemtojustifytheirforcemodelbyscrutinizingtheresulting trajectory.Afterthisinteraction,theybeingtolookforanothertypeofforcetoconstruct incodeonethatiscapableofproducingaclosedorbit.Giventhesetypesofinteractions, weexpecttoseethispracticeinourdata. However,onereasonthatweobserve˝ndingandtestingsolutionslesscommonlywithin ourdatamightbethatindividualsvaryintheirdesiretojustifytheiractionsanimportant 139 characteristicofthispractice.Thistypeofself-justi˝cation(i.e.,beingcoherentandlogically consistent)isratherdi˚cult[54,55],andsoweonlyseethispracticeslesscommonlyrelative totherest.Forexample,Excerpt25fromGroupDshowsonesuchrelativelyrareinstance ofagroupmemberclearlyandcorrectlyjustifyingheractions: SD: No,butwehavetoputitdownhere[pointstocalculationloop]. SA: Whynotjustwithalltheotherstu˙uphere? SD: Because...Becauseithastobeabletochange... SD: Ifithastochangeithastogodownhereinthecalculationloop... Here,StudentDjusti˝esde˝ningtheirforcemodelinsidethecalculationloopgiventhatit needstochangeovertime.Giventhatthisexchangeisatypical,weexpecttosee˝nding andtestingsolutionslesscommonlyrelativetotheotherpractices. Investigatingsystems Thepracticeof investigatingsystems involves questioning and interpreting datagathered fromasystemasawhole.Forexample,agraphofthesetofdatarepresentingthedistance betweentheEarthandthesatellitecanbequestionedaboutitsqualitativetime-dependence (e.g.,ifitisconstant,linear,quadratic,sinusoidal,etc.).Thistypeofpracticewasobserved 13 timesacrossthedata,accountingfor 13% ofthesystemsthinkingpractices,and 4% of allpractices. Investigatingsystemsasawholeisexpectedtoshowupinourdatagiventhelearning goalofplanningandcarryingoutinvestigations(P2).Theactofplanningissca˙oldedby thefour-quadrantsstudentsmustlisttheirknowns,unknowns,assumptions,anddrawout 140 anyrepresentations.Thesequadrantshelpstudentstogeneratequestions(e.g.,are weeventryingto˝gurethatcanbeinvestigatedforanswers.Giventhefocuson questioninginWeintropet.al'sde˝nitionofinvestigatingsystemsasawhole,weexpectto seethispracticeinourdata. Additionally,investigatingsystemsasawholelikelyshowsupinourdatagiventhe learninggoalofanalyzingandinterpretingdata(P3).Manysetsofdataneedtobecreated (seeSec.5.2.1)inthecalculationloop.Ultimately,thesesetsofdataneedtobeanalyzed (e.g.,visuallythroughagraphormanuallythroughaprintstatement).Forexample,consider Excerpt49fromGroupI,whereagraphisusedtogeneratemeaningintheirdata: TA: Soifyouseethatwobblethere[pointstograph] TA: And,andsowhatdoesthattellyouabouttheorbit? SB: Thatitsnotperfectlycircular? SC: Rightthatitdoesn'tgoinaperfectcircle. Here,thegroupisbeingaskedtoquestionthemeaningofthesinusoidaldatathattheyhave visualizedgraphically.GiventhefocusoninterpretingdatainWeintropet.al'sde˝nition ofinvestigatingsystemsasawhole,weexpecttoseethispractice. However,onereasonthatweobserveinvestigatingsystemsasawholelesscommonly withinourdatamightbethatindividualsvaryintheirlevelsofcuriosityacrucialchar- acteristicforthispractice.Manygroupsstrugglewiththedetailsoftheproblemandspend mostoftheirtimefocusingonthemwithouttakingastepbacktoquestionhowtheyrelate tothesystemasawhole.Forexample,Excerpt26fromGroupFshowsanatypicalexchange whereagroupmemberistakingastepbacktocheckthesystemasawhole: 141 SC: Butwaitisthatgonnaworkupherethen? SC: Wontthatbreaktheprogram? SC: Becausewealreadyhaveitde˝ned... SB: Nonowerenotde˝ningitagain SB: Wearejustusingit Here,StudentCisconcernedaboutde˝ninganabstractioninthewronglocationandasks aclari˝cationquestionaboutitsrelationtotheprogramasawhole. Understandingrelationships Thepracticeof understandingrelationships inasysteminvolves identifying theindivid- ualelementsofthesystemand explaining theirrelationshipstooneanother.Forexample,a groupmightidentifythemassandlocalaccelerationduetogravityastheindividualelements ofthesystemofthelocalgravitationalforce.Thegroupcouldthenexplaintherelationship betweenthesetwoelementsastheyrelatetotheforce(i.e.,theforceisproportionaltoboth themassandacceleration).Thistypeofpracticewasobserved 13 timesacrossthedata, accountingfor 13% ofthesystemsthinkingpractices,and 4% ofallpractices. Understandingrelationshipsmostlikelyshowsupinourdatagiventhelearninggoal ofbeingabletodevelopanduseamodel(P1).Developingamodelinvolvesaniterative processofcreatingthemodel,makingpredictionswithit,andvalidatingthemodelbased onitsresults[4].Throughoutthisprocess,theindividualelementsofthesystemmustbe identi˝edandcorrectlyrelatedtooneanother.Forexample,considerExcerpt37fromGroup 142 Hwheretheyarevalidatingtheirmodelbasedontherelationshipbetweentheforceandthe separationdistance: SC: Sorightnowtheforceisjustalwaysactinginthiswayinthenegativex-direction. SC: Buttheforceneedstoeventuallypointthiswayinthenegativey-direction... SB: Okay... SC: Soifweputthesatellitedotpositiondownhere... SC: Wecouldmakeitdothat... SA: Rightsolet'sdothatthen. Here,thegrouphasidenti˝edtheindividualelementsoftheforceandthepositionofthe satellite.Further,theyareexplainingthewayinwhichthetwoshouldberelated(i.e.,an inversedependence).Giventhesetypesofinteractions,weexpecttoobservethispracticein ourdata. Additionally,understandingrelationshipslikelyshowsupinourdatagiventhatthecourse isdesignedtocovermultipleforcemodelswithwidelyvaryingcomplexity.Speci˝cally,the ˝rstweekfocusesonaconstantvelocitymotion,witharelativelysimpleconstantzero forcemodel.Thesecondweekfocusesonconstantaccelerationmotion,withaslightlymore complicatedconstantlocalgravitationalforcemodel.Thethirdweekfocusesonnon-constant forces,likethecentripetalforceandtheNewtoniangravitationalforce,whicharegeneral, complex,anddi˚culttograpplewith.Giventhecomplexityofthemodelsusedinthethird week,ittakesasigni˝cantamountoftimeanddiscussiontodevelopastrongunderstanding whichwecanthenobserve. 143 However,onereasonthatweobserveunderstandingrelationshipslesscommonlywithin ourdatamightbethatsomeindividualsaremoreandsomearelessmathematicallyinclined oneofthebiggestfactorsinsuccessinphysics[56,57].Havingastrongmathematical backgroundwithadeepunderstandingofmathematicalrelationshipsingeneral(e.g.,an inversesquarerelationship)isrelativelyrareattheintroductoryphysicslevel,andsowe onlyseethispracticeslesscommonlyrelativetotherest.Forexample,Excerpt30from GroupIshowsarelativelyrareexchange: SA: Solikeifyouthinkaboutmakingthedistancereallybig... SA: Sinceit'sinthedenominator,ifyoumakethatreallybig... SA: Thenthistheforcebecomesreallysmall. SA: Whichitshouldright? SB: Yeahokaythatmakessense. Here,StudentAisexplainingtheconceptofalimitintermsofthewaythattheforceshould dependondistance(i.e., F / 1 =d 2 ).Giventherarityofthistypeofinteraction,weexpect toseethispracticelesscommonlyrelativetotherest. 144 AppendixC Unobservedpractices Thepracticeof manipulatingdata isnotexpectedtoshowupinourdatagiventhat itsde˝nition(seeSec.2.1)focusesonthereshapingofdata(e.g.,˝lteringasetofdataor mergingtwosetsofdataintoone).Studentsarenotrequiredtoreshapedatainthiswayin P 3 (e.g.,byusingthepandaspackagetomergetwodatasets).Rather,theyarerequiredto createthedataalgorithmically,whichcanthenbevisualizedoranalyzed. Anymanipulationofdata,initsmostgeneroussense,happensatthelevelofthemodel oralgorithmthatiscreatingthedata.Accordingly,excerptsthatmightgenerouslybe consideredmanipulatingdataarebetterclassi˝edascreatingdata(seeSec.5.2.1).Overall, wedon'texpectstudentsin P 3 tobeabletoreshape/clean-uplargeofsetsofdata,rather, wewantthemtobeabletocorrectlycreatethoselargesetsofdatausingmathematicaland computationalmodels. Thepracticeof choosingcomputationaltools isnotexpectedtoshowupinourdata giventhatthetooltheyarerequiredtouseitprovidedtothem.The˝rstthreeMWPsare allimplementedthroughVPythonandrequirenoadditionaltools.Accordingly,bythethird problem,studentsarefamiliarwiththetoolandknowtotakeadvantageofit. Itshouldbenotedthatnothingprecludesstudentsfromusingothertools(e.g.,Microsoft Excel)tosolvetheproblem,howeverwehavenotobservedthisinourdata.Thisislikely duetothelackofalearninggoalfocusingontoolselection.Overall,wewantstudentsin P 3 145 tobecomepro˝cientwiththetoolofVPythonformodelingmotion,ratherthanbeingable tochoosebetweencompetingtools. Thepracticeof developingmodularsolutions isnotexpectedtoshowupinour datagiventhatthecomputationalproblemsfromweektoweekaresu˚cientlydi˙erent thatnewmodelsmustalwaysbeused.Thisdoesnotallowformuchcross-overorreuse betweensolutions.Speci˝cally,the˝rstprobleminvolvesnonetforce,thesecondproblem involvesapiecewiseconstantnetforce,andthethirdinvolvesanon-constantnetforceall beingsu˚cientlydi˙erenttowarranttheconstructionofuniquecomputationalmodels.This repeateddesign,construction,andassessmentofnewmodels,writtenfromscratch,aligns wellwiththelearninggoalofdevelopingandusingmodels(P1).Overall,westudentsin P 3 tobeabletodesign,construct,andassessnewmodelsfromscratch,ratherthanbeableto reuseoldmodels. Thepracticeof preparingproblemsforcomputationalsolutions isnotexpected toshowupinourdatagiventhattheproblemhasalreadybeencastinaformthatis amenabletoacomputationalsolution.Infact,thisisthethirdproblemthattheyhave seenlikethis,sotheyalreadyknowtoapproachitcomputationally.Anypreparationofa problem,initsmostgeneroussense,happensatthedesignstage.Accordingly,excerptsthat mightgenerouslybeconsideredpreparationofaproblemarebetterclassi˝edasdesigninga computationalmodel(seeSec.5.2.3).Overall,wedon'texpectstudentsin P 3 togenerate theirownproblems(asidefromthecreateyourownproblemday...),rather,weexpectthem tobeabletosolvewell-de˝nedproblems. Thepracticeof de˝ningsystemsandmanagingcomplexity isnotexpectedtoshow upinourdatagiventhatmoststudentshaveverylittlepriorprogrammingexperience.This ispossiblyduetothefactthattherearenocomputationalprerequisitesfor P 3 .Giventhis 146 lackofpriorprogrammingexperience,interactionwiththeprogrammingsystemasawhole isrestrictedbydesign.Althoughthereisaninstructorgeneratedsystemthatstudentsare using(i.e.,aMWPinPythonthatinterfaceswithPhysUtilandtheVisualmodule),its complexityandmanagementarebeyondthescopeofthecourse.Thisisre˛ectedinthe absenceofalearninggoal(seeSec.3.4)focusingonthesystemasawhole.Additionally,the problemstatementitselfdoesnotexplicitlyrequirestudentstointeractwiththesystemas awhole,andtutorswilldissuadethisaction. 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