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Electronic Theses and Dissertations Graduate School
2019
Leading Learners to Level Up in a High School Mathematics Leading Learners to Level Up in a High School Mathematics
Classroom Classroom
Jennifer Carnes Wilson University of Mississippi
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LEADINGLEARNERSTOLEVELUPINAHIGHSCHOOLMATHEMATICSCLASSROOM
ADissertationpresentedinpartialfulfillmentofrequirements
forthedegreeofDoctorofEducationwithanemphasisinSecondaryMathematicsEducation
intheSchoolofEducationTheUniversityofMississippi
by
JENNIFERC.WILSON
May2019
ii
ABSTRACT
Howoftenareteacherssurprisedtofindoutattheendofalearningepisodethat
studentshavenotactuallylearned?ThefirstMathematicsTeachingPracticefromNCTM’s
PrinciplestoActionsassertsthat“effectiveteachingofmathematicsestablishescleargoals
forthemathematicsthatstudentsarelearning,situatesgoalswithinlearningprogressions,
andusesthegoalstoguideinstructionaldecisions”(NCTM,2014,p.10).Unfortunately,
manyteachersstruggletoestablishcleargoalstofocuslearning,andmanystudents
struggletomeetthosegoals.
Thisresearchstudyconsideredhowwellstudentspredictsuccessonlearning
targetsforanupcomingtestwhentheyaregiventhechancetoratethemselvesbeforethey
takethetestandwhethertreatmentssuchasworkedexamplesandmetacognitive
strategiesmovepredictedlevelsclosertoactualperformanceonthetest.Additionally,the
researchstudyconsideredwhetherthereisadifferenceintheabilitytopredictsuccess
levelbetweenengineeringandnon-engineeringstudentssinceengineeringstudentsuse
learningtargetsinboththeirmathandengineeringclasses.Throughquestionsona
studentGoogleformandforateacherinterview,theresearchersoughttodetermine
studentandteacherperceptionsaroundusinglearningtargetstoinformstudentprogress
inlearning.
Thisresearchstudysoughttodeterminewhetherusinglearningtargets,worked
examples,andmetacognitivestrategiescanensurethatstudentsnotonlyknowwhatis
iv
DEDICATION
This work is dedicated to all of the learners I have had the privilege of learning alongside,
students and teachers and friends, who have made me think about what is important to learn and
how we will know when we have learned it.
v
ACKNOWLEDGEMENTS
This work was inspired by my friend and teacher, Jill Gough, who has been my lead book
recommender and thought provoker. I am grateful to many colleagues and students whose
questions have taught me not just to wonder why but also figure out why. Thank you to Dr. Allan
Bellman and Dr. Tom Brady for always having another what if. Thank you to Adrienne, Jennifer,
LaVonda, Shawna, and Trisha for always helping find possible responses for then. And thank
you to Stan, Jane, and Kate for making and eating breakfast for supper on so many occasions
because that’s all we had in the house.
vi
TABLE OF CONTENTS
ABSTRACT ............................................................................................................................... ii
DEDICATION .......................................................................................................................... iv
ACKNOWLEDGEMENTS ........................................................................................................ v
LIST OF TABLES ................................................................................................................... xii
LIST OF FIGURES .................................................................................................................. xv
CHAPTER ONE: INTRODUCTION ......................................................................................... 1
STATEMENTOFTHEPROBLEM......................................................................................................................3
PURPOSE......................................................................................................................................................................3
SIGNIFICANCE...........................................................................................................................................................5
RESEARCHQUESTIONS........................................................................................................................................6
CHAPTER TWO: LITERATURE REVIEW .............................................................................. 8
DEFINITIONS.............................................................................................................................................................8
LITERATUREREVIEW..........................................................................................................................................8
Why Learning Targets Are Important .................................................................................. 9
Writing Student-Friendly Learning Targets ....................................................................... 11
Sharing Learning Targets with Students ............................................................................ 12
Learning Targets Inform Assessment ................................................................................ 14
vii
Learning Targets Embedded in Learning Progressions ...................................................... 16
Affecting Student Learning ............................................................................................... 17
CONNECTINGTORESEARCHPROJECT.....................................................................................................19
CHAPTER THREE: METHODOLOGY .................................................................................. 20
PURPOSEANDRESEARCHQUESTIONS....................................................................................................20
POPULATIONANDSAMPLING.......................................................................................................................23
INSTRUMENTATION...........................................................................................................................................24
Procedure and Time Frame ............................................................................................... 26
Analysis Plan .................................................................................................................... 28
Question1......................................................................................................................................................30
Question2......................................................................................................................................................31
Question2.1.............................................................................................................................................31
Question2.2.............................................................................................................................................31
Question2.3.............................................................................................................................................32
Question2.4.............................................................................................................................................32
Question3......................................................................................................................................................33
Question4......................................................................................................................................................33
Validity and Reliability ..................................................................................................... 33
viii
Scope and Limitations ....................................................................................................... 34
CHAPTER FOUR: RESULTS ................................................................................................. 36
PURPOSE...................................................................................................................................................................36
POPULATIONANDSAMPLING.......................................................................................................................36
VALIDITYANDRELIABILITY..........................................................................................................................37
TESTRESULTS.......................................................................................................................................................40
Question 1......................................................................................................................... 40
Question 2......................................................................................................................... 44
Question2.1..................................................................................................................................................44
Question2.2..................................................................................................................................................46
Question2.3..................................................................................................................................................47
Question2.4..................................................................................................................................................48
SubgroupWhoSelf-ReportedLevel3andPredictedLevel4..........................................48
SubgroupWhoSelf-ReportedLevel2andPredictedLevel3..........................................50
SubgroupWhoSelf-ReportedLevel2orLevel3andPredictedtheSameLevel...52
SubgroupWhoSelf-ReportedLevel4andPredictedtheSameLevel.........................54
SURVEYRESULTS.................................................................................................................................................56
Question 3......................................................................................................................... 56
ix
INTERVIEWRESULTS........................................................................................................................................58
Question 4......................................................................................................................... 58
SUMMARY................................................................................................................................................................60
CHAPTER FIVE: DISCUSSION ............................................................................................. 61
CONCLUSIONS........................................................................................................................................................64
Question 1......................................................................................................................... 64
Question 2......................................................................................................................... 65
Question2.1..................................................................................................................................................65
Question2.2..................................................................................................................................................67
Question2.3..................................................................................................................................................68
Question2.4..................................................................................................................................................68
Question 3......................................................................................................................... 71
Question 4......................................................................................................................... 72
LIMITATIONS.........................................................................................................................................................74
RECOMMENDATIONSFORFUTURERESEARCH..................................................................................76
CONCLUSIONS........................................................................................................................................................78
CHAPTER SIX: AN INFORMAL ADDENDUM .................................................................... 80
PURPOSEANDRESEARCHQUESTIONS....................................................................................................80
x
METHODOLOGY....................................................................................................................................................83
RESULTS...................................................................................................................................................................87
Question 1......................................................................................................................... 87
Question 2......................................................................................................................... 88
Question 3......................................................................................................................... 91
DISCUSSION............................................................................................................................................................93
Question 1......................................................................................................................... 93
Question 2......................................................................................................................... 95
Question 3......................................................................................................................... 97
SCOPEANDLIMITATIONS...............................................................................................................................98
FUTURERESEARCH.........................................................................................................................................101
LIST OF REFERENCES ........................................................................................................ 103
APPENDICES ........................................................................................................................ 107
APPENDIXA.........................................................................................................................................................108
APPENDIXB.........................................................................................................................................................109
APPENDIXC.........................................................................................................................................................110
APPENDIXD.........................................................................................................................................................112
APPENDIXE.........................................................................................................................................................113
xi
APPENDIXF..........................................................................................................................................................115
APPENDIXG.........................................................................................................................................................119
APPENDIXH.........................................................................................................................................................120
APPENDIXI...........................................................................................................................................................123
APPENDIXJ...........................................................................................................................................................124
APPENDIXK.........................................................................................................................................................126
APPENDIXL..........................................................................................................................................................128
APPENDIXM........................................................................................................................................................131
APPENDIXN.........................................................................................................................................................132
APPENDIXO.........................................................................................................................................................133
APPENDIXP.........................................................................................................................................................139
APPENDIXQ.........................................................................................................................................................144
APPENDIXR.........................................................................................................................................................146
APPENDIXS..........................................................................................................................................................147
APPENDIXT.........................................................................................................................................................149
VITA ...................................................................................................................................... 156
xii
LISTOFTABLES
Table3.1:StudentDemographics 24
Table3.2:SampleStudentData,Unit7Raw 28
Table3.3:SampleStudentData,Unit7LearningTargetsSelf-ReportedChangeandPredictedChange
29
Table3.4:SampleStudentData,UnitSelf-reportedChangeandPredictedChange 29
Table4.1:StudentDemographics 37
Table4.2:One-WayAnalysisofVarianceofFinalAlgebra2GradebyPrecalculusSection
38
Table4.3:One-WayAnalysisofVarianceofSemester1PrecalculusGradesbyPrecalculusSection
38
Table4.4:One-WayAnalysisofVarianceofSemester2PrecalculusGradesbyPrecalculusSection
38
Table4.5:DescriptiveStatisticsofTestGradesbyUnit,Ms.BairdandMs.Dolf 39
Table4.6:One-WayAnalysisofVarianceofTestGradesbyUnit,Ms.BairdandMs.Dolf
39
Table4.7:DescriptiveStatisticsforSelf-ReportedChangeandPredictedChangebyLearningTarget
41
Table4.8:DescriptiveStatisticsforSelf-ReportedChangeandPredictedChangebyUnit
42
Table4.9:One-WayAnalysisofVarianceofPredictedChangebyUnit 45
Table4.10:Tukey-KramerHSDComparisonforPredictedChangebyUnit 45
xiii
Table4.11:Resultsoft-testandDescriptiveStatisticsforPredictedChangebyWorkedExamplesandMetacognitiveTreatment
47
Table4.12:ActualPerformanceofStudentsSelf-ReportingLevel3andPredictingLevel4
49
Table4.13:ActualPerformanceofStudentsSelf-ReportingLevel2andPredictingLevel3
51
Table4.14ActualPerformanceofStudentsSelf-ReportingLevel2orLevel3andPredictingtheSameLevel
53
Table4.15:ActualPerformanceofStudentsSelf-ReportingLevel4andPredictingtheSameLevel
55
Table6.1:SampleStudentData,Unit2Raw 85
Table6.2:SampleStudentData,UnitMeanforPre-test,Predicted,andActualTestLevels 85
Table6.3:SampleStudentData,StudentReflectionBeforetheTest 86
Table6.4:SampleStudentData,StudentReflectionAftertheTest 86
Table6.5:Pre-testandActualTestLevelMeansofAllStudentsbyUnit 87
Table6.6:StudentData,LearningGoalswithStudentPerformanceHigheronPre-testThanonActualTest
88
Table6.7:Pre-testandPredictedLevelMeansofAllStudentsbyUnit 89
Table6.8:ComparisonofPre-testandPredictedLevelsofRatingsforAllStudents 89
Table6.9:StudentPredictionand/orPerformanceGreaterThanPre-testLevel 90
Table6.10:PredictedandActualTestLevelMeansofAllStudentsbyUnit 90
xiv
Table6.11:StudentComparisonofActualTestandPredictedLevelMeansbyUnit
91
Table6.12:StudentReflectionResponsesBeforetheTest/AftertheTest 92
xv
LISTOFFIGURES
Figure3.1:ProcessStudentsCompletedforEachLearningTargetontheTest 25
Figure4.1:TestResultsforSelf-ReportingLevel3andPredictingLevel4 49
Figure4.2:TestResultsforSelf-ReportingLevel2andPredictingLevel3 51
Figure4.3:TestResultsforSelf-ReportingLevel2orLevel3andPredictingtheSameLevel
53
Figure4.4:TestResultsforSelf-ReportingLevel4andPredictingtheSameLevel 55
Figure4.5:StudentperceptionsonUsingLearningTargets,LikertScaleQuestions
56
Figure4.6:StudentperceptionsonUsingLearningTargets,Yes/NoQuestions 57
Figure6.1:ProcessStudentsCompletedforEachLearningTargetontheTest 81
Figure6.2:ProcessStudentsCompletedforEachLearningTargetontheTestintheAddendum
84
1
CHAPTERONE:INTRODUCTION
Considerthesescenariosfamiliartomanystudentsandteachers.
Astudentisasked,“Whatareyoulearningabouttodayinclass?”Howdoesthestudentrespond?
A. “Nothing”B. “Math”C. “Thequestionsonthisworksheet”D. “Decidingiftwofiguresarecongruent”
Duringclass,astudentaskstheteacher,“Isthisgoingtobeonthetest?”Howdoestheteacherrespond?
A. Pretendslikeshedidn’thearthequestionB. WithaneyerollC. “EverythingIsayisgoingtobeonthetest”D. “Let’sseehowwhatwe’redoingisconnectedtotoday’slearning
goals”
Howoftenareteacherssurprisedtofindoutattheendofalearningepisodethat
studentshavenotactuallylearned?Howoftenareteachersfrustratedbystudentswhoask,
“Isthisgoingtobeonthetest?”
Yearsofmathematicseducationresearchshowthatestablishingandsharing
learninggoalsareimportantforbothteachersandstudents.ThefirstofNCTM’s
MathematicsTeachingPracticesfromPrinciplestoActionsisto“establishmathematics
goalstofocuslearning”(2014,p.10).Themathematicalgoalofthelessonshouldnotbea
secretkeptfromstudents.Bothstudentsandteachersneedtoknowwhatmathtolearn,
whytolearnit,howitisconnectedtopreviouslearning,andhowitisconnectedtofuture
learning(NCTM,2014).
2
Oneframeworkforfacilitatingmeaningfulmathematicaldiscourseisthe“5
PracticesforOrchestratingProductiveMathematicsDiscussions”,inwhichteachers
anticipatestudentstrategiesforatask,monitorstudentswhileworking,selectandsequence
studentworktobesharedwiththewholeclass,andthenconnectthestudentworktothe
mathematicallearningthatneedstotakeplaceinthelesson.Beforethe5Practicescanbe
effective,however,teachersmustsetlearninggoalsforinstruction.“Specifyingthe
mathematicalgoalsforthelessonisacriticalstartingpointforplanningandteachinga
lesson”(Smith&Stein,2011,p.13).Whenteachersdonothaveamathematicalgoalfora
lesson,theythinkaboutthelessonintermsoftheactivitiesstudentswilldoinsteadofthe
mathematicsthatstudentswillknowandunderstandasaresultofengaginginthe
activities(Smith&Stein,2011).Nowondermanystudents’answersto“whatareyou
learningabouttodayinclass”aremorefocusedonanactivitytheyaredoingratherthan
themathematicstheyarelearning.
“Clarifying,sharing,andunderstandinggoalsforlearningandcriteriaforsuccess
withlearners”isthefirstofWiliamandThompson’skeystrategiesforeffectiveformative
assessment(2007,p.64).Theydefinelearningintentionsaswhatstudentsshouldlearn
andsuccesscriteriaasawaytomeasurewhetherthelearninghashappened(Wiliam&
Leahy,2015,p.31).
“Thelearningtargetarticulatesforstudentswhattheyaretolearnandatthesame
timeprovidesinsightastohowstudentswillbeassessed”(Kanold&Larson,2012,p.49).
Whatisgoingtobeonthetestshouldnotbeasurprisetostudents.Learningtargetsshould
informteacherswhatcontent-aligneditemstoputonthetestandshouldinformstudents
whatcontent-aligneditemswillbeonthetest.
3
Thisresearchstudybuildsontheimportanceofestablishinglearninggoalsand
clarifyingsuccesscriteriaforstudentstofindouthowteachersmightprovide
opportunitiesforstudentstouselearninggoalsandsuccesscriteriaformativelyinorderto
knowbothwhattheyhavelearnedandwhattheystillneedtoknow.
STATEMENTOFTHEPROBLEM
Severalteachersintheresearcher’sformermathdepartmenthavebeenworkingon
clarifyingandsharinglearninggoalsandlearningtargetsforandwithstudentsforseveral
yearsnow.Afewyearsback,theybegantoincludethelearningtargetsontheunit
assessmentsandorganizetheproblemsbylearningtargets(seeAppendicesEandFfora
beforeandafterprecalculustest).Theteacherswerereadyforanextstepinimproving
studentlearning.Asadepartment,theyreadHattie’sVisibleLearningforMathematics
duringtheyearoftheresearchstudy,andtheywereinterestedintryingsomeofwhatthey
werereadingaseffectivestrategiesformaximizingstudentlearning.
Manyinterventionstoutsuccessinimprovingstudentlearning.Howdoteachers
decidewhichonestotryintheirclassrooms?Hattiehasspentyearsperformingmeta-
analysesofthousandsofresearchstudiesonmillionsofstudentsandusingeffectsizesto
compareinterventions.Mostinterventionshaveaneffectsizeabovezero,andsotheyshow
someeffectonstudentlearning.Inordertothinkaboutwhichinterventionsworkbetter
thanothers,Hattieusedthemeaneffectsizeof0.40toindicategrowthatanormalrateina
schoolyearandeffectsizesabove0.40toindicategrowthaboveanormalrateinaschool
year(Hattie,2012;Hattie,Fisher,&Fray,2017).
PURPOSE
4
Self-reportingprogresstowardslearningtargetsandsettinganexpectationfor
successhasaneffectsizeof1.44,oneofthehighesteffectsizesonstudentachievement.
Hattiesuggeststhatstudentsknowhowtheyaregoingtoperformonatest.Whengiven
theopportunitytoself-reporttheirprogresstowardsalearningtarget,studentssetsafe
expectations.Hegoesontosaythatteachersshouldnothelpstudentsreachtheirpredicted
levelbuthelpthemexceedtheirpredictedlevel(Hattie,May2012).
Theteacherintheresearchstudyplannedtoaskstudentstoself-report(atwhat
leveldoesthestudentthinksheisrightnow?)andpredict(atwhatleveldoesthestudent
expecttobewhentakingthetestduringthenextclass?)oneachlearningtargetasLevel1-
beginning,Level2-progressing,Level3-proficient,orLevel4-exceptionaltheclassperiod
beforetheytakeatest.Sheusedanalogiesofridingabikeanddrivingacartoestablish
whatlearninglooksforbeginning,progressing,proficient,andexceptionallevels(see
AppendixG).Howwelldostudentsself-reportorpredicttheirsuccessforeachlearning
targetcomparedtotheiractualperformanceonthetest?Dothey,infact,knowwherethey
areandsetsafeexpectations,ensuringthattheydonotover-predicthowtheywilldoon
thetest?
Whenstudentsknowwhatthelearningtargetis,theycancomparewheretheythink
theyaretowherethelearningtargetsuggeststheyshouldbe.Whentheyarenotwhere
theyshouldbeyet,theincongruousprogressspursstudentstotakeactionontheir
learning.Whenstudentsknowhowtheywillknowwhentheyreachthelearningtarget,
theyarebetterabletomonitortheirprogresstowardsmeetingit(Hattie,Fisher,&Fray,
2017).Inordertorealizethe1.44effectsizefromself-reportedgrades/student
5
expectations,teachersmustensurethatstudentsnotonlyknowwhatthelearningtargetis
butalsohowtoreachthelearningtarget.
Workedexamplesalsoimprovestudentachievement,withaneffectsizeof0.57.A
workedexampleshowsstudentsthestepsforsolvingamathproblem(Hattie,Fisher,&
Fray,2017).Mightprovidingworkedexamplesofwhateachlearningtargetlookslikeat
Level1-beginning,Level2-progressing,Level3-proficient,andLevel4-exceptionalnotonly
helpstudentsknowwhatthelearningtargetisbutalsohowtoreachit,thushavinga
positiveeffectonhowwellstudentspredicttheirexpectedsuccessoneachlearningtarget
(seeAppendixH)?
Studentsusingmetacognitivestrategiesasaninterventionhasaneffectsizeof0.67.
Establishinganormintheclassroomforalllearnerstosharewhytheyarethinkingwhat
theyarethinkingaboutaproblembuildsthehabitofreflectivelearningforstudents,which
increasesthetendencyforstudentstothinkaboutwhensomethingdoesnotmakesense
andtaketimetofigureoutwhy(Hattie,Fisher,&Frey,2017).Mightdiscussingwith
studentstheimportanceofthinkingabouthowandwhattheyarelearning,alongwith
providingstudentsexplicitopportunitiestoknowthelearningtargetandhowtoreachit,
haveanyeffectonhowwellstudentspredicttheirsuccessoneachlearningtarget?
SIGNIFICANCE
Askingstudentstoratetheirprogressonlearningtargetsbeforeatesttakeslittle
classtimeandisalow-riskrequestforstudentswiththepotentialofimprovinghowthey
thinkaboutwhattheyhavelearned.Theratingprocesscouldbeawake-upcallforsome
studentstorecognizewhattheyhavenotyetlearnedandtakeactiontolearnit.Itwill
likelyrequirestudentstothinkmoreabouttheirprogresstowardseachlearningtargetin
6
theunitmorepurposefullythantheyhavedonesobefore,whichcanleadtomore
deliberatestudyhabitsnotonlyinmathematicsbutalsoinothersubjects.
RESEARCHQUESTIONS
1. Arethestudents’actualperformancelevelsontestdayclosertothestudent
predictedlevelsorclosertothestudents’self-reportedlevels(wheretheythink
theyareonthedaybeforethetest)?
H0:Themeandifferenceinactualperformancelevelandstudentpredictedlevelis
equaltothemeandifferenceinactualperformancelevelandtheself-reportedlevel
(wheretheythinktheyareonthedaybeforethetest).
2. Arethereinterventionsthatimprovestudentpredictionsforhowtheyexpectto
performonatest?
2.1 Doworkedexampleshaveanyeffectonhowclosestudentpredictedlevelisto
actualperformancelevel?
H0:Forstudentswhoreceivedworkedexamples,themeandifferenceinactual
performancelevelandstudentpredictedlevelisequaltothemeandifferencefor
studentswhodidnotreceiveworkedexamples.
2.2 Doworkedexamplesandanemphasisonteachingstudentstheimportanceof
metacognitionhaveanyeffectonhowclosestudentpredictedlevelistoactual
performancelevel?
H0:Forstudentswhoreceivedworkedexamplesandametacognitivetreatment,the
meandifferenceinactualperformancelevelandstudentpredictedlevelisequalto
themeandifferenceforstudentswhodidnotreceiveworkedexamplesanda
metacognitivetreatment.
7
2.3 Isthereadifferencebetweenengineeringandnon-engineeringstudentsonhow
closestudentpredictedlevelistoactualperformancelevel?
H0:Forstudentswhoareinengineering,themeandifferenceinactualperformance
levelandstudentpredictedlevelisequaltothemeandifferencefornon-engineering
students.
2.4 Doworkedexamplesandmetacognitivestrategieshaveanyeffectonhowclose
studentpredictedlevelistoactualperformancelevelforsubgroupsofstudents,
basedonparticularself-reportedlevelsandpredictedlevels?
H0:Forstudentsinsubgroupsofparticularself-reportedandpredictedlevels,the
meandifferenceinactualperformancelevelandstudentpredictedlevelforthose
whoreceivedworkedexamplesandametacognitivetreatmentisequaltothemean
differenceforthosewhodidnot.
3. Whatarestudentperceptionsaroundusinglearningtargetstoinformstudent
progressinlearning?
4. Whatareteacherperceptionsaroundusinglearningtargetstoinformstudent
progressinlearning?
8
CHAPTERTWO:LITERATUREREVIEW
DEFINITIONS
Learninggoalsorlearningintentionsdescribethemathematicsthatstudentsshould
knowasaresultofalearningepisode.
LearningtargetsorsuccesscriteriaorIcanstatementsrevealwhatstudentsshouldbe
abletodowhentheysuccessfullymeetthelearninggoal.
LITERATUREREVIEW
Toooften,inclassroomseverywhere,studentsdonotknowhowtorespondwhen
theyareasked,“Whatareyoulearningabouttodayinclass?”Toooften,inclassrooms
everywhere,teachersareoffendedbystudentswhoask,“Isthisgoingtobeonthetest?”
Establishingandsharinglearninggoalsandtargetswithstudentscanalleviatesome
ofthetensionthatcomesbetweenstudentsandteachersandtheaforementioned
questions,butteachersoftendonotknowwheretostart.Dependingontheadministrator,
thelessonplanform,andtheteacherevaluationinstrument,teachersareinundatedwith
figuringoutwhatismeantbyallsortsofeducationaljargonsurroundingwhatstudents
shouldlearnandbeabletodo:learninggoal,learningtarget,learningintention,learning
standard,learningoutcome,measurableoutcome,learnerobjective,studentlearning
objective(SLO),instructionalgoal,successcriteria,performancecriteria,thestudentwill
(TSW),thestudentwillbeableto(TSWBAT),Icanstatement,curricularaim,essential
question,focusquestion,etc.
9
Yearsofmathematicseducationresearchshowthatestablishingandsharing
learninggoalsisimportantforbothteachersandstudents.In2014,theNationalCouncilof
TeachersofMathematics(NCTM)publishedPrinciplestoActions,aresearch-infused
endeavortoupdateNCTM’sprinciplesforteachingandlearningmathematicsandtolay
outaction-basedpracticesforallmathematicsleaders–informingteachers,coaches,
administrators,andcurriculumspecialistshowtheymightensureallstudentsexperience
aneffective,high-qualitymathematicseducation.
ThefirstofNCTM’sMathematicsTeachingPracticesisto“establishmathematics
goalstofocuslearning”(2014,p.10).Themathematicalgoalofthelessonshouldnotbea
secretkeptfromstudents.Bothstudentsandteachersneedtoknowwhatmathtolearn,
whytolearnit,howitisconnectedtopreviouslearning,andhowitisconnectedtofuture
learning(NCTM,2014).
WhyLearningTargetsAreImportant
Whenteachersestablishmathematicsgoalstofocuslearning,theynotonlyshare
lessongoalswithstudentsbutalsohelpstudentsunderstandalearningtrajectoryover
time.Teachersensurestudentsknowhowtheirworkonthelessontasksandactivities
connectstothelearninggoal,andtheyusethelearninggoalstomakedecisionsaboutwhat
todonextthroughoutthelesson.Simultaneously,studentsusethelearninggoalstomake
connectionstopreviousandupcominglearning.Theyusethegoalstofocusonthemath
content,self-assesstheirlearning,andseekhelpwhenneeded(NCTM,2014).
Whenteachersknowthemathematicalgoalsofthelesson,theyarebetterequipped
toenactotherMathematicsTeachingPractices,suchasselectingataskthatpromotes
reasoning,facilitatemeaningfulmathematicaldiscourse,anduseevidenceofstudent
10
thinking(NCTM,2014).Oneframeworkforfacilitatingmeaningfulmathematicaldiscourse
isthe“5PracticesforOrchestratingProductiveMathematicsDiscussions”,inwhich
teachersanticipatestudentstrategiesforatask,monitorstudentswhileworking,selectand
sequencestudentworktobesharedwiththewholeclass,andthenconnectthestudent
worktothemathematicallearningthatneedstotakeplaceinthelesson.Beforethe5
Practicescanbeeffective,however,teachersmustsetlearninggoalsforinstruction.
“Specifyingthemathematicalgoalsforthelessonisacriticalstartingpointforplanningand
teachingalesson”(Smith&Stein,2011,p.13).
Whenteachersdonothaveamathematicalgoalforalesson,theythinkaboutthe
lessonintermsoftheactivitiesstudentswilldoinsteadofthemathematicsthatstudents
willknowandunderstandasaresultofengagingintheactivities(Smith&Stein,2011).No
wondermanystudents’answersto“whatareyoulearningabouttodayinclass?”aremore
focusedonanactivitytheyaredoingratherthanthemathematicstheyarelearning.
“Clarifying,sharing,andunderstandinggoalsforlearningandcriteriaforsuccess
withlearners”isthefirstofWiliamandThompson’skeystrategiesforeffectiveformative
assessment(2007,p.64).By2015,Wiliamrewordedthestrategyas“clarifying,sharing,
andunderstandinglearningintentionsandsuccesscriteria”(Wiliam&Leahy,2015,p.27).
Wiliam&Leahyalsobemoanteacherswhotalkabouttheirlessonintermsofwhat
studentsaregoingtodoratherthanwhatstudentsshouldlearn.Theydefinelearning
intentionsaswhatstudentsshouldlearnandsuccesscriteriaasawaytomeasurewhether
thelearninghashappened(Wiliam&Leahy,2015).
Clearlearninggoalshelpinformteacherswhenplanningthetasksthatwillbe
appropriateforstudentstoengageinthemathematics.Clearlearninggoalsinform
11
formativeassessmentmovesforalesson,givingteachersinsightduringthelessontomake
instructionaldecisionsthatmovethelearningforward(Boston,et.al,2017).“Innovations
thatincludestrengtheningthepracticeofformativeassessmentproducesignificantand
oftensubstantiallearninggains”(Black&Wiliam,1998,p.141).Teacherswhouse
formativeassessmentregularlyfocusmoreonwhatthestudentislearningthanonwhat
thestudentisdoing(Wiliam&Thompson,2007).
Clearlearninggoalshelpstudentsembracelearning.Astudent’sbrainiswiredto
learnwhenthestudent’sbrainfindsmeaninginthatlearning.Meaningoccurswhen
learningisconnectedtogoals.Whenstudentscantellthatactivitiesandtasksare
connectedtothelearninggoals,theirbrainsaremorelikelytoallowworkonthetask,and
theycompletethetaskmorequickly(Sousa&Tomlinson,2011).
Sharinglearninggoalswithstudentscommunicatesteacherbeliefandagrowth
mindsetthatallstudentsareabletomeetthegoals.“Goalscansupportequitable
instructionbysettingclearandhighexpectations”(Boston,etal.,2017,p.25).
WritingStudent-FriendlyLearningTargets
“Touseknowledgeflexibly,studentsneedtounderstandwhattheyarelearning”
(Horn,2012,p.36).Learningtargetsshouldbesharedwithstudentsusingstudent-friendly
language(Bailey&Jakicic,2012).Theyshouldbe“statedclearlyinage-appropriate
language”,andteachersshould“clarifyanyquestionsstudentsmayhaveaboutthem”
(Sousa,2015,p.92).Whilestudent-friendlylanguageisimportant,teachersshouldbesure
thattheoriginalintentofthestandardisnotlostwhenrewritingforstudents(Ainsworth,
2015).Academiclanguagecanbeincludedinalearningtargetwrittenforstudents,but
teachersshouldensurethatstudentsunderstandtheacademiclanguage.BaileyandJakicic
12
haveestablishedthatwritinglearningtargetsintheformof“Ican…”statementsincrease
studentownershipofthelearning.Writinglearningtargetssothatstudentswillknowhow
toshowtheyaresuccessfulhelpsstudentsself-assesstheirprogresstowardssuccessfully
meetingthelearningtargets(2012).
Inordertoensurethatlearningtargetsarewrittensothatstudentsunderstand,the
teachercouldaskafewstudentstoquietlyreadthetargetanddescribethelearningtarget
intheirownwords.Theteachercanusewhatstudentshavewrittentocalibratetheir
understandingofthelearningtarget.Ifstudentshavewrittenwidelyvarieddescriptionsof
thetarget,thentheteachershouldlikelyrewordthelearningtargettoensurestudent
understanding(Popham,2008).Studentunderstandingofthelearningtargetisessential,
asstudentswhodonotunderstandthelearningtargetareunabletoassesstheirprogress
towardsmeetingthetarget(Heritage,2018).
SharingLearningTargetswithStudents
Educatorsdonotalwaysagreeonwhenandhowlearningtargetsshouldbeshared.
Forexample,somethinkthatlearningtargetsshouldbepostedintheclasssothatstudents
canseethemandreferencethemwhiletheclassactivitiesandtasksarefocusedonthose
targets(Popham,2008).Othersbelievethat“sometimestellingthestudentswheretheyare
goingcompletelyspoilsthejourney!”(Wiliam,2011,p.57).Manyteacherevaluationforms
haveacheckboxforteacherssharingandpostingthelearningtargetatthebeginningof
class,whichoftenresultsinaperfunctoryattemptbyteachersofensuringstudentsknow
whattheyaretolearn.Teachersshoulddiscernwhensharingthelearningtargetatthe
beginningofthelessonwillsupportstudentlearningandwhenitwilltaintstudent
learning,andshareaccordingly(Wiliam,2011).
13
Intheresearcher’sformerschool,teacherswererequiredtopostthelearningtarget
atthebeginningofclassandkeepitvisiblethroughouttheclass.However,theresearcher
found,likeWiliam,thatthelearningtargetoftengaveawaywhatstudentswereinvitedto
figureoutasaresultoftheclasstasksandactivities.Sharingthelearningtargetbeforethe
lessonwouldbelikesharingthepunchlinetoajokeatthebeginningofthejokeinsteadof
theend.Theresearcherworkedwiththeassistantprincipalonacompromisethatresulted
insharingatthebeginningofclassthemathpracticegoalthatstudentswouldlikelyuse,
suchasIcan“lookforandexpressregularityinrepeatedreasoning”(NGA,2010,p.8),to
engageinthemathcontentthatwouldberevealedbytheendofclass,suchasIcan“derive
theequationofacircleofgivencenterandradiususingthePythagoreanTheorem”(NGA,
2010,p.78).Duringthelearningepisode,studentsareprovidedtheopportunitytomake
connectionsbetweenarighttrianglewithahypotenusethatistheradiusofacircle,the
PythagoreanTheorem,andtheequationofthecircleinsteadofbeingtoldatthebeginning
ofthelessonthattheequationofthecircleisrelatedtothePythagoreanTheorem.
TheIllustrativeMathematics6–8Mathcurriculumalleviatestheproblemofspoiling
thejourneybyincludingbothstudent-facinglearninggoalsandstudent-facinglearning
targets.Learninggoalsarewrittenintheformof“Let’s...”toinvitestudentsintothework
tobedoneandtofocuslearningatthebeginningofclasswithoutrevealingthe
mathematicalrelationshipsthatwillbeuncoveredduringthelesson.Learningtargetsare
writtenintheformofactionable“Ican...”statementstohelpstudentsconnectthegoalto
themaththeyarelearning.Thecool-downforeachlessongivesstudentstheopportunity
toshowandassesstheirprogressinreachingthetarget(OpenUpResources,2017a).For
example,thelearninggoalforaneighth-gradelessononcongruentfigurespolygonsis
14
“Let’sdecideiftwofiguresarecongruent.”Thelearningtargetis“Icandecideusingrigid
transformationswhetherornottwofiguresarecongruent.”Whilethelearninggoalfocuses
thelearningondeterminingwhethertwofiguresarecongruent,itdoesnotrevealhowto
determinewhethertwofiguresarecongruent,whichisuncoveredthroughtheactivitiesin
thelesson(OpenUpResources,2017b).
LearningTargetsInformAssessment
“Thelearningtargetarticulatesforstudentswhattheyaretolearnandatthesame
timeprovidesinsightastohowstudentswillbeassessed”(Kanold&Larson,2012,p.49).
Learningtargets“drivethecreationofunitassessments(pre-,post-,andquickprogress
checks)”(Ainsworth,2015,p.21).Whatisgoingtobeonthetestshouldnotbeasurprise
tostudents.ThefirstindicatoronKanold&Larson’sassessmentevaluationtoolis
“identificationandemphasisonlearningtargets”(seeAppendixI).Level1(notpresent)
suggeststhat“learningtargetsareunclearorabsentfromtheassessmentinstrument.Too
muchattentionisgiventoonetarget.”Level4(fullypresent)suggeststhat“clearlystated
learningtargetsareontheassessmentandconnectedtotheassessmentquestions”
(Kanold&Larson,2012,p.94).Learningtargetsshouldinformteacherswhatcontent-
aligneditemstoputonthetestandshouldinformstudentswhatcontent-aligneditemswill
beonthetest.
Teacherscanhelpstudentsbetterunderstandlearningtargetsbysharingwith
studentshowthelearningtargetwillbeassessed.Sharingexampletestproblemsisan
idealwaytoimprovestudentunderstandingofthelearningtarget.Sharingthetypesoftest
itemsthatmightbeusedtoassessalearningtargetandwhythattypeofitemwaschosen
addstostudentunderstandingofthelearningtarget.Sharinganoviceworkedexample
15
alongsideaproficientworkedexamplecanalsoilluminatestudentunderstandingofthe
learningtarget(Popham,2008).Manyteachersobjecttoworkedexamplesbecause
studentsreadthroughthemwithouttryingtounderstandthem.Learningwithworked
examplesismoreeffectivewhenstudentsareencouragedtoself-explainthestepsinthe
problem.Teachersareintegraltotrainingstudentshowtoself-explain(Renkl,2014).
“Withthisbrain-friendlyapproach,formativeassessmentsbecomepractice-for-mastery
activitiesratherthananxiety-producingepisodes”(Sousa,2015,p.92).
Studentsmustpartnerwiththeteacherinreachingtowardsthelearningtarget,and
theycanalsohelpeachotherbetterunderstandlearningtargets.“Ithelpstomakethe
studentsfullyawareofthelearningintentionsandsuccesscriteria,ofthevalueof
deliberatepractice,andofwhattodowhentheydonotknowwhattodo”(Hattie,2012,p.
111).Wiliamcallsouttheseprocessesintwoofhisfivekeystrategiesofformative
assessment:“activatinglearnersasinstructionalresourcesforeachother”and“activating
learnersasownersoftheirlearning”(2011,p.2).Studentsbecomemoreinterestedin
learningwhentheycangaugetheirprogresstowardsmeetingthelearninggoalandknow
whatstepstotaketoimprove(Sousa,2015).
Exitticketsareonewayforstudentsandteacherstogatherinformationaboutwhat
studentscanknowanddoasaresultofengaginginalearningepisode.Exitticketsare
usuallygivenattheendofalessonasanopportunityforstudentstoreflectontheir
learningandforteacherstohaveinformationtomakeinstructionaladjustmentsbasedon
studentlearning.Askingstudentstocompletesuchpromptsas“Ilearned…“and“My
questionis…”and“WhatIlearnedtodayisimportantbecause…”givesstudentsand
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teachersinsightintothelearningthathasoccurredandwhatlearningshouldcomenext
(Baron,2016).
LearningTargetsEmbeddedinLearningProgressions
Ultimately,thelearningtargetsforonelessonshouldnotbeisolatedfromthe
learningtargetsforanotherlesson.Overtime,teachersandstudentsneedtohaveanidea
ofthebigpictureoflearning(Wiliam&Leahy,2015).Overarchinglearninggoalsgive
insightintowhatstudentsshouldlearnthroughoutacourse;unitlearninggoalsgive
insightintowhatstudentsshouldlearnduringaunit;andlessonlearninggoalsgiveinsight
intowhatstudentsshouldlearningduringalesson(HiebertandStigler,2017).
Studentswillhaveabetterideaofwhattheyaretolearnwhenlearningtargetsare
embeddedwithinlearningprogressions(Popham,2008).Workingtowardsalearning
targetisnotalinearprocessforallstudents,buttheplansurroundingalearningtarget
shouldbeinclusiveofallstudents(Hattie,2012).Teacherscanusealearningprogression
toanalyzestudentstrategiesforsolvingatask,makeinstructionaladjustmentsbasedon
thoseresponses,andmoveallstudentstowardsproceduralfluency(Ebby&Pettit,2017).
Knowingwherethelearningtargetfallswithintheprogressionoflearninghelpsstudents
makedecisionsaboutwhattheydonotyetknowandthusadjusthowandwhatthey
practiceinordertoreachthelearningtarget.Learningprogressionscanprovide
informationabouttheskillsneededtoreachatargetaswellasenrichmentopportunities
forthosewhohavealreadyreachedthetarget.Knowinghowthelearningtargetis
connectedtopriorandfuturelearningisessential(Popham,2008).
Learningprogressionsnotonlyinformtheformativeassessmentprocess,theyhelp
teachersplantheformativeassessmentprocess.Teachersusethelearningprogressionto
17
determinewhatquestionstoask,whentoaskthem,andwhattodonextdependingon
studentresponses.“Ifashipwithoutarudderis,bydefinition,rudderless,thenformative
assessmentwithoutalearningprogressionoftenbecomesplan-less”(Popham,2011,p.
24).
Studentscanbehelpfulinco-constructingandrevisinglearningprogressionsas
theybecomemoreawareoftheirlearning.Teachersshouldrememberthatlearning
progressionsarenotone-size-fits-all.Learningprogressionsvaryfromstatetostateand
fromonesetofcurricularmaterialstoanother.Studentsmayormaynotengageina
learningprogressioninthegivensequence,asmanyfactors,priorknowledgeinparticular,
affecthowandwhatstudentslearn(Wiliam&Leahy,2015).
Writinglearningprogressionsischallenging,time-consumingworkforteachers.Not
alllearningtargetsneedtobesituatedinalearningprogression.Whetherthelearning
targetisaskillthatwilltakelongerthanoneclasstolearn,whetherthelearningtargetwill
beusedinadditionalunitsorcoursesandconnectedtoreal-worldsituations,whetherthe
skillwillbeassessedonhigh-stakestests,andultimatelywhetherthelearningtargetis
reallyimportanttostudentlearningshouldallbetakenintoconsiderationwhenateacher
decideswhethertowritealearningprogression(Popham,2011).
AffectingStudentLearning
Manyinterventionstoutimprovingstudentlearning.Howdoteachersdecidewhich
onestotryintheirclassrooms?Hattiehasspentyearsperformingmeta-analysesof
thousandsofresearchstudiesonmillionsofstudentsandusingeffectsizestocompare
interventions.Mostinterventionshaveaneffectsizeabovezero,andsotheyshowsome
effectonstudentlearning.Inordertothinkaboutwhichinterventionsworkbetterthan
18
others,Hattieusedthemeaneffectsizeof0.40toindicategrowthatanormalrateina
schoolyearandeffectsizesabove0.40toindicategrowthaboveanormalrateinaschool
year(Hattie,2012;Hattie,Fisher,&Fray,2017).
Self-reportingprogresstowardslearningtargetsandsettinganexpectationfor
successhasaneffectsizeof1.44,oneofthehighesteffectsizesonstudentachievement
(Hattie,2017).Studentswhoareabletoratetheirprogressonlearningtargetsas
beginningorproficientshowhowwelltheyunderstandthelearningtargetandtheir
progresstowardsmeetingit.“Whenthereisagapbetweenwheretheyareandwherethey
wanttobe,itcreatescognitivedissonance”,pushingstudentstolearnmoreandwork
hardersothattheycanclosethegap.Teachersshouldprovidestudentsclearindicationsof
whatitmeanstomeetalearningtargetsothatstudentswillknowhowtoimprove(Hattie,
Fisher,&Frey,2017,p.57).
Workedexamplesalsoimprovestudentachievement,withaneffectsizeof0.57.A
workedexampleshowsstudentsthestepsforsolvingamathproblem.Teachersshould
makeitcleartostudentswhethertheworkedexamplesarecorrectorincorrectsolutions
totheproblemsothatstudentsdonotunintentionallylearnincorrectmethodsforsolving
problems.Analyzingworkedexamplescanhelpstudentsthinkaboutwhytheproblemis
solvedthewayitisandmovestudentstowardsexplanationsforhowtosolvetheproblem
insteadofonlyfocusingontheanswer(Hattie,Fisher,&Frey,2017).
Studentsusingmetacognitivestrategiesasaninterventionhasaneffectsizeof0.67.
Establishinganormintheclassroomforalllearnerstosharewhytheyarethinkingwhat
theyarethinkingaboutaproblembuildsthehabitofreflectivelearningforstudents,which
increasesthetendencyforstudentstothinkaboutwhensomethingdoesnotmakesense
19
andtaketimetofigureoutwhy.Somestudentswillmorenaturallythinkabouttheir
learningthanotherstudents(Hattie,Fisher,&Frey,2017).Inoneresearchstudy,learning
expertswhosharedtheirthinkingwhilelearningwerefoundtofrequentlyreflectonhow
welltheywerelearning,whattheystillneededtoknow,andhowwellwhattheywere
learningjivedwithwhattheyalreadyknew(Bransford,Brown,&Cocking,2001).Teachers
needtopurposefullyteachmetacognitivestrategiestotheclassandprovidedeliberate
opportunitiesforreflectingonlearningsothatallstudentscanadvantageouslyuse
metacognitivestrategiestoimprovelearning(Hattie,Fisher,&Frey,2017).Whenthe
teachermodelstheuseofmetacognitivestrategiesanddiscussesthestrategieswith
studentsastheylearntousethem,studentseventuallyusethestrategiesthemselves
withoutbeingpromptedbytheteacher(Bransford,Brown,&Cocking,2001).
CONNECTINGTORESEARCHPROJECT
Howoftenareteacherssurprisedtofindoutattheendofalearningepisodethat
studentshavenotactuallylearned?Howoftenareteachersfrustratedbystudentswhoask,
“Isthisgoingtobeonthetest?”
Thisresearchstudybuildsontheimportanceofestablishinglearninggoalsand
clarifyingsuccesscriteriaforstudentstofindouthowteachersmightprovide
opportunitiesforstudentstouselearninggoalsandsuccesscriteriaformativelyinorderto
knowbothwhattheyhavelearnedandwhattheystillneedtoknow.
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CHAPTERTHREE:METHODOLOGY
PURPOSEANDRESEARCHQUESTIONS
Thepurposeofthismixedmethodsresearchstudyistolookathowwellstudents
predicttheirexpectedlevelofsuccessonlearningtargetsforanupcomingtestwhenthey
aregiventhechancetoratethemselvesbeforetheytakethetestandwhethertreatments
suchasworkedexamplesandmetacognitivestrategiesmovepredictedlevelscloserto
actualperformanceonthetest.AccordingtoHattie,self-reportingprogresstowards
learningtargetsandsettinganexpectationforsuccesshasahigheffectonstudent
achievement.Hattiesuggeststhatstudentsknowhowtheyaregoingtoperformonatest.
Whengiventheopportunitytoself-reporttheirprogresstowardsalearningtarget,
studentssetsafeexpectations(Hattie,May2012).
Theteacherwhoparticipatedintheresearchstudyaskedstudentstorate
themselvesasLevel1-beginning,Level2-progressing,Level3-proficient,orLevel4-
exceptionaloneachlearningtargetbeforetheytookatest.Inorderforstudentstohave
somemeasureforeachrating,sheusedanalogiesofridingabikeanddrivingacarto
establishwhatlearninglooksforbeginning,progressing,proficient,andexceptional(see
AppendixG).
Whenstudentsknowwhatthelearningtargetis,theycancomparewheretheythink
theyaretowherethelearningtargetsuggeststheyshouldbe.Whentheyarenotwhere
theyshouldbeyet,theincongruousprogressspursstudentstotakeactionontheir
21
learning.Whenstudentsknowhowtheywillknowwhentheyreachthelearningtarget,
theyarebetterabletomonitortheirprogresstowardsmeetingit(Hattie,Fisher,&Fray,
2017).Inordertorealizethe1.44effectsizefromself-reportedgrades/student
expectations,teachersmustensurethatstudentsnotonlyknowwhatthelearningtargetis
butalsohowtoreachthelearningtarget.
AccordingtoHattie,workedexamplesalsoimprovestudentachievement(2017).
Theresearcherconsideredwhetherprovidingstudentswithworkedexamplesofwhat
eachlearningtargetlookslikeatLevel1-beginning,Level2-progressing,Level3-proficient,
andLevel4-exceptionalnotonlyhelpedstudentsknowwhatthelearningtargetisbutalso
howtoreachit,thushavingapositiveeffectonhowwellstudentspredicttheirsuccesson
eachlearningtarget(seeAppendixH.)
Hattiealsosuggeststhatmetacognitivestrategiesimprovestudentachievementand
thatprovidingstudentsopportunitiestoreflectontheirlearningcanfurthermetacognition
(2017,p.39).Theresearcheralsoconsideredwhethertheteacherdiscussingwithstudents
theimportanceofthinkingabouthowandwhattheyarelearningandalsoproviding
studentsexplicitopportunitiestoknowthelearningtargetandhowtoreachithadany
effectonhowwellstudentspredicttheirsuccessoneachlearningtarget.
Inconsideringhowwellstudentsself-reportedtheirprogresstowardsthelearning
targetthefollowingresearchquestionswereexamined.
1. Arethestudents’actualperformancelevelsontestdayclosertothestudent
predictedlevelsorclosertothestudents’self-reportedlevels(wheretheythink
theyareonthedaybeforethetest)?
22
H0:Themeandifferenceinactualperformancelevelandstudentpredictedlevelis
equaltothemeandifferenceinactualperformancelevelandtheself-reportedlevel
(wheretheythinktheyareonthedaybeforethetest).
2. Arethereinterventionsthatimprovestudentpredictionsforhowtheyexpectto
performonatest?
2.1 Doworkedexampleshaveanyeffectonhowclosestudentpredictedlevelisto
actualperformancelevel?
H0:Forstudentswhoreceivedworkedexamples,themeandifferenceinactual
performancelevelandstudentpredictedlevelisequaltothemeandifferencefor
studentswhodidnotreceiveworkedexamples.
2.2 Doworkedexamplesandanemphasisonteachingstudentstheimportanceof
metacognitionhaveanyeffectonhowclosestudentpredictedlevelistoactual
performancelevel?
H0:Forstudentswhoreceivedworkedexamplesandametacognitivetreatment,the
meandifferenceinactualperformancelevelandstudentpredictedlevelisequalto
themeandifferenceforstudentswhodidnotreceiveworkedexamplesanda
metacognitivetreatment.
2.3 Isthereadifferencebetweenengineeringandnon-engineeringstudentsonhow
closestudentpredictedlevelistoactualperformancelevel?
H0:Forstudentswhoareinengineering,themeandifferenceinactualperformance
levelandstudentpredictedlevelisequaltothemeandifferencefornon-engineering
students.
23
2.4 Doworkedexamplesandmetacognitivestrategieshaveanyeffectonhowclose
studentpredictedlevelistoactualperformancelevelforsubgroupsofstudents,
basedonparticularself-reportedlevelsandpredictedlevels?
H0:Forstudentsinsubgroupsofparticularself-reportedandpredictedlevels,the
meandifferenceinactualperformancelevelandstudentpredictedlevelforthose
whoreceivedworkedexamplesandametacognitivetreatmentisequaltothemean
differenceforthosewhodidnot.
3. Whatarestudentperceptionsaroundusinglearningtargetstoinformstudent
progressinlearning?
4. Whatareteacherperceptionsaroundusinglearningtargetstoinformstudent
progressinlearning?
POPULATIONANDSAMPLING
ThisresearchstudytookplaceatNorthwestRankinHighSchool,asuburbanschool
inRankinCountySchoolDistrictnearJackson,Mississippi.Sixty-fiveofthesixty-six
studentsenrolledinMs.Baird’sthreesections(A4,B2,andB3)ofAdvancedMathPlus
(precalculus)tookpartinthestudy.StudentswhoenrolledinMs.Baird’sprecalculusclass
haveshowninterestintakingAdvancedPlacement(AP)Calculustheirsenioryear.
NorthwestRankinHighSchooltakesseriouslythestanceoftheCollegeBoardonaccess
andequitybyofferingopenenrollmentforallAPandpre-APcourses.Whileitshouldbe
notedthatmanyofMs.Baird’sstudentsrankatthetopoftheirclass,itshouldalsobe
notedthatanystudentcouldself-electtoparticipateintheclass.
Ms.Bairdandherstudentswereselectedtoparticipateinthestudybecauseofthe
progressMs.Bairdmadewithstudentsonsharinglearningtargetsbothinclassandonunit
24
assessmentsthroughoutthefirstsemester.Additionally,Ms.Bairdhasshowninterestin
takinganextstepinclarifyingandsharinglearningtargetswithstudentsandwaswilling
toprovideopportunitiesforstudentstoratetheirprogressonlearningtargetsduringthe
secondsemester.Studentswerenotaskedtoidentifythemselvesinanymannerandthus
theiranonymityhasbeenprotected.
AllofMs.Baird’sstudentswereclassifiedasjuniors.45%arefemale.79%arewhite.
30%havetakenatleastoneclassintheNWRHSEngineeringAcademy.SeeTable3.1fora
breakdownofstudentdemographicsbysection.
Table3.1
StudentDemographics
Section Number(n) Female/Male% White/Black/Hispanic/Asian% EngineeringAcademy%
1(A4) 18 44%/56% 72%/22%/6%/0% 11%
2(B2) 25 40%/60% 80%/16%/0%/4% 40%
3(B3) 23 52%/48% 82%/9%/0%/9% 35%
Total 66 45%/55% 79%/15%/1%/5% 30%
INSTRUMENTATION
Thisresearchstudyusedamixedmethods,sequentialexplanatorydesigntostudy
thesuccessofstudentsself-reportingandpredictinglevelsonlearningtargets,withand
withoutleveledworkedexamples,withandwithoutemphasizingmetacognitivestrategies.
Quantitativedatawerecollectedbeforequalitativedata.
StudentscompletedaGoogleFormatthebeginningoftheclassperiodbeforethe
testforthreetestsduringthesecondsemester.Theyrecordedthestudentnumber
assignedtothembytheirteacherandratedboththeirself-reportedlevel(Level1-
25
beginning,Level2-progressing,Level3-proficient,orLevel4-exceptional)foreachlearning
target(atwhatleveldoesthestudentthinkheisrightnow?)andtheirpredictedlevelfor
testday(atwhatleveldoesthestudentexpecttobewhentakingthetestduringthenext
class?)(seeAppendixJ).
Theteacherdeterminedwhatpercentagecorrectconstitutedbeginning,
progressing,proficient,orexceptionalforeachlearningtarget.Forexample,onaparticular
learningtarget,0-50%couldbeconsideredLevel1-beginning,51-70%Level2-
progressing,71-90%Level3-proficient,and91-100%Level4-exceptional.Thepercentages
mightbedifferentonanotherlearningtarget.Ms.Bairdcompletedaspreadsheetafter
gradingeachtest,recordingthestudentnumberandactuallevelforeachlearningtarget
(atwhatleveldidthestudentactuallyperformonthetest?).Figure3.1showstheprocess
thatstudentscompletedforeachlearningtargetonthetest.
Figure3.1
ProcessStudentsCompletedforEachLearningTargetontheTest
Afterthethirdtest,studentscompletedaGoogleFormsurveytosharehowthey
usedlearningtargetsandwhetherratingtheirprogressand/ortheworkedexampleswere
helpfulintheirlearning(seeAppendixK).TheresearcherinterviewedMs.Bairdattheend
oftheresearchstudytofindoutherthoughtsontheresearchstudyandtoseewhat
26
aspects,ifany,shemightcontinueduringanothersemester,class,orschoolyear(see
AppendixM).
ProcedureandTimeFrame
Ms.Bairdcollecteddatafor3unitassessmentsduringthesecondsemesterofthe
2017-2018schoolyear.
Beforethefirsttestofthesecondsemester,studentsself-reportedtheleveloftheir
currentprogresstowardseachlearningtargetandpredictedtheleveloftheirsuccesson
thetestasLevel1-beginning,Level2-progressing,Level3-proficient,orLevel4-
exceptional(seeAppendixJ).Studentsweregivenanalogiesforwhatismeantby
beginning,progressing,proficient,andexceptional(seeAppendixG).Theteacherreported
theactuallevelofsuccessonthetestforeachlearningtarget.
Forthesecondtest,theteacherprovidedexampleproblemsofwhateachlearning
targetlookslikeateachlevelalongwiththeworkedsolutions(seeAppendixH.)During
thisphase,theteacherdiscussedtheimportanceofmetacognitionwithoneofthesections
butnottheothers.Studentsself-reportedtheleveloftheircurrentprogresstowardseach
learningtargetandpredictedtheleveloftheirsuccessonthetest.Theteacherreportedthe
actuallevelofsuccessonthetestforeachlearningtarget.
Forthethirdtest,theteacherprovidedleveledexampleproblemswithworked
solutionsanddiscussedtheimportanceofmetacognitionwithallthreesections.Students
self-reportedtheleveloftheircurrentprogresstowardseachlearninggoalandpredicted
theleveloftheirsuccessonthetest.Theteacherreportedtheactuallevelofsuccessonthe
testforeachlearningtarget.
27
Sincetheproblemsonthetestarealreadysortedbylearningtarget,theteacherwas
easilyabletodeterminethetotalpointsearnedoutofthetotalpointpossibleforeach
studentforeachlearningtargetandthencorrelatethetotalpointsearnedwithLevel1-
beginning,Level2-progressing,Level3-proficient,andLevel4-advanced.Theresearcher
thenconsideredthedifferenceoftheactuallevelofstudentsuccessonthetestandtheself-
reportedlevelofsuccessforeachlearningtarget(denotedself-reportedchange)andalso
thedifferenceoftheactuallevelofstudentsuccessonthetestandthestudentpredicted
levelofsuccessforeachlearningtarget(denotedpredictedchange).
Eachstudentwasassignedastudentnumberforuseduringtheresearchstudyto
ensurethatstudentanonymitywaspreservedthroughoutthestudy.Studentsrecorded
theirstudentnumberinthethreeself-assessmentsurveys,andtheteacherreportedthe
testdatatotheresearcherusingthesamestudentnumbersothattheresearchercould
determineanystatisticalsignificancebetweenthestudents’self-reportedlevel,predicted
level,andtheiractuallevelofsuccessonthetest.
Attheendofthethreetests,studentscompletedananonymoussurveyaboutthe
processtofindoutwhethertheyusedtheself-reportedratingsandworkedexamplesand
whethertheirconfidenceleveland/orachievementimprovedbasedontheiruse(see
AppendixK).Finally,theresearcherinterviewedtheteacherabouttheprocesstosee
whethershehadanyanecdotalevidenceforwhethertheself-reportedratings,worked
examples,andemphasisonmetacognitionmadeanydifferenceonstudentlearningand
mighthaveanyeffectonherfuturepracticesintheclassroom(seeAppendixM).
28
AnalysisPlan
Foreachoftheunits(7,8,and9)andeachofthelearningtargets(7_1,7_2,7_3,7_4,
8_1,etc.)thestudentssubmittedaself-reportedlevelfrom1to4(atwhatleveldoesthe
studentthinksheisrightnow?)andapredictedlevelfrom1to4(atwhatleveldoesthe
studentexpecttobewhentakingthetestduringthenextclass?).Theteacherreportedan
actuallevelfrom1to4(atwhatleveldidthestudentactuallyperformonthetest?).No
instrumentwasusedtoverifytheaccuracyofthestudentself-reportedlevel;itwasbased
onlyoneachstudent’sevaluationofwherehethoughthewasatthattime.Table3.2shows
rawsamplestudentdataforUnit7.
Table3.2
SampleStudentData,Unit7Raw
StudentNumber
7_1self-reportedlevel
7_1predictedlevel
7_1actuallevel
7_2self-reportedlevel
7_2predictedlevel
7_2actuallevel
7_3self-reportedlevel
7_3predictedlevel
7_3actuallevel
7_4self-reportedlevel
7_4predictedlevel
7_4actuallevel
100 3 4 3 2 4 4 3 4 4 2 3 4
101 4 4 3 4 4 2 4 4 3 3 4 3102 3 4 3 2 3 3 3 4 4 2 4 3
Todeterminehowwellstudentsself-reportedandpredictedtheirprogresstowards
thelearningtarget,theresearcherfoundthedifferenceforeachlearningtargetoftheself-
reportedlevel(atwhatleveldoesthestudentthinksheisrightnow?)andtheactuallevel
ofstudentsuccess(atwhatleveldidthestudentactuallyperformonthetest?),whichfrom
thispointonwillbecalledself-reportedchange,andalsothedifferenceofthepredicted
levelofsuccessforeachlearningtargetandtheactuallevelofstudentsuccess,whichfrom
thispointonwillbecalledpredictedchange(seeTable3.3).
29
Table3.3
SampleStudentData,Unit7LearningTargetsSelf-ReportedChangeandPredictedChangeStudentNumber
7_1self-reportedchange
7_1predictedchange
7_2self-reportedchange
7_2predictedchange
7_3self-reportedchange
7_3predictedchange
7_4self-reportedchange
7_4predictedchange
100 0 -1 2 0 1 0 2 1101 -1 -1 -2 -2 -1 -1 0 -1
102 0 -1 1 0 1 0 1 -1Note.Self-reportedchangeisthedifferencebetweenself-reportedlevelandactualperformancelevelontest;predictedchangeisthedifferencebetweenpredictedlevelandactualperformancelevelontest.
Avalueof0indicatesthatthestudentactuallyperformedonthetestatthesame
leveltheyself-reportedorpredicted.Avalueof-1indicatesthatthestudentperformedone
levellowerthanthelevelself-reportedorpredicted;-2indicatesanactualperformance
twolevelslowerthanthelevelself-reportedorpredicted.Avalueof1indicatesthatthe
studentperformedonelevelhigherthanthelevelself-reportedorpredicted;2indicatesan
actualperformancetwolevelshigherthanthelevelself-reportedorpredicted.
Themeanofthedifferencesbetweenself-reportedandactuallevelsaswellasthe
meanofthedifferencesbetweenpredictedandactuallevelsforthelearningtargetson
eachunittestwereusedtodetermineasingleself-reportedchangeandasinglepredicted
changeforeachstudentbyeachunit(seeTable3.4).
Table3.4
SampleStudentData,UnitSelf-reportedChangeandPredictedChangeStudentNumber
Unit7self-reportedchange
Unit7predictedchange
Unit8self-reportedchange
Unit8predictedchange
Unit9self-reportedchange
Unit9predictedchange
100 1.25 0 0.6 -0.2 0.83 -0.17
101 -1 -1.25 -0.8 -1.8 0.67 -0.33
102 0.75 -0.5 0 -0.6 0.83 0.17Note.Self-reportedchangeisthedifferencebetweenself-reportedlevelandactualperformancelevelontest;predictedchangeisthedifferencebetweenpredictedlevelandactualperformancelevelontest.
30
Avalueof0indicatesthat,onaverage,thestudentactuallyperformedaroundthe
samelevelself-reportedorpredictedforalllearningtargetsonthetest.Anegativevalue
indicatesthat,onaverage,thestudentactuallyperformedlowerthantheself-reportedor
predictedlevel.Apositivevalueindicatesthat,onaverage,thestudentactuallyperformed
higherthantheself-reportedorpredictedlevelforalllearningtargetsonthetest.
Studentswhowereabsentonthedaybeforeanyoneofthethreetestsanddidnot
completetheGoogleFormwereremovedfromthequantitativedataanalysis.Outofthe65
studentswhoagreedtoparticipateintheresearchstudy,theresearcherwasabletocollect
allquantitativedatafor41studentswhowerepresentforallthreereviewdaysandall
threetestdays.Toconsiderhowwellstudentsself-reportedtheirprogresstowardsthe
learningtarget,theresearcheruseddescriptivestatistics,t-tests,one-wayANOVAtests,
andchi-squaretestsofindependencetoanalyzethequantitativedata.
Question1
Arethestudents’actualperformancelevelsontestdayclosertothestudentpredicted
levelsorclosertothestudents’self-reportedlevels(wheretheythinktheyareontheday
beforethetest)?
Apairedt-testwithacriticalalphalevelof0.05showedanystatisticalsignificance
betweenstudentsself-reportedlevelandpredictedlevel.Theself-reported,predicted,and
actualsuccesslevelsforUnits7,8and9wereusedforthet-test.Theresearcherusedthet-
testtodeterminewhetherthedifferencebetweenself-reportedchangeandpredicted
changeisstatisticallysignificant.
31
Question2
Arethereinterventionsthatimprovestudentpredictionsforhowtheyexpecttoperform
onatest?Interventionssuchasworkedexamplesandametacognitivetreatmentwere
considered.
Question2.1
Doworkedexampleshaveanyeffectonhowclosestudentpredictedlevelistoactual
performancelevel?
Aone-wayANOVAwithacriticalalphalevelof0.05showedanystatistical
significanceforstudentspredictingtheirlevelofsuccesswhenstudentshadthe
opportunitytoassesstheirprogressthroughouttheunitusingworkedexamples.The
researcherusedtheANOVAcomparisonfollowedbyaTukey-KramerHSDComparisonto
determinehowclosestudentpredictedlevelistoactualperformancelevelforstudents
whoreceivedworkedexamplescomparedtostudentswhodidnotreceiveworked
examples.
Question2.2
Doworkedexamplesandanemphasisonteachingstudentstheimportanceof
metacognitionhaveanyeffectonhowclosestudentpredictedlevelistoactual
performancelevel?
AllstudentsreceivedworkedexamplesinUnit8andUnit9.Forthemetacognitive
treatment,theteacherhadconversationsinclasswiththestudentsabouthowresearch
showsthatmetacognitionimprovesstudentachievement.Onesection,B2,hadthe
metacognitivetreatmentforUnit8.Allthreesectionshadthemetacognitivetreatmentfor
Unit9.AnANOVAcomparisonwithacriticalalphalevelof0.05showedanystatistical
32
significanceforhowclosestudentpredictedlevelistoactualperformancelevelfor
studentswhoreceivedworkedexamplesandametacognitivetreatmentcomparedto
studentswhodidnotreceiveworkedexamplesandametacognitivetreatment.
Question2.3
Isthereadifferencebetweenengineeringandnon-engineeringstudentsonhowclose
studentpredictedlevelistoactualperformancelevel?
Engineeringstudentshaveusedlearningtargetsnotonlyinmathbutalsoin
engineering.Isthereadifferenceonhowclosestudentpredictedlevelistoactual
performancelevelforstudentswhotakeengineeringclassescomparedtostudentswhodo
nottakeengineeringclasses?AnANOVAcomparisonwithacriticalalphalevelof0.05
showedanystatisticalsignificanceforengineeringstudentsuccessinpredictedlevelwhen
comparedtonon-engineeringstudents.
Question2.4
Doworkedexamplesandmetacognitivestrategieshaveanyeffectonhowclosestudent
predictedlevelistoactualperformancelevelforsubgroupsofstudents,basedon
particularself-reportedlevelsandpredictedlevels?
Forsubgroupssuchasstudentswhoself-reportedLevel3andpredictedLevel4,orself-
reportedLevel2andpredictedLevel3,datawereanalyzedusingachi-squaretestof
significancebydecomposingresultsineachcategoryintoperformedatalowerlevel,
performedatself-reportedlevel,performedatpredictedlevel,orperformedatahigher
leveltodeterminewhetherthereisanydifferenceonhowclosepredictedprogressisto
actualperformanceforstudentswhoreceivedtheworkedexamplesandmetacognitive
treatmentcomparedtothosewhodidnotreceivethetreatment.
33
Question3
Whatarestudentperceptionsaroundusinglearningtargetstoinformstudentprogressin
learning?
Question4
Whatareteacherperceptionsaroundusinglearningtargetstoinformstudentprogressin
learning?
Theanonymoussurveythatstudentscompletedattheendoftheresearchstudyand
theteacherinterviewproducedqualitativedatathattheresearcherusedtogaugestudent
andteacherimpressionsontheeffectofstudentspredictingtheirlevelonlearningtargets
withorwithoutworkedexamples,andwithorwithoutanemphasisonmetacognition.
ResponsesfromLikertscalequestionswerecollectedonabargraphtodeterminewhether
anyresponsesaresignificant.Theresearcheranalyzedopen-endedqualitativequestions
bycoding.Theresearchercomparedstudentandteacherimpressionstotheresultsofthe
analysisonthequantitativedata.
ValidityandReliability
AllbutoneofMs.Baird’sprecalculusstudentsagreedtoparticipateinthestudy.
Whileitwasconvenienttosurveystudentswhohavethesameteacher,itwasalso
purposeful.Ms.Bairdusedlearningtargetsinallthreesectionsofherprecalculusclasses
duringthefirstsemesterbysharingthelearningtargetswithstudentsforeachunitand
includingthemonalltests.Consequently,theresultsfromthisresearchstudyarenot
generalizabletoallmathstudents.Includingstudentsfromadifferentteacherwhowere
notalreadyusinglearningtargetstoinformlearningandassessmentcouldhavevery
differentresults.
34
Thethreesectionsofprecalculuswerealltaughtatthesameschoolandbythesame
teacher.Theresearcherusedanon-pairedt-testwithacriticalalphalevelof0.05withthe
numericalAlgebra2coursegradeandthenumericalprecalculussemestergradesto
determinewhetherthethreesectionswerereasonablycomparableandensurethatthe
resultsofthestudyarereliable.Theteacherworkedwithallthreesectionsinthesame
mannerthroughouttheresearchstudyexceptforthesecondphaseofthestudy.During
thisphase,shebothusedworkedexamplesanddiscussedtheimportanceofmetacognition
tohelppredictsuccesswithhersecondsection(B2)butnottheothers.Doesanemphasis
onteachingstudentstheimportanceofmetacognitionhaveanyeffectonhowclose
predictedprogressistoactualperformance?Onthethirdtest,theteacherdiscussedthe
importanceofmetacognitionandusingworkedexamplestohelppredictsuccesswithall
threesections.
Mathematicseducatorclassmatesoftheresearcherprovidedfeedbackonthe
surveyquestionsthatwereusedwithstudentsandinterviewquestionsthatwereused
withtheteacher.Ms.BairdcalibratedwhatpercentagecorrectconstitutesLevel1-
beginning,Level2-progressing,Level3-proficient,orLevel4-exceptionalforeachlearning
targetwithaformerNorthwestRankinHighSchoolprecalculusteacher.
ScopeandLimitations
ThestudentsinMs.Baird’sclasseswerejuniors.Allbutfivestudentshada
geometryclassthatusedlearningtargetsinclassandonthetestwhentheywereinthe
ninthgrade.NostudenthadanAlgebra2classthatusedlearningtargetsonthetestwhen
theywereinthe10thgrade.Someofthemwereengineeringstudentswhohaveused
learningtargetsinengineeringclassesaswellasmathclasses.Eventhoughstudents
35
startedprecalculuswithvaryingexperiencesofusinglearningtargetsindividuallyandin
previousclasses,allofthestudentsusedlearningtargetsinclassandonthetestduringthe
entireyearofprecalculus.Itcouldbeinterestingtorepeatthisstudyinthesameschool
withadifferentteacheroradifferentcourseandinadifferentschoolwithstudentswho
hadnotpreviouslyfocusedonlearningtargetsduringclassorhadlearningtargetsontheir
tests,butthatisbeyondthescopeofthisresearchproject.
Inthisstudy,studentsself-reportedtheleveltheythoughttheywerethedaybefore
thetestandpredictedtheleveltheyexpectedtobewhentakingthetestduringthenext
class.Noinstrumentwasusedtoverifytheaccuracyofthestudentself-reportedlevel;it
wasbasedonlyoneachstudent’sevaluationofwherehethoughthewasatthattime.It
couldbeinterestingtorepeatthisstudywiththesamestudentsusingsomesortof
instrumenttoverifytheaccuracyofthestudents’self-reportedlevels.
36
CHAPTERFOUR:RESULTS
PURPOSE
Thisresearchstudysoughttodeterminehowwellstudentspredictsuccesson
learningtargetsforanupcomingtestwhentheyaregiventhechancetoratethemselves
beforetheytakethetestandwhethertreatmentssuchasworkedexamplesand
metacognitivestrategiesmovepredictedlevelsclosertoactualperformanceonthetest.
Additionally,theresearcherconsideredwhetherthereisadifferenceintheabilityto
predictsuccesslevelbetweenengineeringandnon-engineeringstudentssinceengineering
studentsuselearningtargetsinboththeirmathandengineeringclasses.Through
questionsonastudentGoogleformandforateacherinterview,theresearchersoughtto
determinestudentandteacherperceptionsaroundusinglearningtargetstoinformstudent
progressinlearning.
POPULATIONANDSAMPLING
ThisresearchstudytookplaceatNorthwestRankinHighSchool,asuburbanschool
inRankinCountySchoolDistrictnearJackson,Mississippi.Sixty-fiveofthesixty-six
studentsenrolledinMs.Baird’sthreesections(A4,B2,andB3)ofAdvancedMathPlus
(precalculus)tookpartinthestudy.
AllofMs.Baird’sstudentswereclassifiedasjuniors.45%arefemale.79%arewhite.
30%havetakenatleastoneclassintheNWRHSEngineeringAcademy.SeeTable4.1fora
breakdownofstudentdemographicsbysection.
37
Table4.1
StudentDemographics
Section Number(n)
Female/Male%
White/Black/Hispanic/Asian%
EngineeringAcademy%
1(A4) 18 44%/56% 72%/22%/6%/0% 11%
2(B2) 25 40%/60% 80%/16%/0%/4% 40%
3(B3) 23 52%/48% 82%/9%/0%/9% 35%
Total 66 45%/55% 79%/15%/1%/5% 30%
VALIDITYANDRELIABILITY
Theresearcherusedstudents’finalAlgebra2gradesandtheirtwosemester
precalculusgradestodeterminewhetherthethreeprecalculussectionsinthestudywere
reasonablycomparabletoensurethattheresultsofthestudyarereliable.92%ofthe
studentshadthesameAlgebra2teacher.AllstudentswereinMs.Baird’sprecalculus
sections,andthussheassignedallstudentgradesthroughouttheyear.Thedatawere
analyzedusingaone-wayANOVAwithstatisticalsignificancesetatanalphalevelof0.05to
measuretheinfluenceoftheindependentvariable,classsection,onthedependent
variables,finalAlgebra2grade(seeTable4.2),semesteroneprecalculusgrade(seeTable
4.3),andsemestertwoprecalculusgrade(seeTable4.4).Theresultingp-valuesshowno
statisticallysignificantdifferencebetweenMs.Baird’sthreeprecalculussections.
38
Table4.2
One-WayAnalysisofVarianceofFinalAlgebra2GradebyPrecalculusSection
Source df SS MS F p
Betweengroups 2 78.26 39.13 0.36 0.699
Withingroups 56 6021.88 107.53
Total 58 6100.14
Table4.3
One-WayAnalysisofVarianceofSemester1PrecalculusGradesbyPrecalculusSection
Source df SS MS F p
Betweengroups 2 155.09 77.54 0.57 0.568
Withingroups 62 8508.45 137.23
Total 64 8663.54
Table4.4
One-WayAnalysisofVarianceofSemester2PrecalculusGradesbyPrecalculusSection
Source df SS MS F p
Betweengroups 2 202.18 101.09 0.75 0.477
Withingroups 62 8314.07 134.10
Total 64 8516.25
TheresearcherexaminedtestscoresfromthepreviousthreeyearsforUnits7,8,
and9todeterminewhetherstudentscoresonthetestswerereasonablycomparableto
39
eliminatethepossibilitythatthedifficultyofthecontentinfluencedstudentperformance.
Ms.Bairdtaughtprecalculuslastyear;Ms.Dolftaughtprecalculusthetwoyearsbefore
that.BothteachersusedthesameinstrumenttoassessstudentseachyearforUnits7,8,
and9.DescriptivestatisticsfortestscoresareincludedinTable9.
Table4.5
DescriptiveStatisticsofTestGradesbyUnit,Ms.BairdandMs.Dolf
Unit Minimum Maximum Mean SD
7 50 100 85.27 11.21
8 50 100 84.83 12.31
9 49 100 82.21 13.77
Thedatawereanalyzedusingaone-wayANOVAwithstatisticalsignificancesetat
analphalevelof0.05tomeasuretheinfluenceoftheindependentvariable,unitnumber,
onthedependentvariable,testgrade.Testgradesfromthecurrentandprevious
precalculusteachers,Ms.BairdandMs.Dolf,forthepastthreeyearsshowednostatistical
differencebetweentests(seeTable4.6).
Table4.6
One-WayAnalysisofVarianceofTestGradesbyUnit,Ms.BairdandMs.Dolf
Source df SS MS F p
Betweengroups 2 639.61 319.81 2.06 0.13
Withingroups 358 54143.49 155.58
Total 350 54783.10
40
TESTRESULTS
Thebroadresearchquestionconsidershowwellstudentspredictthelevelatwhichthey
expecttoperformonalearningtargetonatestandwhetherthereareinterventionsthat
improvestudentpredictions.
Question1
Arethestudents’actualperformancelevelsontestdayclosertothestudentpredicted
levelsorclosertothestudents’self-reportedlevels(wheretheythinktheyareontheday
beforethetest)?
H0:Themeandifferenceinactualperformancelevelandstudentpredictedlevelisequalto
themeandifferenceinactualperformancelevelandtheself-reportedlevel(wherethey
thinktheyareonthedaybeforethetest).
Foreachoftheunits(7,8,and9)andeachofthelearningtargets(7_1,7_2,7_3,7_4,
8_1,etc.)thestudentssubmittedaself-reportedlevelfrom1to4(atwhatleveldoesthe
studentthinkheisrightnow?)andapredictedlevelfrom1to4(atwhatleveldoesthe
studentexpecttobewhentakingthetestduringthenextclass?).Theteacherreportedan
actuallevelfrom1to4(atwhatleveldidthestudentactuallyperformonthetest?).
Completequantitativedataweresecuredfor41ofthe65studentswhoagreedto
participateintheresearchstudy.
AppendixNhasalistofeachcontentlearningtargetbyunit.Table4.7shows
descriptivestatisticsforself-reportedchangeandpredictedchangebylearningtarget.
41
Table4.7
DescriptiveStatisticsforSelf-ReportedChangeandPredictedChangebyLearningTarget
Learning
Target
Minimum
Self-ReportedChange
Maximum
Self-ReportedChange
Mean
Self-ReportedChange
SD
Self-ReportedChange
Minimum
PredictedChange
Maximum
PredictedChange
Mean
PredictedChange
SD
PredictedChange
7_1 -2 1 -0.22 0.82 -3 2 -0.63 0.80
7_2 -3 2 0.12 0.95 -3 1 -0.27 0.81
7_3 -1 2 0.37 0.83 -1 2 -0.10 0.80
7_4 -2 2 0.32 1.06 -2 1 -0.24 0.89
8_1 -1 2 0.17 0.77 -2 1 -0.41 0.84
8_2 -3 1 -0.22 1.01 -3 1 -0.68 0.93
8_3 -2 2 -0.02 0.99 -2 1 -0.59 0.89
8_4 -2 1 -0.61 0.80 -3 1 -1.15 0.94
8_5 -2 2 -0.29 1.23 -3 1 -0.90 1.24
9_1 -1 2 0.49 0.68 -1 1 -0.05 0.55
9_2 -1 2 0.46 0.71 -1 1 -0.07 0.61
9_3 -1 1 0.20 0.71 -2 0 -0.39 0.54
9_4 -2 2 0.07 0.75 -3 1 -0.56 0.78
9_5 -1 1 0.39 0.63 -1 1 -0.15 0.57
9_6 -1 2 0.27 0.84 -2 1 -0.32 0.72Note.Self-reportedchangeisthedifferencebetweenself-reportedlevelandactualperformancelevelontest;predictedchangeisthedifferencebetweenpredictedlevelandactualperformancelevelontest.Ameanvalueof0indicatesthat,onaverage,thestudentsactuallyperformedaroundthe
sameleveltheyself-reportedorpredictedforthatlearningtargetonthetest.Forexample,
42
themeanvalueof-0.05indicatesthat,onaverage,studentsactuallyperformedaroundthe
sameleveltheypredictedforlearningtarget9_1.Anegativemeanvalueindicatesthat,on
average,thestudentsactuallyperformedlowerthanthelevelself-reportedorpredicted.
Forexample,themeanvalueof-1.15indicatesthat,onaverage,studentsactually
performedlowerthanpredictedforlearningtarget8_4.Apositivemeanvalueindicates
that,onaverage,thestudentsactuallyperformedhigherthanthelevelself-reportedor
predicted.
Table4.8showsdescriptivestatisticsforself-reportedchangeandpredictedchange
byunit.
Table4.8
DescriptiveStatisticsforSelf-ReportedChangeandPredictedChangebyUnit
Unit Mean
Self-ReportedChange
SD
Self-ReportedChange
Mean
PredictedChange
SD
PredictedChange
7 0.15 0.63 -0.31 0.55
8 -0.20 0.58 -0.75 0.64
9 0.31 0.47 -0.26 0.41
All 0.09 0.60 -0.44 0.59
Note.Self-reportedchangeisthedifferencebetweenself-reportedlevelandactualperformancelevelontest;predictedchangeisthedifferencebetweenpredictedlevelandactualperformancelevelontest.
ForUnit7,theself-reportedchangemeanof0.15indicatesthat,onaverage,
studentsactuallyperformedjusthigherthanthemeanlevelatwhichtheyself-reported.
Thepredictedchangemeanof-0.31indicatesthat,onaverage,studentsactuallyperformed
justlowerthanthelevelatwhichtheypredictedforthatlearningtargetonthetest.Self-
43
reportedratings(atwhatleveldoesthestudentthinksheisrightnow?)duringtheclass
beforethetestwereclosertotheactualperformanceonthetestthanpredictedscores(at
whatleveldoesthestudentexpecttobewhentakingthetestduringthenextclass?).
ForUnit8,theself-reportedmeanof-0.20indicatesthat,onaverage,students
actuallyperformedjustlowerthanthemeanlevelatwhichtheyself-reported.The
predictedmeanof-0.75indicatesthat,onaverage,studentsactuallyperformedalmostone
levellowerthanthelevelatwhichtheypredictedforthatlearningtargetonthetest.Self-
reportedlevels(atwhatleveldoesthestudentthinkheisrightnow?)duringtheclass
beforethetestwereclosertotheactualperformanceonthetestthanpredictedlevels(at
whatleveldoesthestudentexpecttobewhentakingthetestduringthenextclass?).
ForUnit9,theself-reportedmeanof0.31indicatesthat,onaverage,students
actuallyperformedjusthigherthanthelevelatwhichtheyself-reported.Thepredicted
meanof-0.26indicatesthat,onaverage,studentsactuallyperformedjustlowerthanthe
levelatwhichtheypredictedforthatlearningtargetonthetest.Studentswerebetterat
self-reportingthanpredicting.ByUnit9,predictedratings(atwhatleveldoesthestudent
expecttobewhentakingthetestduringthenextclass?)wereclosertotheactual
performanceonthetestthanpredictedscores(whereisthestudentduringtheclassperiod
beforethetest?).
Apairedt-testwasconductedtocompareUnit7self-reportedchange(M=0.15,
SD=0.63)andpredictedchange(M=-0.31,SD=0.55)conditions;t(40)=6.83,p<0.0001.A
pairedt-testwasconductedtocompareUnit8self-reportedchange(M=-0.20,SD=0.58)
andpredictedchange(M=-0.75,SD=0.64)conditions;t(40)=8.89,p<0.0001.Apairedt-
testwasconductedtocompareUnit9self-reportedchange(M=0.31,SD=0.47)and
44
predictedchange(M=-0.26,SD=0.41)conditions;t(80)=5.87,p<0.0001.Apairedt-test
wasconductedtocompareallself-reportedchange(M=0.09,SD=0.60)andpredicted
change(M=-0.44,SD=0.59)conditions;t(122)=14.6,p<0.0001.
Foreachunitindividuallyandforallunitstogether,thedifferenceinpredicted
changeandself-reportedchangewasstatisticallysignificant.Thenullhypothesisis
rejected.Onaverage,studentself-reportedlevelswereclosertoactualperformance.
Studentsovershottheirpredictedlevelbyaboutone-halflevel.
Question2
Arethereinterventionsthatimprovestudentpredictionsforhowtheyexpecttoperform
onatest?
Question2.1
Doworkedexampleshaveanyeffectonhowclosestudentpredictedlevelistoactual
performancelevel?
H0:Forstudentswhoreceivedworkedexamples,themeandifferenceinactual
performancelevelandstudentpredictedlevelisequaltothemeandifferenceforstudents
whodidnotreceiveworkedexamples.
ForUnits8and9,theteacherprovidedexampleproblemsofwhateachlearning
targetlookslikeateachlevelalongwiththeworkedsolutions(seeAppendixH.)The
predictedratingswereanalyzedusingaone-wayANOVAwithstatisticalsignificancesetat
analphalevelof0.05tomeasuretheinfluenceoftheindependentvariable,unitnumber,
onthedependentvariable,predictedchange(seeTable4.9).
45
Table4.9
One-WayAnalysisofVarianceofPredictedChangebyUnit
Source df SS MS F p
Betweengroups 2 5.92 2.96 10.12 0.000087
Withingroups 120 35.08 0.29
Total 122 41.00 Note.Predictedchangeisthedifferencebetweenpredictedlevelandactualperformancelevelontest.Sincep<.01,pairsofgroupswereanalyzedusingaTukey-KramerHSDPost-HocTest,
indicatingstatisticalsignificancebothforTest7vsTest8andalsoTest8vsTest9(see
Table4.10).
Table4.10
Tukey-KramerHSDComparisonforPredictedChangebyUnit
95%CI
ComparisonsAvsB
MeanGradeDifference(A–B)
Std.Error
LowerBound
UpperBound
Test7vsTest8 0.44* 0.17 0.15 0.72
Test8vsTest9 -0.49* 0.06 0.21 0.77
Note.Predictedchangeisthedifferencebetweenpredictedlevelandactualperformancelevelontest.*p<.01
ForUnit7,thepredictedchangemeanof-0.31indicatesthat,onaverage,students
actuallyperformedjustlowerthantheleveltheypredicted.ForUnit8,thepredicted
changemeanof-0.75indicatesthat,onaverage,studentsactuallyperformedalmostone
levellowerthantheleveltheypredicted.ForUnit9,thepredictedchangemeanof-0.26
indicatesthat,onaverage,studentsactuallyperformedjustlowerthanthelevelthey
46
predicted.Thecloserthepredictedchangeisto0indicates,thecloseractualperformance
wastothestudentprediction.Actualperformancewasclosertostudentpredictionson
Unit7whencomparedtoUnit8,andonUnit9whencomparedtoUnit8,buttherewasno
significantdifferenceonUnit7whencomparedtoUnit9.Testresultsindicateafailureto
rejectthenullhypothesis.
Question2.2
Doworkedexamplesandanemphasisonteachingstudentstheimportanceof
metacognitionhaveanyeffectonhowclosestudentpredictedlevelistoactual
performancelevel?
H0:Forstudentswhoreceivedworkedexamplesandametacognitivetreatment,themean
differenceinactualperformancelevelandstudentpredictedlevelisequaltothemean
differenceforstudentswhodidnotreceiveworkedexamplesandametacognitive
treatment.
AllstudentsreceivedworkedexamplesforUnit8andUnit9.Inaddition,oneclass,
sectionB2,receivedthemetacognitivetreatmentforUnit8.AnANOVAcomparisonon
predictedchangeonUnit8forstudentswhoreceivedthemetacognitivetreatmentversus
thosewhodidnotshowednostatisticalsignificance(p=0.116).
AllstudentsreceivedthemetacognitivetreatmentforUnit9,whichwasasecond
doseforthegroupwhoreceivedthetreatmentinUnit8.AnANOVAcomparisonon
predictedchangeonUnit9forstudentswhoreceivedthemetacognitivetreatmentinboth
Unit8andUnit9versusthosewhodidnotshowednostatisticalsignificance(p=0.168).
Insteadofonlycomparingstudentswithinindividualunits,theresearcher
comparedallpredictedchangeresultswherestudentsreceivedthemetacognitive
47
treatmenttoallpredictedchangeresultswherestudentsdidnotreceivethemetacognitive
treatment.AllstudentsforUnit9andsectionB2forunit8receivedthemetacognitive
treatment.Allremainingresults,whichincludedallstudentsforUnit7andsectionsA4and
B3forUnit8,didnotreceivethemetacognitivetreatment(seeTable4.11).
Table4.11
Resultsoft-testandDescriptiveStatisticsforPredictedChangebyWorkedExamplesandMetacognitiveTreatment
No Yes 95%CIforMean
Difference
M SD n M SD n t df
-0.52 0.64 66 -0.34 0.49 57 -0.18,0.21 -1.78* 121Note:predictedchangeisthedifferencebetweenpredictedlevelandactualperformancelevelontest.*p<.05
Aone-tailedt-testindicatesstatisticalsignificancebetweenstudentswhoreceivedthe
metacognitivetreatmentandthosewhodidnot.Thenullhypothesisisrejected.Students
whoreceivedtheworkedexamplesandametacognitivetreatmentpredictedcloserto
actualperformancewhencomparedtostudentswhodidnotreceivetheworkedexamples
andmetacognitivetreatment.
Question2.3
Isthereadifferencebetweenengineeringandnon-engineeringstudentsonhowclose
studentpredictedlevelistoactualperformancelevel?
H0:Forstudentswhoareinengineering,themeandifferenceinactualperformancelevel
andstudentpredictedlevelisequaltothemeandifferencefornon-engineeringstudents.
Engineeringstudentshaveusedlearningtargetsnotonlyinmathbutalsoin
engineering.Atwo-tailedt-testcomparisonforpredictedchangeonUnit7forstudents
whowereinengineeringclassesversusthosewhowerenotshowednostatistical
significance(p=0.481).Atwo-tailedt-testcomparisonforpredictedchangeonUnit8for
48
studentswhowereinengineeringclassesversusthosewhowerenotshowednostatistical
significance(p=0.779).Atwo-tailedt-testcomparisonforpredictedchangeonUnit9for
studentswhowereinengineeringclassesversusthosewhowerenotshowednostatistical
significance(p=0.526).Atwo-tailedt-testcomparisonforpredictedchangeforstudents
whohadreceivedtheworkedexamplesandmetacognitivetreatmentshowednostatistical
significancebetweenengineeringstudentsandnon-engineeringstudents(p=0.591).Test
resultsindicateafailuretorejectthenullhypothesis.
Question2.4
Doworkedexamplesandmetacognitivestrategieshaveanyeffectonhowclosestudent
predictedlevelistoactualperformancelevelforsubgroupsofstudents,basedon
particularself-reportedlevelsandpredictedlevels?
H0:Forstudentsinsubgroupsofparticularself-reportedandpredictedlevels,themean
differenceinactualperformancelevelandstudentpredictedlevelforthosewhoreceived
workedexamplesandametacognitivetreatmentisequaltothemeandifferenceforthose
whodidnot.
SubgroupWhoSelf-ReportedLevel3andPredictedLevel4
Therewere615self-reportedandpredictedratingsusedinthestudy.28%ofthose
ratingswerestudentswhoself-reportedLevel3andpredictedLevel4.Theresearcher
determinedwhetherthosestudentsperformedlowerthantheself-reportedLevel3,
performedatthereportedLevel3,orperformedatthepredictedLevel4.Achi-squaretest
ofindependencewascalculatedcomparingthefrequencyofthosewhoperformedlower
thantheself-reportedLevel3,performedattheself-reportedLevel3,orperformedatthe
predictedLevel4,forUnit7&Unit8toUnit9(seeTable4.12andFigure4.1).
49
Table4.12
ActualPerformanceofStudentsSelf-ReportingLevel3andPredictingLevel4
Unit PerformedatLowerLevel1orLevel2
PerformedatSelf-Reported
Level3
PerformedatPredictedLevel4
Total
7&8 23(24%) 46(47%) 28(29%) 97(57%)
9 5(7%) 35(48%) 33(45%) 73(43%)
Total 28(16%) 81(48%) 61(36%) 170(100%)
Note.c2=10.29*,df=2.Numbersinparenthesesindicatecolumnpercentages.*p=.005823
Figure4.1
TestResultsforSelf-ReportingLevel3andPredictingLevel4
Chi-squareresultsshowastatisticallysignificantdifferenceinself-reportedratingsof3
andpredictedratingsof4.Thenullhypothesisisrejected.ForUnit9,whenallstudents
receivedboththeworkedexamplesandmetacognitivetreatment,thechi-squaretest
showsthat36%ofstudentswereexpectedtoreachLevel4,but45%ofstudentsactually
50
reachedit.ButonUnit7andUnit8,withoutbothtreatments,only29%ofstudentsactually
reachedLevel4whenthechi-squaretestshowsthat36%wereexpectedtodoso.Onthe
Unit9test,thechi-squaretestshowsthat16%ofstudentswereexpectedtoperformlower
thanLevel3,butonly7%ofstudentsactuallyperformedlower.ForUnit7andUnit8,
whenstudentsdidnotreceivetheworkedexamplesandmetacognitivetreatment,thechi-
squaretestshowsthatonly16%ofstudentswereexpectedtoperformlowerthanLevel3,
but,infact,24%ofstudentsdidperformlowerthanLevel3.
SubgroupWhoSelf-ReportedLevel2andPredictedLevel3
Therewere615self-reportedandpredictedratingsusedinthestudy.21%ofthose
ratingswerestudentswhoself-reportedLevel2andpredictedLevel3.Theresearcher
determinedwhetherthosestudentsperformedlowerthantheself-reportedLevel2,
performedattheself-reportedLevel2,performedatthepredictedLevel3,orperformedat
Level4,whichwashigherthanpredicted.Achi-squaretestofindependencewascalculated
comparingthefrequencyofthosewhoperformedstudentsperformedlowerthantheself-
reportedLevel2,performedattheself-reportedLevel2,performedatthepredictedLevel
3,orperformedhigherthanpredictedatLevel4,forUnit7&Unit8toUnit9(seeTable
4.13andFigure4.2).
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Table4.13
ActualPerformanceofStudentsSelf-ReportingLevel2andPredictingLevel3
Unit PerformedLowerLevel1
PerformedatSelf-Reported
Level2
PerformedatPredictedLevel3
PerformedHigherLevel4
Total
7&8 8(11%) 15(21%) 36(50%) 13(18%) 72(55%)
9 2(3%) 9(16%) 40(69%) 7(12%) 58(45%)
Total 10(8%) 24(18%) 76(58%) 20(15%) 130(100%)
Note.c2=5.67*,df=3.Numbersinparenthesesindicatecolumnpercentages.*p=.1289
Figure4.2
TestResultsforSelf-ReportingLevel2andPredictingLevel3
EventhoughahigherpercentageofstudentsperformedattheirpredictedLevel3or
performedhigherthanpredictedatLevel4duringUnit9,whenstudentshadtheworked
examplesandmetacognitivetreatment,chi-squareresultsshownostatisticalsignificance
forstudentswhoself-reportedLevel2andpredictedLevel3betweenthosewhoreceived
52
treatmentsandthosewhodidnot.Testresultsindicateafailuretorejectthenull
hypothesis.
Theresearcherelectednottoexaminemorecloselythetenratingsthatself-
reportedLevel1andpredictedLevel2orLevel3,sincetheycomprisedlessthan2%ofall
ratings.Eightofthoseratingsweremadebythesametwostudents.
SubgroupWhoSelf-ReportedLevel2orLevel3andPredictedtheSameLevel
Therewere615self-reportedandpredictedratingsusedinthestudy.26%ofthose
ratingswerestudentswhoself-reportedLevel2orLevel3andpredictedthesamelevel.
Foreachunittest,theresearcherdeterminedwhetherthosestudentsperformedlower
thantheself-reportedlevel,performedattheself-reportedlevel(whichwasequivalentto
performingatthepredictedlevel),orperformedatahigherlevel).Achi-squaretestof
independencewascalculatedcomparingthefrequencyofthosewhoperformedlowerthan
theself-reportedlevel,performedattheself-reportedlevel(whichwasthesameasthe
predicitedlevel),orperformedatahigherlevel,forUnit7&Unit8toUnit9(seeTable4.14
andFigure4.3).
53
Table4.14
ActualPerformanceofStudentsSelf-ReportingLevel2orLevel3andPredictingtheSameLevel
Unit PerformedatLowerLevel
PerformedatSelf-Reported/PredictedLevel
PerformedatHigherLevel
Total
7&8 27(29%) 44(48%) 21(23%) 92(59%)
9 7(11%) 47(72%) 11(17%) 65(41%)
Total 34(22%) 91(58%) 32(20%) 157(100%)
Note.c2=10.66*,df=3.Numbersinparenthesesindicatecolumnpercentages.*p=.004843
Figure4.3
TestResultsforSelf-ReportingLevel2orLevel3andPredictingtheSameLevel
Chi-squareresultsshowastatisticallysignificantdifferenceinresultsforself-reported
ratingsofLevel2orLevel3andpredictedratingsofthesamelevel.Thenullhypothesisis
rejected.While29%ofstudentswhodidnotreceivethetreatmentsperformedatalevel
lowerthanself-reportedorpredicted,thechi-squaretestshowsthatonly22%were
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expectedtodoso.Whileonly11%ofstudentswhodidreceivethetreatmentsperformedat
alevellowerthanself-reportedorpredicted,thechi-squaretestshowsthat22%were
expectedtodoso.
Therewereonly3ratingsforstudentswhoself-reportedLevel1andpredictedthe
samelevel.Theresearcherelectednottofurtheranalyzethoseratingsbecausethey
comprisedlessthan1%ofallratings.
SubgroupWhoSelf-ReportedLevel4andPredictedtheSameLevel
Therewere615self-reportedandpredictedratingsusedinthestudy.20%ofthose
ratingswerestudentswhoself-reportedLevel2orLevel3andpredictedthesamelevel.
Foreachunittest,theresearcherdeterminedwhetherthosestudentsperformedlower
thanthereportedlevel,performedatthereportedlevel(whichwasequivalentto
performingatthepredictedlevel),orperformedatahigherlevel.Achi-squaretestof
independencewascalculatedcomparingthefrequencyofthosewhoperformedlowerthan
theself-reportedlevel,performedattheself-reportedlevel,orperformedatahigherlevel,
forUnit7&Unit8toUnit9(seeTable4.14andFigure4.3).
55
Table4.15
ActualPerformanceofStudentsSelf-ReportingLevel4andPredictingtheSameLevel
Unit PerformedatLowerLevel
PerformedatSelf-Reported/PredictedLevel4
Total
7&8 49(58%) 35(42%) 84(69%)
9 11(29%) 27(71%) 38(31%)
Total 60(49%) 62(51%) 122(100%)
Note.c2=9.04*,df=1.Numbersinparenthesesindicatecolumnpercentages.*p=.002642
Figure4.4
TestResultsforSelf-ReportingLevel4andPredictingtheSameLevel
Chi-squareresultsshowastatisticallysignificantdifferenceinresultsforself-reported
ratingsofLevel4andpredictedratingsofthesamelevel.Thenullhypothesisisrejected.
While58%ofstudentswhodidnotreceivethetreatmentsperformedatalevellowerthan
self-reportedorpredicted,thechi-squaretestshowsthatonly49%wereexpectedtodoso.
Whileonly29%ofstudentswhodidreceivethetreatmentsperformedatalevellowerthan
56
self-reportedorpredicted,thechi-squaretestshowsthat50%wereexpectedtodoso.
Additionally,while71%ofstudentswhodidreceivethetreatmentsreachedthepredicted
level,thechi-squaretestshowsthatonly50%wereexpectedtodoso.
SURVEYRESULTS
Question3
Whatarestudentperceptionsaroundusinglearningtargetstoinformstudentprogressin
learning?
Ofthe65studentswhoagreedtoparticipateintheresearchstudy,47respondedto
aGoogleFormsurveytosharehowtheyusedlearningtargetsandwhetherratingtheir
progressand/ortheworkedexampleswerehelpfulintheirlearning(seeAppendixK).
StudentresponsestoLikertScalequestionsaroundtheirperceptionsonusing
learningtargetsareincludedinFigure4.5.
Figure4.5
StudentperceptionsonUsingLearningTargets,LikertScaleQuestions
Studentresponsestoyesornoquestionsaroundtheirperceptionsonusinglearning
targetsareincludedinFigure4.6.
57
Figure4.6
StudentperceptionsonUsingLearningTargets,Yes/NoQuestions
ThelearningtargetsareincludedoneachtestinMs.Baird’sprecalculusclass(see
AppendixF).Whenstudentswereaskedhowtheyusethelearningtargetswhilethey
testing,92%ofstudentsusetheminsomeway–toknowwhatkindofproblemswillbein
theupcomingsection,toknowwhichskillstousefortheproblemsinthesection,toknow
whatthegoalofthesectionis.Onestudentnotedthatthelearningtargetsmakeitmore
clearwhattheteacherislookingforineachsection.Anothernotedthatthelearningtargets
arearemindertothestudentthatshehasanunderstandingofwhattheylearnedduring
class.
Whenasked,“whatmighthelpyouinyourlearning?”overhalfofthestudents
acknowledgedthatstudyingandpracticingwouldbehelpful.Somecompletely
contradictedeachotherintheirneeds:moregroupwork,morelecture,moreindividual
assistancefromtheteacher.Afewstudentswrotespecificallyaboutlearningtargets.One
studentsaidheshouldstudymoretorealizewhichskillneedsattentionandthenlearn
moreaboutthatskill.Anothersuggestedthatteachingthelessonsintheorderofthe
58
learningtargetswouldbehelpful.Anotherstudentsaidthathavingthelearningtargets
categorizedonthetestwashelpfulbutwantedmoreclarityaroundwhichlearningtarget
wasthefocusofwhichlesson.Onestudentsuggestedthatstudyingstrategiesforsolving
problemscorrespondingtoeachlearningtargetwouldhelpinherunderstandingofthe
material.Onestudentnotedthatitwouldbehelpfultoknowhowtoimprovehisskillsfor
eachsectionafteraquiz.SeeAppendixLforcompletesurveyresponsestothetwoopen-
endedquestions.
INTERVIEWRESULTS
Question4
Whatareteacherperceptionsaroundusinglearningtargetstoinformstudentprogressin
learning?
TheresearcherinterviewedMs.Bairdbyphoneaftercollectingallstudentdatato
findoutherthoughtsontheresearchstudyandtoseewhataspects,ifany,shemight
continueduringanothersemester,class,orschoolyear(seeAppendixM).Sheshared
anecdotalevidencearoundstudentsratingtheirprogresstowardslearningtargets,noting
thatmanystudentsdiscussedtheirratingswitheachotherandsaid,“Ireallyneedtolook
atthisbeforenexttime[testday].”Studentsnotonlyhadtherealizationthattheywerenot
wheretheyneededtobebutalsodiscussedwhattheyneededtodotogetthere.In
particular,shenoticedthatseveralstudentsratedthemselvesasLevel3butwantedtobe
Level4.TheyendedupgettingLevel4onthetest,sotheyeitherwenthomeandstudied
whattheyneededtoknowortheyhadnotgiventhemselvescreditforwhattheyalready
knew.
59
Ms.BairdpostedtheworkedexamplesforUnit8inCanvas,thelearning
managementsystemfortheclass.Shementionedtothestudentsthattheywereavailable,
butshedidnotemphasizetheirimportanceinpreparingfortheunittest.Whenstudents
workedontheirtestcorrectionsforUnit8inclassafterthetest,shesuggestedthatthey
pulluptheworkedexamplesandusethemastheycorrectedtheirtests.ForUnit9,several
studentsprintedouttheworkedexamplesandusedtheminclasseachday.Afewstudents
askedquestionsabouttheworkedexamplesduringzeroblock.Somestudentsusedthe
workedexampleswhiletheywereself-reportingandpredictingtheirlearningtargetlevels.
Shenoticedthattheworkedexampleshelpedsomestudentswhodidnototherwiseknow
wheretostarttolearnwhattheyneededtoknow.
Ms.BairdemphasizedtheimportanceofmetacognitionwithoneclassduringUnit8
andwithallclassesduringUnit9.Sheandothermembersofthemathematicsdepartment
oftenaskstudentstothinkaboutwheretheyareintheirlearning,butshenoticedthatself-
reportingandpredictingtheirlearningtargetlevelsmadestudentthinkingmorespecific.
Theratingspinpointedforstudentswheretheywereandwheretheywantedtobeand
helpedthemrealizethattheystillhadtimetodosomethingabouttheirrating.Ms.Baird
overheardstudentsaskingeachotherwheretheywereandwheretheywantedtobe.
Ms.Bairdplanstocontinueaskingstudentstoratetheirprogresstowardslearning
targets,andshebelievesthatstartingoutwithUnit1willhelpstudentsbecome
accustomedtothepracticeandtakeitmoreseriously.Shenotedthatwritingleveled
workedexamplesforeachunittakesalotoftime,butshethinksthatthetimeisworthit.In
precalculus,shealreadyhasworkedexamplesforUnits8and9,andsosheplanstowrite
themforotherunitsnextyear.
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OneoftheAlgebra1teacherswhoworkswithMs.Bairdmentionedthather
students’gradeswerenotgreat,andsoMs.Bairdsuggestedthatsheconsiderhaving
studentsratethemselvestogivethemtimeandspacetoreflectontheirlearningandthink
aboutimproving.
SUMMARY
Thischapterreportedtheresultsofthismixedmethodsstudy,seekinginsighton
whetherstudents’actualperformancelevelsontestdayareclosertothestudentpredicted
levelsorclosertostudents’self-reportedlevelsforalearningtargetandwhether
treatmentssuchasworkedexamplesandanemphasisonmetacognitionimprovethe
predictions.Thenextchapterwillsummarizetheresults,considerwhatconclusionscanbe
madeandwhy,considerlimitationsofthestudy,andmakesuggestionsaboutfurther
researchonthistopic.
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CHAPTERFIVE:DISCUSSION
Toooften,inclassroomseverywhere,studentsdonotknowhowtorespondwhen
theyareasked,“Whatareyoulearningabouttodayinclass?”Toooften,inclassrooms
everywhere,teachersareoffendedbystudentswhoask,“Isthisgoingtobeonthetest?”
Establishingandsharinglearninggoalsandtargetswithstudentscanalleviatesomeofthe
tensionthatcomesbetweenstudentsandteachersandtheaforementionedquestions,but
teachersandstudentsoftendonotknowwheretostart.
Thisresearchstudysoughttodeterminehowwellstudentspredicttheirexpected
successforlearningtargetsonatest.Self-reportingprogresstowardslearningtargetsand
settinganexpectationforsuccesshasaneffectsizeof1.44,oneofthehighesteffectsizes
onstudentachievement(Hattie,Fisher,&Fray,2017).Theteacherintheresearchstudy
askedstudentstoself-report(atwhatleveldoesthestudentthinksheisrightnow?)and
predict(atwhatleveldoesthestudentexpecttobewhentakingthetestduringthenext
class?)oneachlearningtargetasLevel1-beginning,Level2-progressing,Level3-
proficient,orLevel4-exceptionaltheclassperiodbeforetheytakeatest(seeAppendixJ).
Whenstudentsknowwhatthelearningtargetis,theycancomparewheretheythink
theyaretowherethelearningtargetsuggeststheyshouldbe.Whentheyarenotwhere
theyshouldbeyet,theincongruousprogressspursstudentstotakeactionontheir
learning.Whenstudentsknowhowtheywillknowwhentheyreachthelearningtarget,
theyarebetterabletomonitortheirprogresstowardsmeetingit(Hattie,Fisher,&Fray,
62
2017).Inordertorealizethe1.44effectsizefromself-reportedgrades/student
expectations,teachersmustensurethatstudentsnotonlyknowwhatthelearningtargetis
butalsohowtoreachthelearningtarget.
Thestudyhypothesizedthattreatmentssuchasworkedexamplesandanemphasis
onteachingstudentstheimportanceofmetacognitionnotonlyhelpstudentsknowwhat
thelearningtargetisbutalsohowtoreachit,thushavingapositiveeffectonstudent
successpredictingtheirexpectedsuccessonalearningtarget,and,infact,confirmedthat
workedexamplesandmetacognitivestrategiesdocontributetohowwellstudentspredict
theirexpectedsuccess.ThroughquestionsforastudentGoogleformandateacher
interview,theresearcheralsosoughttodeterminestudentandteacherperceptionsaround
usinglearningtargetstoinformstudentprogressinlearning.Inconsideringhowwell
studentspredictedthelevelatwhichtheyexpectedtoperformforeachlearningtargetand
whattreatmentsmightimprovepredictedsuccess,aseriesofrelatedquestionswas
examined.
1. Arethestudents’actualperformancelevelsontestdayclosertothestudent
predictedlevelsorclosertothestudents’self-reportedlevels(wheretheythink
theyareonthedaybeforethetest)?
H0:Themeandifferenceinactualperformancelevelandstudentpredictedlevelis
equaltothemeandifferenceinactualperformancelevelandtheself-reportedlevel
(wheretheythinktheyareonthedaybeforethetest).
2. Arethereinterventionsthatimprovestudentpredictionsforhowtheyexpectto
performonatest?
63
2.1 Doworkedexampleshaveanyeffectonhowclosestudentpredictedlevelisto
actualperformancelevel?
H0:Forstudentswhoreceivedworkedexamples,themeandifferenceinactual
performancelevelandstudentpredictedlevelisequaltothemeandifferencefor
studentswhodidnotreceiveworkedexamples.
2.2 Doworkedexamplesandanemphasisonteachingstudentstheimportanceof
metacognitionhaveanyeffectonhowclosestudentpredictedlevelistoactual
performancelevel?
H0:Forstudentswhoreceivedworkedexamplesandametacognitivetreatment,the
meandifferenceinactualperformancelevelandstudentpredictedlevelisequalto
themeandifferenceforstudentswhodidnotreceiveworkedexamplesanda
metacognitivetreatment.
2.3 Isthereadifferencebetweenengineeringandnon-engineeringstudentsonhow
closestudentpredictedlevelistoactualperformancelevel?
H0:Forstudentswhoareinengineering,themeandifferenceinactualperformance
levelandstudentpredictedlevelisequaltothemeandifferencefornon-engineering
students.
2.4 Doworkedexamplesandmetacognitivestrategieshaveanyeffectonhowclose
studentpredictedlevelistoactualperformancelevelforsubgroupsofstudents,
basedonparticularself-reportedlevelsandpredictedlevels?
H0:Forstudentsinsubgroupsofparticularself-reportedandpredictedlevels,the
meandifferenceinactualperformancelevelandstudentpredictedlevelforthose
64
whoreceivedworkedexamplesandametacognitivetreatmentisequaltothemean
differenceforthosewhodidnot.
3. Whatarestudentperceptionsaroundusinglearningtargetstoinformstudent
progressinlearning?
4. Whatareteacherperceptionsaroundusinglearningtargetstoinformstudent
progressinlearning?
CONCLUSIONS
Thestudyconfirmsthehypothesisthattreatmentssuchasworkedexamplesand
metacognitivetreatmentcanhaveapositiveimpactonstudentsuccesspredictingtheir
progresstowardsalearningtarget.
Question1
Arethestudents’actualperformancelevelsontestdayclosertothestudentpredicted
levelsorclosertothestudents’self-reportedlevels(wheretheythinktheyareontheday
beforethetest)?
Hattiesuggeststhatstudentsknowhowtheyaregoingtoperformonatest.When
giventheopportunitytoself-reporttheirperformancelevelonalearningtarget,students
setsafeexpectations.(Hattie,May2012).ThestudentsinthisstudyperformedasHattie
suggests.Studentsself-reported(atwhatleveldoesthestudentthinkheisrightnow?)and
predicted(atwhatleveldoesthestudentexpecttobewhentakingthetestduringthenext
class?)theirlevelforeachlearningtargetonthedaybeforethetest.Themeanofallself-
reportedchangeforstudentsinthisstudyis0.09,whichindicatesthat,onaverage,
studentsactuallyperformedaroundthesamelevelthattheyself-reported.Themeanofall
predictedchangeforstudentsinthisstudyis-0.44,whichindicatesthat,onaverage,
65
studentsactuallyperformedataboutone-halfofalevellowerthantheypredicted(see
Table4.8).Self-reportedchangeiscloserto0withoutperforminglowerthanreported.
Thepredictedchangeshowsthatstudentsexpectedtoimprovetheirprogress
towardsmeetingthelearningtargets.Didthestudentsknowhowtochange?Teachers
shouldprovidestudentsclearindicationsofwhatitmeanstomeetalearningtargetsothat
studentswillknowhowtoimprove(Hattie,Fisher,&Frey,2017,p.57).
Question2
Arethereinterventionsthatimprovestudentpredictionsforhowtheyexpecttoperform
onatest?
Theresearcherconsiderednextwhetherprovidingstudentswithworkedexamplesmight
improvetheirpredictions.
Question2.1
Doworkedexampleshaveanyeffectonhowclosestudentpredictedlevelistoactual
performancelevel?
Teacherscanhelpstudentsbetterunderstandlearningtargetsbysharingwith
studentshowthelearningtargetwillbeassessed.Sharingexampletestproblemsisan
idealwaytoimprovestudentunderstandingofthelearningtarget.Sharinganovice
workedexamplealongsideaproficientworkedexamplecanalsoilluminatestudent
understandingofthelearningtarget(Popham,2008).
ForUnits8and9,theteacherprovidedexampleproblemsofwhateachlearning
targetlookslikeateachlevelalongwiththeworkedsolutions(seeAppendixH).Actual
performancewasclosertostudentpredictionsonUnit7whencomparedtoUnit8,andon
Unit9whencomparedtoUnit8,buttherewasnosignificantdifferenceonUnit7when
66
comparedtoUnit9.ForUnit8,Ms.BairddistributedtheworkedexamplesviaCanvas,the
classlearningmanagementsystem.Shealertedstudentsatthebeginningoftheunitthat
workedexampleswereposted,butshedidnotovertlyencouragestudentstousethe
workedexamples.Shedidnotnoticemanystudentstakeadvantageofusingtheworked
examplesthroughoutUnit8tobetterunderstandthelearningtargets
WhenstudentsreceivedtheirUnit8testsbacktocorrectthem,Ms.Baird
encouragedstudentstousetheworkedexamples.Studentsengagedinself-explainingthe
stepsintheworkedexamplesastheycomparedtheexamplestothemissedproblemson
thetestandcorrectedthemissedproblems.AttheendofUnit8,studentssawthe
advantageofusingtheworkedexampleswhencorrectingtheUnit8test,whichcould
explainwhytherewasstatisticalsignificancebetweenpredictedchangeonUnit9when
comparedtoUnit8,butitdoesnotexplainwhytherewasnosignificantdifferencein
predictedchangefromUnit7toUnit9andwhythedifferenceinpredictedchangefrom
Unit7toUnit8wasreversed.Theresearchershowedearlierthatthethreetestswerenot
statisticallydifferent(seeTables4.5and4.6),butMs.Bairdwasonprofessionalleaveaway
fromclassmorethanonedayduringUnit8,whichcouldexplaintheanomalyofresultsfor
Unit8.
Workedexampleshavebeenshowntoimprovestudentlearning,butthisresearch
studydoesnotshowdefinitivelythatworkedexamplesimprovestudentsuccesspredicting
theiractualtestperformance.Itcouldbethatstudentsneedtobemoredeliberatelytaught
howtouseworkedexamplesandnotjustprovidedworkedexamplesforthemtoimprove
studentpredictions.Isthereaninterventionthatmightworkalongsideprovidingworked
examplestopositivelyaffectstudentsuccesspredictingtheirsuccessonatest?
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Question2.2
Doworkedexamplesandanemphasisonteachingstudentstheimportanceof
metacognitionhaveanyeffectonhowclosestudentpredictedlevelistoactual
performancelevel?
AllstudentsreceivedworkedexamplesforUnit8andUnit9.Oneclass,sectionB2,
receivedthemetacognitivetreatmentforUnit8,andallstudentsreceivedthe
metacognitivetreatmentforUnit9.Studentswhoreceivedtheworkedexamplesanda
metacognitivetreatmentpredictedclosertoactualperformance(aboutone-thirdofalevel
lowerthanactualperformance)whencomparedtostudentswhodidnotreceivethe
workedexamplesandmetacognitivetreatment(morethanone-halfofalevellowerthan
actualperformance).
Establishinganormintheclassroomforalllearnerstosharewhytheyarethinking
whattheyarethinkingaboutaproblembuildsthehabitofreflectivelearningforstudents,
whichincreasesthetendencyforstudentstothinkaboutwhensomethingdoesnotmake
senseandtaketimetofigureoutwhy.Somestudentswillmorenaturallythinkabouttheir
learningthanotherstudents(Hattie,Fisher,&Frey,2017).Teachersneedtopurposefully
teachmetacognitivestrategiestotheclassandprovidedeliberateopportunitiesfor
reflectingonlearningsothatallstudentscanadvantageouslyusemetacognitivestrategies
toimprovelearning(Hattie,Fisher,&Frey,2017).
Asshowninthisresearchstudy,learningwithworkedexamplesismoreeffective
whenstudentsareencouragedtoself-explainthestepsintheproblem.Teachersare
integraltotrainingstudentshowtoself-explain(Renkl,2014).Whentheteachermodels
theuseofmetacognitivestrategiesanddiscussesthestrategieswithstudentsastheylearn
68
tousethem,studentseventuallyusethestrategiesthemselveswithoutbeingpromptedby
theteacher(Bransford,Brown,&Cocking,2001).
Question2.3
Isthereadifferencebetweenengineeringandnon-engineeringstudentsonhowclose
studentpredictedlevelistoactualperformancelevel?
Engineeringstudentshaveusedlearningtargetsnotonlyinmathbutalsoin
engineering.JustlikeinMs.Baird’sprecalculusclasses,engineeringteachersclarifyand
sharelearningtargetswithstudentsandincludelearningtargetsontheassessment,
connectedtotheassessmentitems.Engineeringstudentshadmoreextensiveexperience
usinglearningtargets,andsoitseemsthattheywouldout-predicttheirpeerswhowere
notinengineeringclasses.However,theengineeringstudentsdidnotout-predicttheir
peers.Theinterventionsusedintheirprecalculusclass–self-reportingtheleveltheythink
theyareandthenpredictingthelevelofsuccesstheyexpecttobeonatest,worked
examples,andmetacognitivestrategies–supersededanypreviouseffectthatusinglearning
targetsinmultipleclassesmighthavehad.
Question2.4
Doworkedexamplesandmetacognitivestrategieshaveanyeffectonhowclosestudent
predictedlevelistoactualperformancelevelforsubgroupsofstudents,basedon
particularself-reportedlevelsandpredictedlevels?
Duringtheteacherinterview,Ms.Bairdreportedanecdotallythatseveralstudents
ratedthemselvesasLevel3butwantedtobeLevel4.Herobservationwasthattheyended
upgettingLevel4onthetest,sotheyeitherwenthomeandstudiedwhattheyneededto
knowortheyhadnotgiventhemselvescreditforwhattheyalreadyknew.Infact,the
69
studentsweremorelikelytoreachtheirpredictedLevel4onUnit9,whentheyhadboth
theworkedexamplesandmetacognitivetreatment.Studentsweremorelikelytoperform
lowerthantheirself-reportedLevel3onUnit7andUnit8,whentheydidnothaveboth
treatments,andlesslikelytoperformlowerthantheirself-reportedLevel3onUnit9,
whichtheydidhavebothtreatments(seeFigure4.1).AsMs.Bairdnoted,itcouldbethat
thesestudentsdidnotgivethemselvescreditforwhattheyknewwhentheyself-reported
theirlevel,butthestatisticalsignificanceofwhathappenedinUnit9whencomparedto
Unit7andUnit8indicatesthattheworkedexamplesandmetacognitivetreatments
studentsmadeadifferenceforstudentswhowantedtoperformataLevel4-exceptionalon
thetest.
Whenstudentsarenotwheretheyshouldbeyet,theincongruousprogressspurs
studentstotakeactionontheirlearning.Whenstudentsknowhowtheywillknowwhen
theyreachthelearningtarget,theyarebetterabletomonitortheirprogresstowards
meetingit(Hattie,Fisher,&Fray,2017).Thepositiveresultsforthisgroupofstudents
raisethequestionofwhytheywerebetterabletoreachtheirpredictedlevelthanother
groupsofstudents.ThesestudentswerenotsatisfiedwithLevel3-proficient.Theywanted
tobeLevel4-exceptional.Itcouldbethatthemetacognitivetreatmentspurredthisgroup
toreflectonwhattheydidnotknowandtakeactiontodosomethingabouttoimprove
theirlearning.ItcouldbethattheLevel4workedexamplesprovidedjustenoughofa
challengeforthisgrouptoworkalittlehardertobetterunderstandthelearningtarget.
Therewasnostatisticalsignificanceinself-reportedratingsof2andpredicted
ratingsof3forstudentswhoreceivedtheworkedexamplesandmetacognitivetreatment
(seeTable4.13andFigure4.2),whichraisesseveralquestions.Didthesestudentstryto
70
makeuseoftheworkedexamples,butdidnotknowhow?Howmanyofthemsoughtout
extrahelpfromtheteacherorotherstudents?Wiliamcallsout“activatinglearnersas
instructionalresourcesforeachother”and“activatinglearnersasownersoftheirlearning”
twoofhisfivekeystrategiesofformativeassessment(2011,p.2).Whatadditional
interventionsimprovesuccessforstudentswhoself-reportedLevel2andpredictedLevel
3?
Ofthestudentswhoself-reportedratingsofLevel2orLevel3andpredictedratings
ofthesamelevel,studentswhoreceivedtheworkedexamplesandmetacognitive
treatmentwerelesslikelytoscorelowerthanself-reportedandpredicted.Thosewhodid
notreceivetheworkedexamplesandmetacognitivetreatmentweremorelikelytoscore
lowerthanself-reportedandpredicted(seeTable4.14andFigure4).Whengiventhe
opportunitytoself-reporttheirprogresstowardsalearningtarget,studentssetsafe
expectations(Hattie,May2012).AssumingHattie’sassertion,itappearsthattheworked
examplesandmetacognitivetreatmentplayedaroleinensuringthatthestudentsmetor
exceededthatsafelevel,ratherthanfallingbelowthesafelevel.
Studentsweremorelikelytoreachtheirself-reportedandpredictedLevel4onUnit
9,whentheyhadboththeworkedexamplesandmetacognitivetreatment.Itappearsthat
theworkedexamplesandmetacognitivetreatmentplayedaroleinensuringthatthe
studentswhoself-reportedandpredictedLevel4actuallyperformedatthatlevel.Students
willhaveabetterideaofwhattheyaretolearnwhenlearningtargetsareembeddedwithin
learningprogressions(Popham,2008).Itcouldbe,morespecifically,thatthesestudents
hadabetterideathattheyhadreachedLevel4-exceptionalbecauseoftheprogressionof
workedexamplesfromlevels1through4.
71
Question3
Whatarestudentperceptionsaroundusinglearningtargetstoinformstudentprogressin
learning?
“Thelearningtargetarticulatesforstudentswhattheyaretolearnandatthesame
timeprovidesinsightastohowstudentswillbeassessed”(Kanold&Larson,2012,p.49).
Overhalfofthestudentsreportedthattheyfrequentlyoralwayspayattentiontothe
learningtargetsineachunit(seeFigure4.5).
Whatisgoingtobeonthetestshouldnotbeasurprisetostudents.Learningtargets
shouldinformteacherswhatcontent-aligneditemstoputonthetestandshouldinform
studentswhatcontent-aligneditemswillbeonthetest.Almostallstudentshavenoticed
thatthelearningtargetsareincludedonthetest,andabouthalfreportedthathavingthe
learningtargetsonthetestfrequentlyoralwayshelpsthemwhiletheyaretakingthetest
(seeFigure4.5).
Teacherscanhelpstudentsbetterunderstandlearningtargetsbysharingwith
studentshowthelearningtargetwillbeassessed.Sharingexampletestproblemsisan
idealwaytoimprovestudentunderstandingofthelearningtarget.Sharinganovice
workedexamplealongsideaproficientworkedexamplecanalsoilluminatestudent
understandingofthelearningtarget(Popham,2008).Ms.Bairdsharedleveledworked
examplesforeachlearningtargetduringUnit8andUnit9(seeAppendixH).
ForUnit8,Ms.BairdpostedtheleveledworkedexamplesthroughCanvas,theclass
learningmanagementsystem,butshedidnottalkwithstudentsabouthowtheleveled
workedexamplesmightbehelpfultothemintheirlearning.Shereportedduringthe
teacherinterviewthatshedidnotthinkmanystudentsusedthemforUnit8.However,
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whenstudentsworkedoncorrectingtheUnit8test,sheencouragedstudentstousethe
workedexamples.Thatencouragementandapurposefulmetacognitivetreatmentwith
studentsforUnit9,causedmorestudentstomakeuseoftheworkedexamplesduringUnit
9.Bytheendofthestudy,three-fourthsofstudentsreportedthattheyusedtheworked
examplesinUnits8and9,andthree-fourthsofstudentsreportedthattheworked
exampleswerehelpfulinpreparingforUnits8and9.Nine-tenthsofstudentsreportedthat
havingworkedexamplesforallunitswouldbehelpful.
Self-reportingprogresstowardslearningtargetsandsettinganexpectationfor
successhasaneffectsizeof1.44,oneofthehighesteffectsizesonstudentachievement.
Whenstudentsknowwhatthelearningtargetis,theycancomparewheretheythinkthey
aretowherethelearningtargetsuggeststheyshouldbe.Whentheyarenotwherethey
shouldbeyet,theincongruousprogressspursstudentstotakeactionontheirlearning.
Whenstudentsknowhowtheywillknowwhentheyreachthelearningtarget,theyare
betterabletomonitortheirprogresstowardsmeetingit(Hattie,Fisher,&Fray,2017).
WhilethestudydoesnotaffirmthatstudentsmadealloftheconnectionsthatHattiehas
betweenstudentsknowingthelearningtarget,ratingsuccessonit,andactionstotaketo
improvesuccess,overthree-fourthsofMs.Baird’sstudentsreportedthatratingtheir
progressonthelearningtargetsbeforetheUnits7,8,and9testswashelpful,andthree-
fourthsreportedthatratingtheirprogressbeforeeveryunittestwouldbehelpful.
Question4
Whatareteacherperceptionsaroundusinglearningtargetstoinformstudentprogressin
learning?
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Studentswillhaveabetterideaofwhattheyaretolearnwhenlearningtargetsare
embeddedwithinlearningprogressions.Learningprogressionscanprovideinformation
abouttheskillsneededtoreachatargetaswellasenrichmentopportunitiesforthosewho
havealreadyreachedthetarget(Popham,2008).Writinglearningprogressionsis
challenging,time-consumingworkforteachers(Popham,2011).Duringtheteacher
interview,Ms.BairdechoedPopham’sassertionthatwritinglearningprogressionsis
challengingandtime-consuming.WhileMs.Bairddoesnotknowthatshewillhavethetime
towriteaprogressionofleveledworkedexamplesforeachlearningtargetineachunitthat
sheteachers,shedoesplantowritemoreofthem.Inparticular,Ms.Bairdexpressed
concernaboutnotbeingabletowriteaprogressionofleveledworkedexamplesforon-
levelcalculus,whichwillbeanewprepforher.WhenshefoundoutthattheAPCalculus
teacherhasbeenusingleveledworkedexamples,shethoughtthatshemightbeableto
startwiththatteacher’sworkandreviseasneededforherownstudents.Nothavingto
starttheprogressionfromscratchallayedherconcernofthetimeandexpertiseneededto
writetheprogressionofleveledworkedexamples.
Whengiventheopportunitytoself-reporttheirprogresstowardsalearningtarget,
studentssetsafeexpectations(Hattie,May2012).Ms.Bairdnotedinherinterviewthat
havingstudentspredicttheirlevelaffordedthemtherealizationthattheystillhadtimeto
dosomethingabouthowtheywouldperformonthetest.Self-reportingandpredicting
theirlevelsforeachlearningtargetpinpointedforstudentsnotonlywheretheywerebut
alsowheretheywantedtobe.
Hattiegoesontosaythatteachersshouldnothelpstudentsreachtheirpotential
levelbuthelpthemgobeyondwhattheythinktheycando(Hattie,May2012).Buildingthe
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habitofreflectivelearningforstudentsincreasesthetendencyforstudentstothinkabout
whensomethingdoesnotmakesenseandtaketimetofigureoutwhy(Hattie,Fisher,&
Frey,2017).Studentsbecomemoreinterestedinlearningwhentheycangaugetheir
progresstowardsmeetingthelearninggoalandknowwhatstepstotaketoimprove
(Sousa,2015).Connectingleveledworkedexamplesandmetacognitionnotonlyprovides
studentstheopportunitytorecognizetheirownpotentialbutalsoknowthestepstotake
toreachthatpotential.
LIMITATIONS
Inthisresearchstudy,studentshadtheopportunitytoself-report(atwhatlevel
doesthestudentthinksheisrightnow?)andpredict(atwhatleveldoesthestudentexpect
tobewhentakingthetestduringthenextclass?)oneachlearningtargetasLevel1-
beginning,Level2-progressing,Level3-proficient,orLevel4-exceptionaltheclassperiod
beforetheytakeatest.Amajorlimitationofthisstudyisthatnoinstrumentwasusedto
verifytheaccuracyofthestudentself-reportedlevel;itwasbasedonlyoneachstudent’s
evaluationofwherehethoughthewasatthattime.Theself-reportedratingsaresimply
wherestudentsthoughttheywereonthedaybeforethetest.
Aretherestudentswhothinkthatitlooksbettertoimprove?Howmanystudents
self-reportedalowerlevelthanheactuallythoughthewasatthemomentandpredicteda
higherlevelsothatitappearedthatheimprovedwhenheactuallywasatthehigherlevel
allalong?Becausetheresearcherdidnotcollectdataonstudentimpressionsforsuccessful
rating,andbecausethestudentsself-reportedratings,somedatacouldbeskewedtowards
studentsseemingtoimprovealevelwhentheywerealreadyatthatlevel.
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Studentsmustpartnerwiththeteacherinreachingtowardsthelearningtarget,and
theycanalsohelpeachotherbetterunderstandlearningtargets.“Ithelpstomakethe
studentsfullyawareofthelearningintentionsandsuccesscriteria,ofthevalueof
deliberatepractice,andofwhattodowhentheydonotknowwhattodo”(Hattie,2012,p.
111).Studentsbecomemoreinterestedinlearningwhentheycangaugetheirprogress
towardsmeetingthelearninggoalandknowwhatstepstotaketoimprove(Sousa,2015).
Inthisresearchstudy,thefocuswasonwhetherprovidingtheworkedexamplesand
metacognitivestrategiesimprovedpredictedratings.Theteachermadethestudentsaware
ofthelearningintentionsandsuccesscriteria.Abouthalfofthestudentresponsestowhat
mighthelpintheirlearningassertedthatmorepracticewouldhelp,sotheyareawarethat
deliberatepracticeisneeded.However,nostepsweretakentoensurethatstudentsknow
whattodowhentheydonotknowwhattodo.Ms.Bairdhadnotedthattheratings
pinpointedforstudentswheretheywereandwheretheywantedtobeandhelpedthem
realizethattheystillhadtimetodosomethingabouttheirrating.Onestudentnotedthat
knowinghowtoimprovewouldbehelpful.Soitappearsthattheworkedexamplesand
metacognitivetreatmentsmadestudentsawarethattheyneededtoimprovebutwithout
alwaysknowhowtoimprove.
ManyoftheratingsforUnit8didnotjivewiththeprogressionofratingsfromUnit
7toUnit9.Ms.Bairdwasonprofessionalleaveforseveraldaysduringthatunit,which
couldhaveplayedaroleinstudentsuccessonthatunit.
Halfoftheratingscollectedfromstudentsduringthisresearchstudywerestudents
whoself-reportedandpredictedthesamelevel,suchasself-reportingLevel3and
predictingLevel3orself-reportingLevel2andpredictingLevel2.Thestudentsseemedto
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alreadythinktheywerewheretheywantedtobe.Werethesestudentssatisfiedwithwhat
theythoughttheyalreadyknew?Whydidthosewhoself-reportedLevel2predictthatthey
wouldstayatLevel2ratherthantryingtoreachLevel3?Didtheythinkitwastoolateto
improve?IsthisthegroupHattiemeanswhenhesaysthatteachersshouldnothelp
studentsreachtheirpredictionbutexceedtheirprediction?
Almosthalfofthestudentsself-reportedatLevel2andpredictedLevel3orself-
reportedatLevel3andpredictedLevel4.Didthosestudentsreallyknowwhatitmeantto
beaLevel3orLevel4?Ordidtheyjustthinkthattheyneededtogetbetter?Because
studentsself-reportedtheirlevels,theresearcherhasnowayofknowinghowaccuratethe
self-reportedlevelwas.
RECOMMENDATIONSFORFUTURERESEARCH
“Ithelpstomakethestudentsfullyawareofthelearningintentionsandsuccess
criteria,ofthevalueofdeliberatepractice,andofwhattodowhentheydonotknowwhat
todo”(Hattie,2012,p.111).Studentsbecomemoreinterestedinlearningwhentheycan
gaugetheirprogresstowardsmeetingthelearninggoalandknowwhatstepstotaketo
improve(Sousa,2015).Whilethisresearchstudyensuredthatstudentswereawareofthe
learningtargetsandthevalueofdeliberatepractice,therewasnospecifictreatmentthat
addressedpossibleactionsstudentscouldtakewhentheydidnotknowwhattodo.Future
researchisneededtodeterminetreatmentsthatcanhelpstudentsknowwhatstepstotake
toimprove.Forexample,peertutoringhasaneffectsizeof0.55(Hattie,2012).Peer
tutoringcouldbeatreatmentforstudentstopurposefullyutilizewhentheydonotknow
howtoimprovetheirprogresstowardsalearningtarget.
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“Thelearningtargetarticulatesforstudentswhattheyaretolearnandatthesame
timeprovidesinsightastohowstudentswillbeassessed”(Kanold&Larson,2012,p.49).
Furtherresearchisneededwithavarietyofstudentpopulations,suchasthosewhohave
notusedlearningtargetstothinkaboutwheretheyareintheirlearningandthosewho
havenothadlearningtargetsincludedontheirtesttorealizethattheyprovideinsightinto
theassessment.Becausethestudentsinthisresearchstudyhadbeenusinglearningtargets
throughouttheunitandonthetest,theywerefartheronthepathforrecognizingtheir
importancethanstudentsforwhomlearningtargetsarenew.
Inthisstudy,studentsself-reportedtheleveltheythoughttheywerethedaybefore
thetestandpredictedtheleveltheyexpectedtobewhentakingthetestduringthenext
class.Noinstrumentwasusedtoverifytheaccuracyofthestudentself-reportedlevel;it
wasbasedonlyoneachstudent’sevaluationofwherehethoughthewasatthattime.
Hattieassertsthatteachersshouldnothelpstudentsreachtheirpredictedlevelbuthelp
themexceedtheirpredictedlevelsothatstudentsdobetterthantheythoughttheycould
andrealizethepowertheyhavetoimprovetheirlearning(Hattie,May2012).Infuture
research,usingsomesortofinstrumenttoverifytheaccuracyofthestudents’self-reported
levelscanbetterinformteachersonwaystopushstudentstogaintheconfidencethatthey
needtobelievethattheyhavecontrolovertheirlearning.
OneoftheAlgebra1teacherswhoworkswithMs.Bairdmentionedthather
students’gradeswerenotgreat,andsoMs.Bairdsuggestedthatsheconsiderhaving
studentsratethemselvestogivethemtimeandspacetoreflectontheirlearningandthink
aboutimproving.Again,furtherresearchisneededwithavarietyofstudentpopulationsin
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ordertosaywithconfidencethatworkedexamplesandmetacognitivestrategiesimprove
howwellallstudentspredicttheirprogresstowardsalearningtarget.
CONCLUSIONS
Howoftenareteacherssurprisedtofindoutattheendofalearningepisodethat
studentshavenotactuallylearned?Howoftenareteachersfrustratedbystudentswhoask,
“Isthisgoingtobeonthetest?”
Thisresearchstudybuiltontheimportanceofestablishinglearninggoalsand
clarifyingsuccesscriteriaforstudentstofindouthowteachersmightprovide
opportunitiesforstudentstouselearninggoalsandsuccesscriteriaformativelyinorderto
knowbothwhattheyhavelearnedandwhattheystillneedtoknow.Self-reporting
progresstowardslearningtargetsandsettinganexpectationforsuccesshasaneffectsize
of1.44,oneofthehighesteffectsizesonstudentachievement.Whenstudentsknowwhat
thelearningtargetis,theycancomparewheretheythinktheyaretowherethelearning
targetsuggeststheyshouldbe.Whentheyarenotwheretheyshouldbeyet,the
incongruousprogressspursstudentstotakeactionontheirlearning.Whenstudentsknow
howtheywillknowwhentheyreachthelearningtarget,theyarebetterabletomonitor
theirprogresstowardsmeetingit(Hattie,Fisher,&Fray,2017).Studentsweregiventhe
chancetoself-reporttheleveltheythoughttheywerethedaybeforethetestandpredict
thelevelatwhichtheywouldperformonthetestduringthenextclass.
Thestudyhypothesizedthattreatmentssuchasworkedexamplesandanemphasis
onteachingstudentstheimportanceofmetacognitionnotonlyhelpstudentsknowwhat
thelearningtargetisbutalsohowtoreachit,thushavingapositiveeffectonstudent
successpredictingthelevelatwhichtheywillperformonalearningtargetwhentheytake
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atest,and,infact,confirmedthatworkedexamplesandmetacognitivestrategiesdo
contributetostudentsuccesswhenpredictingthelevelatwhichtheywillperformona
learningtarget.Whenworkedexamplesandmetacognitivestrategiesarecombinedwith
theopportunityforstudentstopredictthelevelatwhichtheywillperformonalearning
target,notonlywillstudentsknowwhatisgoingtobeonthetestandhowtheyaregoing
todoonthetest,theycanusethatinformationandworktoimprovetheirlearning.
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CHAPTERSIX:ANINFORMALADDENDUM
PURPOSEANDRESEARCHQUESTIONS
“Ithelpstomakethestudentsfullyawareofthelearningintentionsandsuccess
criteria,ofthevalueofdeliberatepractice,andofwhattodowhentheydonotknowwhat
todo”(Hattie,2012,p.111).Studentsbecomemoreinterestedinlearningwhentheycan
gaugetheirprogresstowardsmeetingthelearninggoalandknowwhatstepstotaketo
improve(Sousa,2015).Self-reportingprogresstowardslearningtargetsandsettingan
expectationforsuccesshasaneffectsizeof1.44,oneofthehighesteffectsizesonstudent
achievement.Whenstudentsknowwhatthelearningtargetis,theycancomparewhere
theythinktheyaretowherethelearningtargetsuggeststheyshouldbe.Whentheyarenot
wheretheyshouldbeyet,theincongruousprogressspursstudentstotakeactionontheir
learning.Whenstudentsknowhowtheywillknowwhentheyreachthelearningtarget,
theyarebetterabletomonitortheirprogresstowardsmeetingit(Hattie,Fisher,&Fray,
2017).
Whiletheoriginalresearchstudyensuredthatstudentswereawareofthelearning
goalsandthevalueofdeliberatepractice,therewasnospecifictreatmentthataddressed
possibleactionsstudentscouldtakewhentheydidnotknowwhattodo.Futureresearchis
neededtodeterminetreatmentsthatcanhelpstudentsknowwhatstepstotaketo
improve.Forexample,peertutoringhasaneffectsizeof0.55(Hattie,2012).Peertutoring
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couldbeatreatmentforstudentstopurposefullyutilizewhentheydonotknowhowto
improvetheirprogresstowardsalearninggoal.
Intheoriginalresearchstudy,studentsself-reportedtheleveltheythoughtthey
werethedaybeforethetestandpredictedtheleveltheyexpectedtobewhentakingthe
testduringthenextclass.Figure6.1showstheprocessthatstudentscompletedforeach
learningtargetonthetest.
Figure6.1
ProcessStudentsCompletedforEachLearningTargetontheTest
Almosthalfofthestudentsself-reportedatLevel2andpredictedLevel3orself-
reportedatLevel3andpredictedLevel4.Didthosestudentsreallyknowwhatitmeantto
beaLevel3orLevel4?Ordidtheyjustthinkthattheyneededtogetbetter?Amajor
limitationoftheoriginalstudyisthatnoinstrumentwasusedtoverifytheaccuracyofthe
studentself-reportedlevel;itwasbasedonlyoneachstudent’sevaluationofwherehe
thoughthewasatthattime.Theself-reportedratingsweresimplywherestudentsthought
theywereonthedaybeforethetest.
Hattieassertsthatteachersshouldnothelpstudentsreachtheirpredictedlevelbut
helpthemexceedtheirpredictedlevelsothatstudentsdobetterthantheythoughtthey
couldandrealizethepowertheyhavetoimprovetheirlearning(Hattie,May2012).To
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actuallyimprove,studentsmustdomorethanrealizetheycanimprove;theymusttake
actionstoimprove.Revisitingtheresearchstudyallowedforusinganinstrumenttoverify
thestudents’currentlevelbeforethetestinsteadofrelyingonstudents’self-reported
levels,givingstudentstheopportunitytotrulyknowwheretheyneededtoimproveand
givingtheteacherbetterinformationonwaystopushstudentstogaintheconfidencethat
theyneedtobelievethattheyhavecontrolovertheirlearning.
Intheaddendumtothestudy,apre-testonthedaybeforethetesthelpedstudents
determinetheircurrentlearninglevelforeachlearninggoalbeforepredictingthelearning
levelatwhichtheyplannedtoperformontestday.Theteachercalledthepre-testa
“learninggoallevelquiz”,asstudentsweretakingthepre-testtodeterminetheircurrent
leveloneachofthelearninggoals,thoughstudentswerenotassignedagradeforthepre-
test.Studentswerealsoaskedquestionsaroundwhattheyplannedtodotoreachtheir
predictedlevel.Afterthetest,studentswereaskedwhattheyactuallydid,whetherit
worked,andwhattheymightdonexttime.Sincetheoriginalresearchstudyconfirmedthe
hypothesisthattreatmentssuchasworkedexamplesandmetacognitivetreatmentcan
haveapositiveimpactonstudentsuccesspredictingtheirprogresstowardsalearning
goal,theteachercontinuedtousethosetreatmentsduringtheaddendum.Thefollowing
researchquestionswereexamined.
1. Howdoesstudentperformanceonthepre-testcomparetoactualperformanceon
thetest?
2. Howdostudentpredictionscomparetostudentperformance?
3. Whenstudentsknowthattheyneedtoimprovetheirprogresstowardsalearning
goal,howdotheytrytoimprove?
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METHODOLOGY
ThisstudyalsotookplaceatNorthwestRankinHighSchool,asuburbanschoolin
RankinCountySchoolDistrictnearJackson,Mississippi.Theresearcherpartneredwitha
calculusteacherMs.Dolfandherthiry-fiveAPCalculusstudents.
Ms.Dolfassessedthelearninglevelofthestudentsonthedaybeforethetestforfive
testsduringtheschoolyear,utilizingapre-test(seeAppendixO)andaGoogleformwith
branchingquestions(seeAppendixP).Foreachlearninggoal,theformstartsfirstwitha
Level3questionthatbranchestoaLevel4questionifthestudentgetstheLevel3question
correctandaLevel2questionifthestudentgetstheLevel3questionincorrect.Students
determinetheircurrentlearninglevelforeachlearninggoalbasedonthehighestlevel
questionthestudentgetscorrect,withLevel1correspondingtolearninggoalsonwhich
thestudentgetsnocorrectresponse.MsDolfaskedstudentsnottoguessonthepre-test
andtoselectEfor“Idon’tknowhowtodothisproblem”whenneededtopotentiallylimit
thenumberofstudentswhorandomlyselectedthecorrectresponse.
Oncestudentsknewtheircurrentlevel,theyhadtheopportunitytopredictthelevel
theyexpectedtobewhentakingthetestduringthenextclassoneachlearninggoalas
Level1-beginning,Level2-progressing,Level3-proficient,orLevel4-exceptionalonthe
UnitLearningGoalsSelf-assessmentGoogleform.Studentswerealsogiventheopportunity
toselectwhattheyplannedtodotoreachthepredictedlearninglevels,suchasusing
workedexamples,studyingwithanotherstudent,gettinghelpfromateacherortutor,
watchingcalculusvideos,and/orreviewingclassnotes(seeAppendixQ).
Ms.Dolfdeterminedeachstudent’sactuallearningleveloneachlearninggoalfrom
theirperformanceontheunittest.Studentsweregiventhisinformationafterthetest,and
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whentheylookedovertheirtestswiththeirclassmatestomakecorrections,theywere
askedtoreflectonwhatactionstheytooktoreachtheirpredictedlevel,whetherithelped,
andwhattheymightdonexttimeinsteadoforinadditiontowhattheydidthistimeonthe
LearningGoalsReflectionGoogleform(seeAppendixR).Figure6.2showstheprocessthat
studentscompletedforeachlearningtargetonthetest.
Figure6.2
ProcessStudentsCompletedforEachLearningTargetontheTestDuringtheAddendum
Thisstrategywasemployedoverfiveunitsofstudy,Units2,3_1,3_2,4,and5.
LearninggoalsforallunitsarelistedinAppendixS.Table6.1showsrawdataforafew
studentsforUnit2.Therawdatawereusedtodeterminehowoftenstudentsperformed
betteronthepre-testthantheydidonthetestandhowoftenstudentsperformedator
exceededtheirpredictedlevelonthetestwhentheypredictedthattheywouldperform
higherthantheydidonthepre-test.Forexample,student200wasaLevel1onthefirst
learninggoalonthepre-test,predictedthatshewouldbeataLevel3ontestday,and
actuallyperformedatLevel2onthetest.ShewasaLevel3onthethirdlearninggoalon
thepre-test,predictedthatshewouldbeataLevel3ontestday,andactuallyperformedat
Level4onthetest.
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Table6.1
SampleStudentData,Unit2Raw
StudentNumber
2_1pre-testlevel
2_1predictedlevel
2_1actualtestlevel
2_2pre-testlevel
2_2predictedlevel
2_2actualtestlevel
2_3pre-testlevel
2_3predictedlevel
2_3actualtestlevel
2_4pre-testlevel
2_4predictedlevel
2_4actualtestlevel
200 1 3 2 1 3 3 3 3 4 1 2 3
201 3 3 1 2 3 3 3 3 2 1 2 1202 3 4 3 3 3 4 2 3 3 1 2 3
Themeanofthepre-testlevels,predictedlevels,andactualtestlevelswere
calculatedforeachstudentforeachunitandusedtocompareunits(seeTable6.2).For
example,student200hadameanlevelof1.5onthepre-test,predictedameanlevelof
2.75,andactuallyperformedatameanlevelof3onthetest.Individualstudentswere
removedfromthetotalstudentcountinanyunitforwhichtheywereabsentforthepre-
testoriftheyoptedoutofpredictingtheirlearninglevelsonthetest.
Table6.2
SampleStudentData,UnitMeansforPre-test,Predicted,andActualTestLevelsStudentNumber
Unit2pre-testmean
Unit2predictedmean
Unit2actualtestmean
Unit3_1pre-testmean Unit3_1predictedmean
Unit3_1actualtestmean
200 1.5 2.75 3 1.83 — 2.83
201 2.25 2.75 1.75 2.17 2 2.17
202 2.25 3 3.25 2.67 — 2.83
Studentsweregiventheopportunitytoselecttreatmentsthattheyplannedtodo
whilepreparingforthetest(seeTable6.3).
86
Table6.3
SampleStudentData,StudentReflectionBeforetheTestStudentNumber Whatdoyouplantodotoreachyourpredictedlearninglevels?
200 Usetheworkedexamplesmyteacherprovided,Watchcalculusvideos,Reviewclassnotes
201 Usetheworkedexamplesmyteacherprovided,Studywithanotherstudent,Watchcalculusvideos,Reviewclassnotes
202 Usetheworkedexamplesmyteacherprovided,Studywithanotherstudent,Watchcalculusvideos
Afterthetest,studentsweregiventheopportunitytoreflectonwhattreatments
theyactuallytried,whetherthetreatmentshelped,andwhattheymightdothenexttime
theyprepareforacalculustest(seeTable6.4).
Table6.4
SampleStudentData,StudentReflectionAftertheTestStudentNumber
Whatdidyoudotoreachyourpredictedlearninglevels?(selectallthatapply)
Didwhatyoudohelped?Explain. Whatmightyoudonexttimeinsteadoforinadditiontowhatyoudidthistime?
200 Usetheworkedexamplesmyteacherprovided,Watchcalculusvideos,Reviewclassnotes
Ithinkitdidhelpmedomybetterthaniwouldhavewithoutit.Ididn’tdogreatonthetest,butIthinkreviewingeverythingdefinitelyhelpedmegrasptheconceptsbetter.
IwouldreviewlikeIdidbutIwishIwouldhavegonebackoverdefinitionofderivativeproblemsbecauseIstruggledwiththose.
201
202 Usetheworkedexamplesmyteacherprovided,Studywithanotherstudent,Watchcalculusvideos
YesbecauseIwaslessconfidentontheobjectivesuntilIpracticedtheweekend
Todomathxlaweekearlier
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RESULTS
Question1
Howdoesstudentperformanceonthepre-testcomparetoactualperformanceonthetest?
Table6.5showsthemeansofthepre-testlevelmeansandactualtestlevelmeans
forallstudentsbyunit.Studentswhoweremissinganylearninggoallevelsinaunitwere
removedfromthedataforthatunit.Thedifferencebetweentheactualtestlevelmeanand
thepre-testlevelmeanof0.52forUnit3_2indicatesthat,onaverage,studentsperformed
one-halflevelhigherontheactualtestthanonthepre-test.Thedifferenceof-0.05forUnit
4indicatesthat,onaverage,studentsperformedataboutthesamelevelontheactualtest
asonthepre-test.
Table6.5
Pre-testandActualTestLevelMeansofAllStudentsbyUnitUnit Numberofstudents Pre-testLevelMean ActualLevelMean ActualLevelMean–Pre-testLevelMean
2 33 1.87 2.10 0.233_1 21 2.23 2.75 0.52
3_2 19 2.27 2.64 0.374 20 2.11 2.06 -0.05
5 20 2.28 2.61 0.33
All 113 2.12 2.39 0.28
Table6.6drillsdowntoeachindividuallearninggoaltoshowthepercentageof
studentswhohadahigherlevelonthepre-testforthatlearninggoalthedaybeforethetest
thanontheactualtest.ForUnit3_1,59%ofstudentshadahigherlevelonlearninggoal3
onthepre-testthedaybeforethetestthantheydidonthatlearninggoalontheactualtest.
Forlearninggoal3_1-5,10%ofstudentshadahigherlevelonthepre-testthantheydidon
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thatlearninggoalontheactualtest.ForUnit4,36%ofalllearninggoalratingshadahigher
levelonthepre-testthanontheactualtest.
Table6.6
StudentData,LearningGoalswithStudentPerformanceHigheronPre-testThanonActualTestNumberofstudents
Mean%
LG2_1 LG2_2 LG2_3 LG2_4
33 43% 3% 31% 11% 22%
LG3_1-1 LG3_1-2 LG3_1-3 LG3_1-4 LG3_1-5 LG3_1-6 29 10% 17% 59% 14% 10% 21% 22%
LG3_2-1 LG3_2-2 LG3_2-3 LG3_2-4 LG3_2-5
28 14% 11% 21% 25% 36% 21%
LG4-1 LG4-2 LG4-3 LG4-4a LG4-4b LG4-5
32 25% 34% --- --- 25% 59% 36%
LG5-1a LG5-1b LG5-2 LG5-3 LG5-4 LG5-5 LG5-6 28 36% 25% --- 21% --- 18% 39% 28%
Question2
Howdostudentpredictionscomparetostudentperformance?
Table6.7showsthemeansofthepre-testlevelmeansandpredictedlevelmeansfor
allstudentsbyunit.Studentswhoweremissinganylearninggoallevelsinaunitwere
removedfromthedataforthatunit.Thedifferencebetweenthepredictedlevelmeanand
thepre-testlevelmeanof0.77forUnit1indicatesthat,onaverage,studentspredictedthat
theywouldscorethree-fourthsofalevelhigherontheactualtestthantheydidonthepre-
test.Thedifferenceof0.33forUnit3_2indicatesthat,onaverage,studentspredictedthat
theywouldscoreone-thirdofalevelhigherontheactualtestthantheydidonthepre-test.
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Table6.7
Pre-testandPredictedLevelMeansofAllStudentsbyUnitUnit Numberofstudents Pre-testLevelMean PredictedLevelMean PredictedLevelMean–Pre-testLevelMean
2 33 1.87 2.64 0.77
3_1 21 2.23 2.71 0.483_2 19 2.27 2.60 0.33
4 20 2.11 2.55 0.44
5 20 2.28 2.71 0.43All 113 2.12 2.64 0.52
Table6.8showsthepre-testlevelandpredictedlevelfromthetotalnumberof
learninggoalsratedineachunit.18%ofthepredictedratingsforallstudentswerelower
thanthelevelofthestudentsonthepre-test.
Table6.8
ComparisonofPre-testandPredictedLevelsofRatingsforAllStudents
Comparison Count
Predicted<Pre-test 97(18%)Predicted=Pre-testforLevel1orLevel2 64(12%)
Predicted=Pre-testforLevel3orLevel4 99(19%)Predicted>Pre-test 273(51%)
Table6.9showsdatafromthetotalnumberoflearninggoalsratedineachunit,the
numberofthosewherestudentspredictedtheywouldperformhigheronthetestthanthey
didonthepre-test,andthenumberofthosewherestudentsactuallydidperformhigheron
thetestthantheydidonthepre-test.For63%oftheratingsforUnit2,studentspredicted
theywouldperformhigheronthetestthantheydidonthepre-test;however,theyonly
actuallyperformedhigheronthetestthanthepre-testfor37%oftheUnit2ratings.
90
Table6.9
StudentPredictionand/orPerformanceGreaterThanPre-testLevelUnit TotalNumberof
LearningGoalsRatedPredictedLevelGreaterThanPre-testLevel
TotalNumberofLearningGoalsRated
ActualTestLevelGreaterThanPre-testLevel
2 132 83(63%) 140 52(37%)
3_1 126 65(52%) 174 90(51%)3_2 95 39(41%) 140 53(38%)
4 80 37(46%) 128 39(30%)
5 100 49(49%) 140 65(46%)All 533 273(51%) 722 299(41%)
Table6.10showsthemeansofthepredictedlevelmeansandactuallevelmeansfor
allstudentsbyunit.Studentswhoweremissinganylearninggoallevelsinaunitwere
removedfromthedataforthatunit.Thedifferencebetweentheactuallevelmeanandthe
predictedlevelmeanof0.04forUnit3_1andUnit3_2indicatesthat,onaverage,students
performedataboutthesamelevelontheactualtestastheypredictedtheywouldperform.
Thedifferenceof-0.54forUnit2indicatesthat,onaverage,studentsactuallyperformedat
aboutone-halflevelbelowontheactualtestthantheypredictedtheywouldperform.
Table6.10
PredictedandActualTestLevelMeansofAllStudentsbyUnitUnit NumberofStudents PredictedLevelMean ActualLevelMean ActualLevelMean–PredictedLevelMean
2 33 2.64 2.10 -0.54
3_1 21 2.71 2.75 0.043_2 19 2.60 2.64 0.04
4 20 2.55 2.06 -0.49
5 20 2.71 2.61 -0.10All 113 2.64 2.39 -0.25
91
Table6.11showshowmanystudentsineachunithadanactuallevelmeanontheir
testthatwasgreaterorequaltotheirpredictedlevelmean.60%ofthestudentshadan
actuallevelmeanontheirUnit3_2testthatwasgreaterorequaltotheirpredictedlevel
mean.
Table6.11
StudentComparisonofActualTestandPredictedLevelMeansbyUnitUnit NumberofStudents ActualLevelMean≥PredictedLevelMean
2 33 10(30%)
3_1 21 10(48%)3_2 19 12(60%)
4 20 5(25%)
5 20 9(45%)All 113 46(41%)
Therewasonestudentwhoseactuallevelmeanwasgreaterorequaltohis
predictedlevelmeanonallfivetests,andtherewasonestudentwhoseactuallevelmean
wasgreaterorequaltoherpredictedlevelmeanonallfourtestsforwhichshehadallof
thedata.
Question3
Whenstudentsknowthattheyneedtoimprovetheirprogresstowardsalearninggoal,
howdotheytrytoimprove?
TheUnitLearningGoalsSelf-assessmentformsforallfiveunitsreceived113
responsesoutofapossible175responses,a65%studentresponserate,throughoutthe
addendumstudy.TheLearningGoalsReflectionformreceived88responses,a50%
studentresponserate,throughouttheaddendumstudy.Table6.12displaysthetotal
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numberofstudentresponsestothebeforequestion“Whatdoyouplantodotoreachyour
predictedlearninglevels?(selectallthatapply)”andtheafterquestion“Whatdidyoudoto
reachyourpredictedlearninglevels?(selectallthatapply)”DuringUnit2,twenty-nine
studentsplannedtousetheworkedexamplesprovidedbytheteacherandfourteen
reportedafterthetestthattheydidusetheworkedexamples.
Table6.12
StudentReflectionResponsesBeforetheTest/AftertheTestUnit Totalnumber
ofstudentresponses
Before/after
Usetheworkedexamplesmyteacherprovided
Before/after
Studywithanotherstudent
Before/after
Gethelpfromateacherortutor
Before/after
Watchcalculusvideos
Before/after
Reviewclassnotes
Before/after
Other
Before/after
2 33/27 29/14 19/7 7/3 24/16 25/12 Dopracticetest/Absolutelynothing,variouspracticeproblemslikeFreeResponseQuestions(FRQs)
3_1 21/23 18/15 21/6 3/1 17/11 11/13 KhanAcademy,FRQsandLabs,WorkonrelatedPSPs,MathXLhelpsmealottoo/KhanAcademy,Nothing,FRQandlab
3_2 20/10 16/5 12/5 4/1 14/5 8/5 Nothingprobably,FRQsandlabs,DomyMathXL/MathXL
4 19/16 15/12 10/9 5/2 15/7 12/11 FRQsandgeneralpractice/FRQs
5 20/12 15/8 6/2 3/0 9/2 18/7 DoMathXL,FRQs,Makemyownflashcards/MathXL,flashcards
All 113/88 93/54 68/29 22/7 79/41 74/48
Whenstudentswereaskedafterthetestwhetherwhattheydidtoreachtheir
predictedlearninglevelshelped,55%saidyeswithresponsessuchas“Ithinkitdidhelp
93
medomy[sic]betterthani[sic]wouldhavewithoutit.Ididn’tdogreatonthetest,butI
thinkreviewingeverythingdefinitelyhelpedmegrasptheconceptsbetter.”and“Igothelp
fromastudent,andithelpedtohavesomeoneelse’sexplanationaswell.”20%said
somewhatwithresponsessuchas“IthinkitwasbutIwastiredsoIdidn’tfullyabsorbthe
information.”and“Maybe?IdidalotbetteronthetestthanIdidonthepractice.”20%said
nowithresponsessuchas“No,Ididn'thaveagoodgrasponthesubjecttobeginwith,so
theproblemsdidn'thelpmyunderstandingofthisunit.”and“IthoughtwhatIdidhelped
butonceIgottothetestIrealizedmyknowledgewasnotenough.”
Halfoftheresponsesto“Whatmightyoudonexttimeinsteadoforinadditionto
whatyoudidthistime?”hadmoreintheresponse,suchas“studymore”,“lookovermore
examples”,“workmoreproblems.”SeeAppendixTforcompleteresults.
DISCUSSION
Theaddendumtothisresearchstudysoughttodeterminehowapre-testmighthelp
studentsdeterminetheircurrentlearninglevelforeachlearninggoalbeforepredictingthe
learninglevelatwhichtheyplannedtoperformontestday.Theaddendumalsoconsidered
treatmentsthathelpstudentsknowwhatstepstotaketoimprovetheirlearningwhenthey
arenotyetperformingonaproficientlevelforalearninggoal.
Question1
Howdoesstudentperformanceonthepre-testcomparetoactualperformanceonthetest?
ForUnit4,onaverage,studentsperformedataboutthesamelevelontheactualtest
asonthepre-test(seeTable6.5).ThisperformanceraisedaredflagforMs.Dolfandthe
researcher.Whydidn’tstudentsperformbetterontheactualtestthanonthepre-test?It
couldbethatstudentsweresatisfiedwiththeirlevel,itcouldbethatstudentsdidnot
94
actuallyspendtimepreparingforthetestasplanned,itcouldbethatthelearninggoalis
particularlychallenging,oritcouldbethatthepre-testdidnotdoagoodjobofassessing
thestudentlevelbeforethetest.
Alloftheitemsonthepre-testweremultiplechoice.EventhoughtheMs.Dolf
includedchoiceEforstudentstoselectiftheydidnotknowhowtotheproblem,guessing
orevenhavingchoicestoeliminatecouldaccountforsomeofthegreatersuccessonthe
pre-testthanontheactualtest.
Usingteacher-andresearcher-createdpre-testsraisedsomequestionofvalidity,so
theresearcherlookedatstudentperformanceoneachlearninggoalandfoundthatover
halfofthestudentsperformedbetteronthepre-testfortwolearninggoals(seeTable6.6).
Thoselearninggoalswerenotremovedfromthisstudy,butifMs.Dolfusesthepre-tests
nextyear,shecanusethedatafromthisyeartomakeadecisionabouthowtoproceedfor
nextyear.Shemightrewritetheitemsusedonthepre-testtodeterminethelevelatwhich
studentsareperformingbeforethetest,orifthelearninggoalsareparticularlychallenging
forstudents,shemightconsiderteachingthecontentinadifferentwayorprovidinga
differenttypeofpracticeforstudentsthanhasbeentypicalinyearspast.ForUnit4,36%
ofthetotallearninggoalratingshadahigherlevelonthepre-testthanontheactualtest.
Ms.DolfwillneedtolookbackattheUnit4pre-testinitsentiretyandcompareitwiththe
Unit4testtodetermineanydiscrepanciesinassessingthelevelatwhichstudentsare
performing.
95
Question2
Howdostudentpredictionscomparetostudentperformance?
Hattiesuggeststhatstudentsknowhowtheyaregoingtoperformonatest.When
giventheopportunitytoself-reporttheirprogresstowardsalearningtarget,studentsset
safeexpectations(May2012).Intheaddendumtothisresearchstudy,studentshadthe
opportunitytoactuallyknowthelevelatwhichtheywereperformingonalearninggoal
beforetheypredictedthelevelatwhichtheyplannedtoperformonthetest.ForUnit1,on
average,studentspredictedthattheywouldscorethree-fourthsofalevelhigheronthe
actualtestthantheydidonthepre-test.ForUnit3_2,onaverage,studentspredictedthat
theywouldscoreone-thirdofalevelhigherontheactualtestthantheydidonthepre-test.
Unit3_2hadmoreapplicationproblemsthananyotherunit,whichcouldaccountforlower
studentconfidenceonthisunit.Forallfiveunits,onaverage,studentspredictedthatthey
wouldscoreone-halfofalevelhigherontheactualtestthantheydidonthepre-test(see
Table6.7).
Hattiegoesontosaythatteachersshouldnothelpstudentsreachtheirpredicted
levelbuthelpthemexceedtheirpredictedlevel(May2012).Table6.8showsthat19%of
studentspredictedtheywouldperformonthetestatalevellowerthantheyperformedon
thepre-test.HowmightMs.Dolfhavespecificallyhelpedthestudentswho,fromthe
beginning,thoughttheywoulddoworseonthetestthantheydidonthepre-test?31%of
studentspredictedtheywouldperformonthetestatthesamelevelthattheyperformedon
thepre-test,with12%predictingtheywouldstayataLevel1orLevel2and19%
predictingtheywouldstayataLevel3orLevel4.
96
51%ofstudentspredictedtheywouldperformhigheronthetestthanthey
performedonthepre-test.Justbecausestudentspredictedtheywouldperformhigheron
theactualtestthantheydidonthepre-testdoesnotmeantheyactuallydidperform
higher.Table6.9showsthatwhilestudentspredictedtheywouldperformhigheronthe
actualtestonabouthalfofthelearninggoalstheyrated,theyonlyactuallyexceededthat
ratingtwo-fifthsofthetime.
ForUnit3_1andUnit3_2,onaverage,studentsactuallyperformedataboutthe
samelevelontheactualtestastheypredictedtheywould(seeTable6.10).Forallfive
units,onaverage,studentsactuallyperformedataboutone-fourthofalevelbelowonthe
actualtestthantheypredictedtheywouldperform.
Whencomparingactuallevelmeanswiththeirpredictedlevelmeans,theactual
levelmeanwasgreaterorequaltothepredictedlevelmeanforonly41%ofthe113tests
withcompletedata(seeTable6.11).Itcouldbethattakingthepre-testandreflectingon
whatactionstheyplannedtotaketoreachtheirpredictedlearninglevelsmadestudents
overconfidentinnotonlywhattheycouldlearnbetweenthereviewdayandthetestday
butalsoinhowmuchtimetheywouldhavetospendlearning.Mostofthestudentsinthe
addendumtothisstudyalsoparticipatedintheoriginalresearchstudy.Thiswastheir
thirdmathclassinwhichtheyusedandreflectedonlearninggoals,andthesecondmath
classinwhichtheypredictedthelevelatwhichtheyexpectedtoperformonthetest.
Studentsmayhavebecometoocomfortableorevenboredwiththeratingprocess,which
mayhaveskewedtheresults.
97
Question3
Whenstudentsknowthattheyneedtoimprovetheirprogresstowardsalearninggoal,
howdotheytrytoimprove?
OntheUnitLearningGoalsSelf-assessmentbeforethetest,overhalfofthe113
responsesindicatedthatstudentsplannedto“usetheworkedexamplesmyteacher
provided”,“studywithanotherstudent”,“watchcalculusvideos”,and“reviewclassnotes.”
However,ontheLearningGoalsReflectionafterthetest,overhalfofthe88responses
indicatedthatstudentsactually“usedtheworkedexamplesmyteacherprovided”and
“reviewedclassnotes”(seeTable6.12).AccordingtoHattie,workedexampleshavean
effectsizeof0.57;inthisstudy,moststudentssaidtheytookadvantageoftheworked
examplesthatMs.Dolfprovided.Peertutoringhasaneffectsizeof0.55;inthisstudy,
studentssaidthattheyplannedtostudywithanotherstudent,butresponsesafterthetest
didnotindicatethatmanystudentstookadvantageofpeertutoring.Intelligenttutoring
systemshaveaneffectsizeof0.48.MathXL,anonlineintelligenttutoringsystem,wasnot
specificallymentionedtostudentsintheirreflectionbecauseMs.Dolfrequiresthat
studentscompleteMathXLpracticeassignmentsfortheunitpriortoreviewday.Evenso,a
fewstudentsmentionedMathXLassomethingtheyplannedtodoanddiddotopreparefor
thetest.Audio-visualmethodshaveaneffectsizeof0.22,butinteractivevideomethods
haveaneffectivesizeof0.54(Hattie,2012).Studentsplannedtowatchcalculusvideos,and
whiletherewasnotmorespecificityastowhattypeofvideostheywerewatching,several
studentsmentionedKhanAcademy.Responsesafterthetestindicatedthatnotasmany
watchedvideosashadinitiallyplannedtodoso.
98
ByUnit3_2,afewstudentsrequestedtohavethepre-testemailedtothemseveral
daysbeforethereviewdaysothattheycoulddoitbeforeclassandspendmoretimein
classworkingonproblemswitheachother.Thesestudentsrecognizedthevalueofpeer
tutoringandknewthattheydidnothavetimeoutsideofclasstomakethathappen,andso
theyworkedwithMs.Dolftofigureoutbothhowtotakeadvantageofthepre-testandpeer
tutoringduringclass.SomeofthesamestudentsaskedMs.Dolffortheworkedproblemsto
thepre-testsothattheycouldlearnfromtheirmistakes.Sheprovidedthesetoallstudents
attheendofthedaystudentstookthepre-test.
Moststrikinginthestudentresponsesto“Whatmightyoudonexttimeinsteadofor
inadditiontowhatyoudidthistime?”isthatoverhalfofthestudentresponsesindicated
thattheyhadnotdoneenoughandshoulddomorenexttime.
Thereweretwostudentswhoseactuallevelmeanwasgreaterorequaltothe
predictedlevelmeanonatleastfourofthetests.OntheLearningGoalsReflectionForm
(seeAppendixR),oneofthesestudentsnotedafterthefirsttestthat“goingoverproblems
withanotherstudenthelpedmeseetheirstrategyatsolvingtheproblem.”Afterthethird
test,thesamestudentnotedthatshe“didnotunderstandL'Hopital'sRuleorrelatedrate
problemsbeforestudying.”
SCOPEANDLIMITATIONS
Only35studentsparticipatedintheaddendumtotheresearchstudy,and
participationwanedastheschoolyearprogressed.Forexample,therewascompletedata
for33studentsforUnit2,butonly21studentshadcompletedataforUnit3_1.Ms.Dolf
wasoutforthePSATandameeting,andseveralseniorsweregoneforaserviceproject.
Additionally,surveyswereoptional;afterthefirstone,fewerstudentstookthetimeto
99
formallyreflectontheirlearning.Therestoftheunitshoveredaroundthesameamountof
participationwith19studentsforUnit3_2and20studentsforUnit4andUnit5.
Previously,halfoftheratingscollectedfromstudentsduringthisresearchstudy
werestudentswhoself-reportedandpredictedthesamelevel,suchasself-reportingLevel
3andpredictingLevel3orself-reportingLevel2andpredictingLevel2.Thestudents
seemedtoalreadythinktheywerewheretheywantedtobe.Werethesestudentssatisfied
withwhattheythoughttheyalreadyknew?Didtheythinkitwastoolatetoimprove?Is
thisthegroupHattiemeanswhenhesaysthatteachersshouldnothelpstudentsreach
theirpredictionbutexceedtheirprediction?Almosthalfofthestudentsintheoriginal
researchstudyself-reportedatLevel2andpredictedLevel3orself-reportedatLevel3
andpredictedLevel4.DidthosestudentsreallyknowwhatitmeanttobeaLevel3or
Level4?Ordidtheyjustthinkthattheyneededtogetbetter?Becausestudentsself-
reportedtheirlevels,theresearcherhadnowayofknowinghowaccuratetheself-reported
levelwas.Intheaddendumtothisstudy,theresearcherhadamoreobjectiveviewof
studentimprovementbecauseofthepre-testthatwasusedtopre-assessstudents’learning
levelsbeforetheytakethetest.Unfortunately,theteacherandresearcherrealizedthatthe
pre-testwasnotalwaysagreatindicatorofwhatstudentsknewaboutalearninggoal.
Somelearninggoalscannotbereducedtoasingleassessmentitemforeachlevel.For
example,learninggoal4-4ais“IcanusetheFundamentalTheoremofCalculus”(see
AppendixS).UsingtheFundamentalTheoremofCalculuswithpolynomialfunctionsisnot
usuallyaschallengingasusingitwithtrigonometricorrationalfunctions.Usingthe
FundamentalTheoremofCalculuswhenu-substitutionisrequiredismorechallenging
thanwhenu-substitutionisnotrequired.Reducingsuccessonalearninggoaltothree
100
assessmentitemstodeterminethelevelthestudentisperformingpriortothetestdoesnot
alwayswork.Addingadditionalassessmentitemstothepre-testdoesnotreallywork,
either,asspendingthewholeofclasstimeonthedaybeforethetesttakingalong
individualpre-testpreventsstudentsfromlearningtogetherbytalkingandasking
questionsaboutmathematics.
Alllearninggoalsarenotequallyimportant,andtheyarenotassessedatequalrates
onthetest.Forexample,fortwolearninggoalsinUnit4(4-2and4-3)andfortwolearning
goalsinUnit5(5-2and5-4),therewasonlyoneassessmentitemonthetest,anditwasnot
aLevel4item.WhilesomestudentspredictedtheywouldperformataLevel4onthese
learninggoalsonthetest,therewasnowaytomeasuretheirperformancebeyondLevel3.
Similarly,therewasnowaytomeasuretheirperformanceatLevel2,asmissingtheitem
resultedinaLevel1forthatlearninggoalonthetest.Theteacherandtheresearcher
decidedtoremovethesefourlearninggoalsfromallcalculationsintheaddendum.Inthe
future,considerationshouldbegiventowhetheralearninggoalshouldbealearninggoalif
thereisonlyoneitemonthetest.Itcouldbethatthelearninggoalwillberevisitedin
futureunitswithadditionaltypesoffunctions,inwhichcaseitmightbebesttoreserve
masteryofthelearninggoalforthelaterunitonlyinsteadofincludinginbothunits.Ifthe
learninggoalshouldbeassessedinbothunits,theteachershouldconsideraddingmultiple
itemstothetestsothatstudentperformanceonthegoalcanbeassessedbeyondonly
Level1andLevel3.
Theoriginalresearchstudyfollowedthreeunitsinprecalculusforwhichtestgrades
fromthecurrentandpreviousprecalculusteachers,Ms.BairdandMs.Dolf,forthepast
threeyearsshowednostatisticaldifferencebetweentests.Thesamecannotbesaidforthe
101
fiveunitsofcalculusstudyintheaddendum,andsonocomparisonscouldbemadefrom
unittounittoseewhetherstudentswereimprovingovertimeintheirpredictionsorhow
theyplannedtoreachtheirpredictedlearninglevels.
FUTURERESEARCH
Writingapre-testtodeterminethelevelatwhichstudentsareperformingwithonly
threeitemsperlearninggoalcouldbelesschallenginginanAlgebra1classwherestudents
studyasmallnumberoffunctiontypesthanitisinanAPcalculusclasswherestudents
studyconceptswithmultiplefunctiontypes.RepeatingthisstudyinanAlgebra1class
couldgiveinsightintowhetherthepre-testworksforsomeclassesortopicsbutnotothers.
Intheaddendumtothisstudy,whilestudentspredictedtheywouldperformhigher
ontheactualtestonabouthalfofthelearninggoalstheyrated,theyonlyactuallyachieved
orexceededthatratingtwo-fifthsofthetime(seeTable6.9).InAPCalculus,students
constantlyrevisitcontentviadifferenttypesoffunctionsandapplications.Repeatingthis
studyinaclasswheretopicsofstudyarelessconnectedandmoreproceduralcouldgive
moreinformationaboutstudentconfidenceintheirpredictionandhowteachersmight
supportstudentstoexceedtheirpredictedlevel.
Researchfollowingstudentsusinglearninggoalsandpredictingtheirsuccessover
multipleclassesandmultipleschoolyearscouldgiveinsightintostudents’useof
metacognitivestrategies.Ifstudentsareactivelythinkingabouttheirlearningintheirmath
classbecauseofactionsinitiatedbytheteacher,howmightthatlookinahistoryclass
whentheteacherisnotinitatingareflectiononlearning?
Finally,whiletheaddendumgavetheresearchermoreinformationabouthow
studentsplannedtoreachtheirpredictedlearninglevelswhentheyhadnotdonesoyet,
102
futureresearchisneededtodeterminewhichtreatmentsactuallyhelpedstudentsimprove.
Inparticular,theteachermightstartbyinterviewingthetwostudentswhoseactuallevel
meanwasgreaterorequaltothepredictedlevelmeanonatleastfouroftheteststofind
outwhathelpedthemexceedtheirpredictions.
104
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APPENDIXE
PrecalculusAssessmentBefore
Unit8Piecewise,Composite,&InverseFunctions.NoCalculator.SignandPledge:IpledgethatIamturninginmyownwork.1.Giventhegraphofh(x).a.Writeapiecewisefunctionforh(x).b.TrueorFalse:h(x)isconstanton(-2,2).
2.Rewritethefollowingabsolutevaluefunctionsaspiecewisefunctions.a. b. 3.Ifpossible,usethetablebelowtoevaluatethefollowing.x j(x) n(x)-1 5 00 3 0.21 -2 -42 4 7a. b. c.
4.Usethegraphsoff(x)andg(x)toevaluatethefollowing.a. b. c.
5.Determinewhetherthefollowingfunctionsareonetoone.a. b.
6.If find andgiveitsdomain.
f x( ) = 3x − 4 +1( ) 5 3f x x= - -
j on( ) −1( )
j − n( ) 2( )
no j( ) 1( )
f + g( ) 0( )
g o f( ) −1( )
f o g( ) 1( )
( ) 2 4f x x= + ( ) 32 5f x x= -
( ) 42xf x
x=
-( )1f x-
114
7.Findthevalueofeachofthefollowingifthegivenfunctionsareasfollows:
a.Domainof b. c.
d.Domainof e. f.Domainof 8.Given .a.Whatis ?b.Completethefollowingtable.
slope x-intercept y-intercept
f(x)
f-1(x)
c.Makeaconjectureabouttheslopesoftwolinearfunctionsthatareinversesofeachother.d.Makeaconjectureaboutthex-andy-interceptsoftwolinearfunctionsthatareinversesofeachother.Willthisbetrueforallfunctionsandtheirinverses?Explainyourreasoning.9.a.Determineafunctionf(x)suchthat .b.Whatmustbetrueaboutafunctionthatisitsowninverse?
10.Thedifferencequotientofafunctionfisgivenby .Findthedifference
quotientforthefunction .Simplifyyouranswer.
11.Findthedifferencequotient for .
12.Given and . isgivenby .Findthedifference
quotientandrewriteitbyrationalizingthenumerator.Thenevaluate byevaluatingyourrewrittendifferencequotientat .Youhavefoundtheslopeofthelinetangenttothegraphof at .
13.Given .
a.Sketchagraphofk(x).b.Evaluatethefollowing: .
( ) 3 4f x x= + ( ) 5g x x= - ( ) 2 1h x x= - ( ) 1 5k x x= -
( )f xk
æ öç ÷è ø
( )( )3h f -! ( )( )g h x!
( )( )g h x! ( )( )h g x! ( )( )h g x!
f x( ) = px + r( )1f x-
( ) ( )1f x f x x-= "
( ) ( )f x x f xx
+D -D
( ) 2 4 1f x x x= - +
( ) ( )3 3f h fh
+ - ( ) 21
f xx
=+
( ) 2f x x= + 4a = ( )' 4f ( ) ( )f x f ax a--
( )' 4f4x =
( ) 2f x x= + 4x =
k x( ) =2x, x < 2x −3 +1, 2 ≤ x < 5
5, x ≥ 5
$
%&&
'&&
( ) ( ) ( ) ( ) ( )0 , 2 , 4 , 5 , 8k k k k k
115
APPENDIXF
PrecalculusAssessmentAfter
Unit8Piecewise,Composite,&InverseFunctions Name______________________________NoCalculator Pleasecircleyouranswers!Icanperformvariousoperationsonoffunctions,includingaddition,subtraction,multiplication,division,andcomposition.Icandeterminethepropertiesofthesenewfunctions.1.Ifpossible,usethetablebelowtoevaluatethefollowing.a. b. c. 2.Usethegraphsoff(x)andg(x)toevaluatethefollowing.a. b. c. 3.Findthevalueofeachofthefollowingifthegivenfunctionsareasfollows:
a.Domainof b. c.
d.Domainof e. f.Domainof
j on( ) −1( )
j − n( ) 2( )
no j( ) 1( )
f + g( ) 0( )
g o f( ) −1( )
f o g( ) 1( )
( ) 3 4f x x= + ( ) 5g x x= - ( ) 2 1h x x= - ( ) 1 5k x x= -
( )f xk
æ öç ÷è ø
( )( )3h f -! ( )( )g h x!
( )( )g h x! ( )( )h g x! ( )( )h g x!
x j(x) n(x)-1 5 00 3 0.21 -2 -42 4 7
116
Icandetermineifafunctionisone-to-oneandfindtheinverseofafunction.4.Determinewhetherthefollowingfunctionsareonetoone.a. b. 5.Given find
a. b.Thedomainof 6.Given .a.Whatis ?b.Completethetableattheright.c.Makeaconjectureabouttheslopesoftwolinearfunctionsthatareinversesofeachother.d.Makeaconjectureaboutthex-andy-interceptsoftwolinearfunctionsthatareinversesofeachother.Willthisbetrueforallfunctionsandtheirinverses?Explainyourreasoning.Icangraph,writeequationsfor,anddeterminepropertiesofpiecewisefunctionsincludingwritingabsolutevaluefunctionsaspiecewisefunctions.7.Giventhegraphofh(x),writeapiecewisefunctionforh(x).8.Rewritethefollowingabsolutevaluefunctionsaspiecewisefunctions.a. b.
( ) 2 4f x x= + ( ) 32 5f x x= -
( ) 42xf x
x=
-( )1f x-
f−1(x)
f x( ) = px + r
( )1f x-
f x( ) = 3x − 4 +1 ( ) 5 3f x x= - -
slope x-intercept y-intercept
f(x)
f-1(x)
117
9.Given .
a.Sketchagraphofk(x).b.Evaluatethefollowing:
.Icansimplifythedifferencequotientofapolynomial,rational,orradicalfunction.
10.Thedifferencequotientofafunctionfisgivenby .Findthedifference
quotientforthefunction .Simplifyyouranswer.
11.Findthedifferencequotient for .Simplifyyouranswer.
k x( ) =2x, x < 2x − 3 +1, 2 ≤ x < 5
5, x ≥ 5
⎧
⎨⎪⎪
⎩⎪⎪
( ) ( ) ( ) ( ) ( )0 , 2 , 4 , 5 , 8k k k k k
( ) ( )f x x f xx
+D -D
( ) 2 4 1f x x x= - +
( ) ( )3 3f h fh
+ - ( ) 21
f xx
=+
118
12.Given and . isgivenby .Findthedifferencequotient
andrewriteitbyrationalizingthenumerator.Thenevaluate byevaluatingyourrewrittendifferencequotientat .Youhavefoundtheslopeofthelinetangenttothegraphof at .Icandetermineifafunctioniscontinuous.Icanidentifydiscontinuitiesasremovable(point)ornonremovable(jumporasymptotic).13.(FreeResponseQuestion)Considerthetwopiecewisedefinedfunctions,f(x)andg(x),belowtoanswerthefollowingquestions.
a.Findf(-9),f(-3),f(7),andthedomainoff(x).b.Doesf(x)haveadiscontinuityatx=-3?Ifso,classifyit.Justifyyourreason.c.Forwhatvalue(s)ofaisthegraphofg(x)continuousatx=-2?
( ) 2f x x= + 4a = ( )' 4f ( ) ( )f x f ax a--
( )' 4f4x =
( ) 2f x x= + 4x =
f x( ) =x2 + 2
3x, −9 < x ≤ −3
−2x +1, −3< x < 2x + 3, x > 2
⎧
⎨
⎪⎪⎪
⎩
⎪⎪⎪
g(x)= ax +3,x < −2x2 +2x ,x ≥ −2
⎧⎨⎪
⎩⎪
128
APPENDIXL
StudentFeedbackonLearningTargetsResponsestoOpen-EndedQuestions
Howdoyouusethelearningintentionswhileyouaretesting?• IthinkaboutthemifIgetstuck.• Ilookatthelearningintentionstofurtherunderstandwhatisbeingaskedinthe
questiononthetest.• IusethemtoreaffirmIamsolvingtheproblemscorrectlyandtothehighestability.• Iusethemtoanalyzetheproblemandrelateitbacktorealclassexperienceand
differentpractices!• Ihonestlydon'tlookattheintentionsonthetest.Igostraighttotheproblemsand
assessthemwithoutthinkingaboutthelearningintentions.• ByknowingwhatIamlookingfor• ItallowsmetoknowwhatIamlookingfor.• TheypointmeinadirectionsoIwillknowwhatisneededtoworkaproblem.• ThesehelpmeunderstandthetaskofthemathproblemIamabouttofacewhile
testing.• Thelearningintentionsgivesmeasenseofwhattostudyforthetest.• Ilookatthemtorememberexactlywhatthatproblemistesting• IthelpsmeunderstandwhatIshouldknow.• Toknowwhatthegoalofthesectionis.• IusethemoncertainpartsofthetestinwhichwhenIneedtouseit.• Isetagoalformetoreachonthetest• IuseittodoaquickreviewofallthestuffIshouldknow.• Ineverdo• IusethemtounderstandwhatIamtryingtofigureout.Ihelpthemtorealizewhat
kindofanswerIshouldhave.• tounderstandhowtoanswertheproblems• AswordsontheypagethatgivemeahinttowhatIshoulddo.• Thelearningintentionsduringthetestarehelpfulbecauseitallowsaframeof
referenceonhowtoapproachtheproblem(s)andwhichfieldoftheunititisabout.• IthelpsmetoknowwhatIknowandwhatIdonotknow.ItremindsmethatIhave
anunderstandingofwhatwehavelearned• IusethemtodeterminehowIwillsolvetheproblemgiven.• IamabletothinkbacktoproblemsIworkedoutandusethemasexamplesonthe
test.• Whenreadingtheintentions,IgetpreparedforwhatproblemsIamabouttodoon
thetests• Seeingtheintentionsonthetesthelpstoremindmeofthespecifictypesof
problemsIamabouttohaveonmytest.• IthelpsmeknowwhatIshouldbetryingtoaccomplishineachsectionofthetest.• ItsomewhatgivesawaytoknowwhatI'mdoingandnarrowingdownwhenIgetto
questionsonwhatmethodIneedtodoandwhatwillbebest
129
• Ireadthelearningintentionsasapreviewtowhatthequestionsinthatsectionwillbeasking.IusethelearningintentionstohelpmeunderstandwhatIamdoingaheadoftimesoIdon'treadaquestionandbecomeconfusedastowhatmethodImayneedtousetosolveit.
• ItrytorememberwhattheyweresoIknowwhatstepsIneedtotakethetestin.• Ijustuseittohelpmeknowwhichspecificskillsgoesalongwiththetest.• Iusethemtosolvemywaytogettheproblem• Idon't.• IthelpsmedeterminewhatIneedtodoinordertosolvefortheproblemandget
thecorrectanswer• Thelearningintentionsmakemethinkaboutwhatwehavelearnedsofarinthe
unitandmakemecalmdownwhenIcannotfigureouthowtoworkaproblem.Lookingatthelearningintentionshelpsmetorememberwhichskillstouse.
• Theygiveanideaoftheanswerthey'relookingfor.• Irememberedthemwhenfacedwithadifficultproblem• Iusetheintentionstoguidemywaytoknowingwhattodo• ItnarrowsdownwhatIneedtodoandmakesitmoreclearwhattheteacheris
lookingfor.Whatmighthelpyouinyourlearning?
• moreexamples• Itwouldhelpmetogetstudentstoexplaintoeachotherproblemsontheboard;
becauseteachingothersalwayshelpsmedeeperunderstandthesubject.• Ifthelessonsweretaughtintheorderofthelearningintentionsbeginningwith
intention1andgoingtothelastintention.• Morepracticeandreallifeapplication.• HavingareviewofwhatIshouldknowwasveryhelpful.• RepeatedpracticeofproblemssoIcanbecomeusedtoworkingtheproblemswith
speedandcorrectness.• Visualexamples• Ifpracticeproblemswerelistedwitheachofthesectionsbeforethetest• Moreone-on-oneteachertimetoclearthingsImaybeunsureabout.• Visuallearninghelpsme• Studyingandpracticingwillimprovemylearning• Ithinkmoreinteractiveteachingcouldbeusedinordertohelpwithmylearning• Adesignatedreviewdaytogobackoveranyquestionsorconfusion• studying,practiceproblems• Morepractice• Morepractice• NothavingajobsoIhavemoretimetostudy.• Stuffthathelpsmeisjustgoingthroughallquestionsthedaywelearntheunitand
thenIhaveitfromthere.• Iwouldsaymore"lecture"wouldhelp.Spendingmoretimehearingtheteacher
explainwouldhelpme.
130
• morepracticeproblems/apracticeopenendedquestion• Havingabetterattentionspan.• Ithinkthatmoreabstractreasoninganddrawingconclusionsmyselfwouldhelpme
inmylearning.• Havingthelearningintentionsonthetesthelpsbutwouldhelpmoreifwhen
learningwespliteachintentionuptoalesson.• Morepracticeproblemsbyhandwithouttheuseofalaptop.• Ifmyteachergaveuslesshandoutstofigureoutonourownandtaughtusmorein
depth• Idonotknowofanythingthatwillhelpme.Ididprettywell.• Itwouldbehelpfultoknowhowtoimprovemyskillsforeachsectionafteraquiz.• StudymoreandrealizewhatskillI'mnotproficientatandlearnmoreaboutit.• somegameswouldbeintriguing• morepracticeandexplainations• Studyingwhatmethodsofproblemsolvingcorrespondtoeachlearninggoalwould
helpinmyunderstandingofthematerial.• Homework• Maybeatadbitmoreclaritythroughtheunit.• Moregroupwork• Selflearningorjustdoingpracticeproblems.• Beingconfidentinmyanswersandtrynottolookforasecondopinion• Ilearnthebestworkingpracticeproblemsandusingworkedexamples,soasmany
practiceproblemsaspossiblehelpsmebemoreconfident.• MorepracticeofwhatisonthetestanddeeperunderstandingofwhatIamlearning.• PracticeTests• Workingmathproblemsbeforetakingthetest• Ithinkwhatwouldhelpmeinthelongrunwouldbemorepracticeandoneonone
toseeifhowIfeelaboutthequestions.• Gettingavisualrepresentationofthingshelpsmeunderstandwhereallofthe
factorsarecomingfromratherthanjustbeingtoldtheyexist.• Individualassistance
131
APPENDIXM
InterviewQuestionsforTeacher
• Whatdoyouthinkabouttheimpactofstudentsratingandpredictingtheirprogress
onlearningtargetsbeforeatest?• Whatdoyouthinkabouttheimpactofleveledlearningprogressionswithworked
examplesonstudentlearning?• Whatdoyouthinkabouttheimpactofemphasizingmetacognitionwithstudentson
studentlearning?• Whatplansdoyouhave,ifany,forcontinuinganyofwhatyouhavedonethis
quartero Nextyear?o Forotherunits?o Inotherclasses?
• Whatwouldyoutelland/orrecommendtootherteachersaboutyourexperiencethisquarter?
132
APPENDIXN
PrecalculusLearningTargetsbyUnitTest1–Unit7TransformationsofParentFunctions
7_1 Icangraphandwritetheequationofatransformedparentfunction.
7_2 Icanrecognizetransformationsgivenagraphorequation.7_3 Icandeterminepropertiesofagraphoffunctionsuchasdomain,range,extrema,
increasing,decreasing,constant,intercepts,andsymmetry.7_4 Icanwriteafunctionorgraphwithgivenpropertiesordeterminethatsucha
functioncannotexistandwhy.
Test2–Unit8Piecewise,Inverse,andCompositeFunctions
8_1 Icangraph,writeequationsfor,anddeterminepropertiesofpiecewisefunctions.8_2 Icandetermineifafunctioniscontinuous.Icanidentifydiscontinuitiesas
removable(point)ornonremovable(jumporasymptotic).
8_3 Icanrewriteabsolutevaluefunctionsaspiecewisefunctions.8_4 Icanperformvariousoperationsonoffunctions,includingaddition,subtraction,
multiplication,division,andcomposition.8_5 Icandeterminepropertiesofcompositeandinversefunctionssuchasdomain,
range,andfunctionvalues.
8_6 Icansimplifythedifferencequotientofapolynomial,rational,orradicalfunction.
Test3–Unit9PolynomialFunctions9_1 Icandeterminetheroots(withmultiplicity),extrema,endbehavior,intercepts,
concavity,anddegreeofapolynomialgivenagraphand/orequation.
9_2 Icandeterminewhetherafunctioniseven,odd,orneither.Icandeterminethesymmetryofafunction.
9_3 Icanwritetheequationofapolynomialgivenvariouspropertiessuchasroots,endbehavior,intercepts,anddegree.
9_4 Icansketchthegraphofapolynomialgivenvariouspropertiessuchasroots,endbehavior,intercepts,anddegree.
9_5 Icanfindrootsandfactorsofapolynomialusinglongand/orsyntheticdivision.9_6 Icansolvepolynomialinequalities.
147
APPENDIXS
Addendum:APCalculusLearningTargetsbyUnit
Test1–Unit2Derivatives
2-1 Icanusethedefinitionofthederivative.2-2 Icanevaluatederivativesfromgraphs,tables,andequationsusingdifferenttechniques.
2-3 Icanwritetheequationsoftangentandnormallinesatapointonagraph.
2-4 Icansolverelatedratesproblemswithareal-worldcontext.
Test2–Unit3_1ApplicationsofDifferentiation3_1-1 Icanusethederivativetodeterminemaximaandminimaofafunction.
3_1-2 IcanusetheIntermediateValueTheorem,theExtremeValueTheorem,andtheMeanValueTheorem.
3_1-3 Icananalyzefunctionsfromgraphs,tables,andequationsusingdifferenttechniques.3_1-4 Icanusetheoriginalfunctiontodeduceinformationaboutthefirstandsecond
derivatives.3_1-5 Icanusethederivativetodeduceinformationaboutthesecondderivativeandthe
originalfunction.
3_1-6 Icanusethesecondderivativetodeduceinformationaboutthefirstderivativeandtheoriginalfunction.
Test3–Unit3_2ApplicationsofDifferentiation3_2-1 Icanusethetangentlineatapointtoapproximatevaluesofthefunctionnearthe
pointoftangency.
3_2-2 IcanuseL'Hopital'sRuletoevaluatealimit.3_2-3 Icandeterminethedifferentialforafunction.
3_2-4 Icansolveoptimizationproblemswithareal-worldcontext.
3_2-5 Icansolverelatedratesproblemswithareal-worldcontext.
Test4–Unit4Antidifferentiation4-1 IcanapproximatetheareaunderacurveusingRiemannSums.
4-2 IcanapproximatetheareaunderacurveusingtheTrapezoidalRule.
148
4-3 Icansetupandcalculatetheexactareaunderacurveusingthelimitofthesumoftheareasofaninfinitenumberofrectangles.
4-4a IcanusetheFundamentalTheoremofCalculus.
4-4b IcanusetheSecondFundamentalTheoremofCalculus.4-5 Icanantidifferentiateusingvarioustechniques,includingsubstitutionofvariables.
Test5–Unit5TranscendentalFunctions5-1a Icancalculateandusederivativesofexponentialfunctions.
5-1b Icancalculateandusederivativesoflogarithmicfunctions.
5-2 Icancalculateanduseantiderivativesofexponentialfunctions.5-3 Icancalculateanduseantiderivativesinvolvinglogarithmicfunctions.
5-4 Icancalculateanduseslopesforinversefunctions.5-5 Icancalculateandusederivativesofinversetrigonometricfunctions.
5-6 Icancalculateanduseantiderivativesinvolvinginversetrigonometricfunctions.
149
APPENDIXT
Addendum:LearningGoalsReflectionResponses
Test1–Unit2What did you do to reach your predicted learning levels? (select all that apply) Did what you do helped? Explain.
What might you do next time instead of or in addition to what you did this time?
Watch calculus videos I believe it did Study longer
Use the worked examples my teacher provided, Watch calculus videos, Review class notes
I think it did help me do my better than i would have without it. I didn’t do great on the test, but I think reviewing everything definitely helped me grasp the concepts better.
I would review like I did but I wish I would have gone back over definition of derivative problems because I struggled with those.
Use the worked examples my teacher provided, Study with another student, Watch calculus videos, Review class notes No. I still failed miserably Get a tutor Use the worked examples my teacher provided, Watch calculus videos, Review class notes Yes, I improved on my learning levels. I should get tutored.
Absolutely nothing Sure did help me stay at level 1 Actually study for the tests Use the worked examples my teacher provided
Yes it did because it help me remember what I needed to do. Review and study with others.
Use the worked examples my teacher provided, Get help from a teacher or tutor, Watch calculus videos, Review class notes
I felt like what I did, didn't really help me. I was learning the basics and testing the waters of unit 2. I was able to find the answers to unit 2 questions if they were given to me at face value. On the test, I felt like I had to incorporate a lot of rules and formulas just to find an answer. Doing that confuses me a lot and I think I may have done bad on the test because of that. When we did test corrections, I was able to see what I did wrong
Next time, if something is not clear to me, I will get help immediately. I wont sit around and let the class get ahead of me.
Use the worked examples my teacher provided, Study with another student, Watch calculus videos
Yes because I was less confident on the objectives until I practiced the weekend To do mathxl a week earlier
Use the worked examples my teacher provided, Get help from a teacher or tutor
It helped for some of the learning goals but I should have prepared more for LG1 and LG3
Study with another student, Watch calculus videos
It gave me a broad understanding but I still didn't know how to do everything. Study more
Use the worked examples my teacher provided, Watch calculus videos
I thought what I did helped but once I got to the test I realized my knowledge was not enough maybe study with friends
Study with another student, Watch calculus videos, Review class notes
Obviously not, because I still got a bad grade.
Literally everything because I need it apparently.
150
Study with another student
Yes. Going over problems with another student helped me see their strategy at solving the problem.
I will watch the calculus videos and get help from a teacher or tutor.
Use the worked examples my teacher provided, Watch calculus videos Extremely Review notes too Use the worked examples my teacher provided, Watch calculus videos, Review class notes
It helped a little. It probably would have helped more if I did more of it.
I will probably get help from a teacher during zero block when I have time.
Review class notes Kind of, i got to review what we already went over
So much more, review videos, practice
I did various practice problems, namely the FRQs
Yes, by doing all seven of the FRQs, I was more than prepared for the test
Next time I will more than likely work the self assessments that's available in addition to the FRQs
Review class notes It was enough, so I was not prepared Everything Use the worked examples my teacher provided, Watch calculus videos, Review class notes some of the videos did gelp
Maybe getting more teacher help
Use the worked examples my teacher provided, Get help from a teacher or tutor, Watch calculus videos
Yes, I felt much more prepared for the test than I did on pre-assessment day.
Next time, I will try to work more challenging questions to ensure that I can reach the answer with accurate (aka not forgetting a step/making a mathematical error)
Watch calculus videos, Review class notes It did some, but not much.
Look at worked examples of the problems and get help from a tutor
Study with another student, Watch calculus videos
I was under the impression I knew the material.
I will do more of what I did to study for this test but use more materials and different questions.
Review class notes
Yes, because there were several concepts that could only be learned through rote memorization.
I might also work some problems on Canvas.
Watch calculus videos No. I needed a better understanding that a video could not help with.
Ask teachers or other students with a better understanding about the material.
Use the worked examples my teacher provided, Watch calculus videos Yes Study with another student
Use the worked examples my teacher provided, Review class notes
Kind of, I feel like I could have tried harder to prepare myself but got swamped with work from other classes. I would prepare earlier.
Use the worked examples my teacher provided, Study with another student
I think it did because I did pretty good on some sections of the test that I actually studied hard for.
I could practice and study the videos more.
151
Test2–Unit3_1What did you do to reach your predicted learning levels? (select all that apply) Did what you do helped? Explain.
What might you do next time instead of or in addition to what you did this time?
Review class notes Yes Actually study Use the worked examples my teacher provided, Watch calculus videos
Yes, the videos explained the subjects I did not know. I will do edpuzzles.
Use the worked examples my teacher provided, Watch calculus videos, Review class notes, sibling help
Called my brother to explain everything I didn't understand
study more than the night before the test
Study with another student Kind of? If I didn’t do anything I would’ve failed harder than if I did
Hopefully I’ll actually study more
Use the worked examples my teacher provided, Get help from a teacher or tutor, Review class notes Yes, I understood the major concepts. I should watch calculus videos. Use the worked examples my teacher provided, Watch calculus videos, Review class notes
I thought it did but I did not do well on the test so I guess not. Study more I guess
Use the worked examples my teacher provided, Review class notes
No, I did not study the notes as much as I needed Study routinely
Watch calculus videos, Review class notes
I just understood the information in class better
Actually study. I say I’m going to study every time and I never do. I’m really trying to change that.
Use the worked examples my teacher provided, Review class notes, Learning Goal Quiz
Yes, it made me feel better about recognizing the types of questions and what was expected for the answers.
I might watch some videos to prepare more to make sure that I have everything down.
Use the worked examples my teacher provided, Watch calculus videos, Review class notes
Yes, my levels from the first quiz went up on the test.
Study with another student because they might know an easy way to do something.
Use the worked examples my teacher provided, Study with another student, Review class notes
I feel like looking at the problems that the teacher gave on canvas and through the level quiz helped.
I probably should watch more of the videos on Edpuzzle.
Use the worked examples my teacher provided, Study with another student, Review class notes
I got help from a student, and it helped to have someone else’s explanation as well. Watch the videos.
Use the worked examples my teacher provided, Watch calculus videos, Review class notes
It definitely helped to review the worked examples because they helped me to understand.
Next time, I will also study outside of school with some friends.
Use the worked examples my teacher provided, Study with another student, Review class notes
It was not enough , for the results were not high More
Watch calculus videos, FRQs
It most definitely did help. The FRQs were similar to the questions on the test, so I feel like I had an edge, if you will, and it really helped me recognize similar questions.
Next time, I'd more than likely study with someone else other than trying to do it all by myself.
I worked a FRAPPY and a lab Maybe? I did a lot better on the test than I did on the practice. Nothing comes to mind
152
Watch calculus videos Kind of, I didn’t study too much. Study more Use the worked examples my teacher provided, Study with another student
Yes. Going over several practice problems and making sure I knew what each question was asking definitely helped.
More practice problems. Watch calculus videos.
Use the worked examples my teacher provided, Watch calculus videos
I think it was but I was tired so I didn’t fully absorb the information review with others
Use the worked examples my teacher provided, Study with another student
yes, I was able to work through the type of problems that were on the test
I will study more before the test, rather than just the block before
I ended up not doing anything :(( I am going to try to do what i said i will do.
Use the worked examples my teacher provided, Watch calculus videos Kind of
Be able to do all of the canvas quizzes and watch more videos on fuzzy content and work more examples.
Use the worked examples my teacher provided, Watch calculus videos, Review class notes, The mighty Khan academy
It did indeed. I would not have gotten the curve without it
Get that kid named Lee to tutor me finally.
Test3–Unit3_2What did you do to reach your predicted learning levels? (select all that apply) Did what you do helped? Explain.
What might you do next time instead of or in addition to what you did this time?
Study with another student, Watch calculus videos Yes Get help from the teacher
Watch calculus videos, Review class notes, MathXL Honestly yes it helped so much.
I have definitely got to watch more edpuzzles because those things work wonders.
Use the worked examples my teacher provided, Study with another student, Review class notes
Studying with classmates really cleared up some confusion
Prepare much more and learn related rates
I asked classmates questions Yes, i got things i was confused on kind of made clearer The things i said i would do
Use the worked examples my teacher provided, Watch calculus videos, Review class notes
Yes. I did not understand L'Hopital's Rule or related rate problems before studying. Maybe work more problems.
Use the worked examples my teacher provided, Study with another student, Review class notes A little....as in .001 percent
make more time rather stress my mind out
Use the worked examples my teacher provided, Study with another student, Get help from a teacher or tutor
yes. The self assessment provided me challenging questions so I could solidify my knowledge on the LGs
I would like to do the canvas questions.
No. I got a 50 on the test.
Lee said the FRQs are helpful, and rollover PSP is never a bad thing. I'll probably do most if not all of those.
153
Use the worked examples my teacher provided, Study with another student, Watch calculus videos
It did help a lot. Being able to study with a friend helped me understand where I was going wrong
Do mathxl and canvas practices as the unit progresses
Watch calculus videos, Review class notes
yes, I wouldn't have known how to do the limit problems Steady all the terms on the test.
Test4–Unit4What did you do to reach your predicted learning levels? (select all that apply) Did what you do helped? Explain.
What might you do next time instead of or in addition to what you did this time?
Use the worked examples my teacher provided, Study with another student, Get help from a teacher or tutor, Watch calculus videos, Review class notes
It helped for the most part. I just did not do enough.
I would take more time to study and do more activities.
Use the worked examples my teacher provided, Get help from a teacher or tutor, Watch calculus videos, Review class notes
Yes, it helped me know each skill that I needed for the test. I would like to do the canvas quizzes
Study with another student, Review class notes A little. I didn't try very hard.
If my exam grade was poor, I will work probably the whole review.
Use the worked examples my teacher provided, Review class notes
Use the worked examples my teacher provided, Watch calculus videos, Review class notes
I thought it helped but my grade said it didn’t. It was mostly the trig functions that tripped me up this time.
Probably look over more examples of finding the integrals of different trig functions.
Study with another student
Not for this particular test, but I plan to retake it when we get back to school.
Watch videos and find my notes for class
Use the worked examples my teacher provided, Watch calculus videos
I went on the student document and read through all the papers. I thought that helped me out a lot because during class, I was not able to grasp a complete understanding of the unit.
I might do more self assessments along with the videos and student folder.
Use the worked examples my teacher provided, Study with another student
It helped me understand my mistakes more Exactly the same thing
Use the worked examples my teacher provided, Study with another student, Watch calculus videos
Some of it did, but I feel like I could have done a little more.
I will probably start working a little bit more ahead of time so that I have time to ask more questions that I can't figure out on my own or online.
Use the worked examples my teacher provided, Study with another student, Review class notes
It helped mostly. I didn’t reach my goal but I did improve significantly with my understanding.
I might watch some videos and do a little more of what I did.
154
Use the worked examples my teacher provided, Study with another student, Review class notes
A little. I could have done better if I had more time, but I'm a slow test taker. I might look at calculus videos.
Review class notes It did, a little refresher Study with someone else
Use the worked examples my teacher provided, Study with another student, Review class notes
Yes it did. I got a really good score on the test.. Watch some calculus videos
Use the worked examples my teacher provided, Watch calculus videos, Review class notes
Khan Academy really helped me learn the fundamental theorem of calculus in more detail than I was about to in class. Do more FRQs!
FRQs
Yes doing the AP practice problems are extremely effective as they as the similar types of questions asked on the tests
Next time I’d probably study with a student or look at worked notes
Use the worked examples my teacher provided, Study with another student, Watch calculus videos, Review class notes I think it did More studding
Test5–Unit5What did you do to reach your predicted learning levels? (select all that apply) Did what you do helped? Explain.
What might you do next time instead of or in addition to what you did this time?
Use the worked examples my teacher provided, Watch calculus videos, Review class notes
Nope- I had no idea what I was doing Get help from a tutor
Use the worked examples my teacher provided
Kind of. I didn’t do it to the extent that is necessary. Study more
Made flash cards No. I don't know why, but knowing more things didn't help. ??? MOAR FLASH CARDS
Use the worked examples my teacher provided, Review class notes Decently until I forgot Do it more
Mathxl :( No, cant study with math xl Study notes
Use the worked examples my teacher provided, Review class notes
I reviewed the notes in class, and I felt like that helped me understand a bit more.
I think I might review the notes in the student folder more.
Use the worked examples my teacher provided
No, I didn't have a good grasp on the subject to begin with, so the problems didn't help my understanding of this unit. Watch khan academy
Use the worked examples my teacher provided, Review class notes
I worked some of the problems from class and worked the self assessment and the frqs
The procrastination stressed me out! But I think it made me work harder.
Use the worked examples my teacher provided, Study with another Kind of
Watch more videos on how to do the topic.
155
student, Review class notes, Self assessment
Review class notes
Use the worked examples my teacher provided, Study with another student, Review class notes, MathXL
It did help me because on a good bit of problems I felt I knew where to start at least. Do my homework on time
Use the worked examples my teacher provided, Watch calculus videos
Worked with friends outside of class
learn the material enough to pass the test.
156
VITA
JenniferCarnesWilson
Profile
Myphilosophyofteachingandlearningmathematicscanbesummedupinthenameofmyblog,“EasingtheHurrySyndrome”,andafewTwitterhashtags:#slowmathand#AskDontTell.Ibelievethatallstudentscanlearn,andIseektocreateacommunityoflearnerswherequestionsarenotonlywelcomedbutsoughtthroughopen-endedtasksandinquiry-basedinstruction.IhavelearnedthemostfromtheworkofDylanWiliam(EmbeddedFormativeAssessment),MaryKaySteinandMargaretSchwanSmith(5PracticesforOrchestratingProductiveMathematicsDiscussions),andJamesPopham(TransformativeAssessment,TransformativeAssessmentinAction).Theirresearchonformativeassessmentandstudentdiscourse,coupledwithimplementingtheCommonCoreStandardsforMathematicalPractice,hastransformedmyclassroomovertheyears.Whilemymostrecentworkhasshiftedfromyounglearnerstoadultlearners,Icontinuetobelievethatalllearnerscanlearn,thatlearnersdon’tcometothelearningepisodevoidofknowledge,andthatquestionsandreflectionareintegraltoasharedlearningexperience.
Education&Licensure
Ed.D.Education,emphasissecondarymath,TheUniversityofMississippi expectedMay2019M.S.Mathematics,summacumlaude,MississippiCollege July2003-2005B.S.Mathematics,specialdistinction,MississippiCollege August1990-May1993NationalBoardforProfessionalTeachingStandards#219902258,AYAMath November1999-2029StateofMississippiEducatorLicense,Mathematics7-12,APCalculusBC validthroughJune2025 Experience
IllustrativeMathematicsApril2018-present(full-time),November2013-March2018(contractor)n HighSchoolProfessionalLearningLead,HighSchoolContentWriter,IllustrativeMathematicsMaster
Coach/Facilitatorforvirtualandface-to-facetrainingssuchasUsingMathematicalRoutinesforPurposefulInstruction,5Practices,andIllustrativeMathematicsAlgebra1,Geometry,Algebra2MathCurriculumUnitOverviews,WritingProfessionalDevelopmentModulesforIllustrativeMathematics6–8MathCurriculum,Item&TaskReview,SmarterBalancedDigitalLibraryProject
RankinCountySchoolDistrict,Brandon,MS Aug.2003-Sept.2017(retired),Aug.1996-May2002n CurriculumSpecialist–Mathematics(July2013-September2017):WorkwithK-12teachersto
implementstandards,curriculum,MathPractices,formativeassessment,teachingwithtechnology;deliverprofessionallearning;SREB/MathematicsDesignCollaborative(MDC)CoachtoassistteachersimplementingMathematicsAssessmentProjectFormativeAssessmentLessons
n NorthwestRankinHighSchoolMathematicsDepartmentChair,LeadershipTeam,&Teacher:LasttaughtAPCalculusandgeometry;experienceteachingallhighschoolmathematicscourses;leadingmathematicsProfessionalLearningCommunity
157
WilliamCareyUniversity,Hattiesburg,MS July2009-presentn Currentlyteachonlineandhavetaughtface-to-facegraduatelevelcoursesincorporatingmathematics
teachingandtheappropriateuseoftechnologyintheclassroom
AdditionalExperience
n ClintonHighSchool(10-12),ClintonPublicSchools,August2002-May2003
n ByramAttendanceCenter(8-12),HindsCountyPublicSchools,August1993-May1996
ProfessionalService&Publications
n www.https://easingthehurrysyndrome.wordpress.com,BlogreflectingonlessonsthatIteachandinstructionaladjustmentsthatImakethroughoutthelesson
n www.https://slowmathmovement.wordpress.com,BlogconnectingtheSlowMovement(food,music,exercise,money,travel,…)withmathematicseducation
n TexasInstrumentsTeachersTeachingwithTechnologyInstructorProgram,2007-present
o ProfessionalDevelopmentInstructorandAuthor,WebinarPresenter,numerousspeakingengagementsincludingTIASSMDinner,April2012and2015,TINCSMLuncheon,April2013,TIInternationalConferences,CMC-South,NCSM,NCTM,APAnnualConference;PilotTeacher
n SouthernRegionEducationalBoard(SREB)HighSchoolsThatWork(HSTW)MathematicsDesignCollaborative(MDC)CertifiedLocalTrainer
n CollegeBoardPre-APMathematicsDevelopmentCommittee,APCalculusSyllabusProject,Reader
n MississippiDepartmentofEducation
o CCRSMathTaskForce;GeometryInstituteAuthor&Presenter;SummerBootCampAuthor&Presenter,2014;Publication,SecondaryTechDiscoveryCurriculumRevision,April2006
n MississippiCouncilofTeachersofMathematics
o Treasurer,2004-2012;NCTMRepresentative,1995-2004;Conferencepresenter,2005-2018
n Publication,casestudyforInstructionalLeadershipintheContentAreas:CaseStudiesforCurriculumandInstructionpublishedbyRoutledge/UCEA(NewYork),2018.
n Publication,“BackTalk:EasingtheHurrySyndrome”,PhiDeltaKappanMagazine,May2011
n Publication,CARS(CareerAwareness:RoadwaytoSuccess),throughMDOT&RCUforSTEM,2008
Awards
n PresidentialAwardforExcellenceinMathematics&ScienceTeaching,2011
n T3LeadershipAward,March2014
n STARTeacher,2015,2014,2012,2011,2010,2005
n MississippiCollegeMathematicsAlumnusAward,2012
n 2010YaleEducatorAward
n RankinCountySchoolDistrictTeacheroftheYear,2008;NWRHSTeacheroftheYear,2008,2006;TeacheroftheMonth,April2005