Leading Learners to Level Up in a High School Mathematics

University of Mississippi University of Mississippi

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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

Copyright©2019byJenniferC.Wilson

Allrightsreserved.

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

iii

goingtobeonthetest,butalsoarebetterabletopredicthowtheyaregoingtodoonthe

test.

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”


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



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

16

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).


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




Thisresearchstudybuildsontheimportanceofestablishinglearninggoalsand



knowbothwhattheyhavelearnedandwhattheystillneedtoknow.

20

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).




21






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.




22





performonatest?








performancelevel?









students.

23









progressinlearning?


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_2predictedlevel

7_2actuallevel


7_3predictedlevel

7_3actuallevel


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_2predictedchange


7_3predictedchange


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





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


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


Total 64 8663.54

Table4.4

One-WayAnalysisofVarianceofSemester2PrecalculusGradesbyPrecalculusSection

Source df SS MS F p

Betweengroups 2 202.18 101.09 0.75 0.477


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



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


onatest?

Question2.1


performancelevel?


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


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



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



H0:Forstudentswhoareinengineering,themeandifferenceinactualperformancelevel

andstudentpredictedlevelisequaltothemeandifferencefornon-engineeringstudents.


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




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


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).

51

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%)


Figure4.2

TestResultsforSelf-ReportingLevel2andPredictingLevel3

EventhoughahigherpercentageofstudentsperformedattheirpredictedLevel3or

performedhigherthanpredictedatLevel4duringUnit9,whenstudentshadtheworked

examplesandmetacognitivetreatment,chi-squareresultsshownostatisticalsignificance

forstudentswhoself-reportedLevel2andpredictedLevel3betweenthosewhoreceived

52

treatmentsandthosewhodidnot.Testresultsindicateafailuretorejectthenull

hypothesis.

Theresearcherelectednottoexaminemorecloselythetenratingsthatself-

reportedLevel1andpredictedLevel2orLevel3,sincetheycomprisedlessthan2%ofall

ratings.Eightofthoseratingsweremadebythesametwostudents.

SubgroupWhoSelf-ReportedLevel2orLevel3andPredictedtheSameLevel


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%)


Figure4.3

TestResultsforSelf-ReportingLevel2orLevel3andPredictingtheSameLevel

Chi-squareresultsshowastatisticallysignificantdifferenceinresultsforself-reported

ratingsofLevel2orLevel3andpredictedratingsofthesamelevel.Thenullhypothesisis

rejected.While29%ofstudentswhodidnotreceivethetreatmentsperformedatalevel

lowerthanself-reportedorpredicted,thechi-squaretestshowsthatonly22%were

54

expectedtodoso.Whileonly11%ofstudentswhodidreceivethetreatmentsperformedat

alevellowerthanself-reportedorpredicted,thechi-squaretestshowsthat22%were

expectedtodoso.

Therewereonly3ratingsforstudentswhoself-reportedLevel1andpredictedthe

samelevel.Theresearcherelectednottofurtheranalyzethoseratingsbecausethey

comprisedlessthan1%ofallratings.

SubgroupWhoSelf-ReportedLevel4andPredictedtheSameLevel


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%)


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


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


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.

60

OneoftheAlgebra1teacherswhoworkswithMs.Bairdmentionedthather

students’gradeswerenotgreat,andsoMs.Bairdsuggestedthatsheconsiderhaving

studentsratethemselvestogivethemtimeandspacetoreflectontheirlearningandthink

aboutimproving.

SUMMARY

Thischapterreportedtheresultsofthismixedmethodsstudy,seekinginsighton

whetherstudents’actualperformancelevelsontestdayareclosertothestudentpredicted

levelsorclosertostudents’self-reportedlevelsforalearningtargetandwhether

treatmentssuchasworkedexamplesandanemphasisonmetacognitionimprovethe

predictions.Thenextchapterwillsummarizetheresults,considerwhatconclusionscanbe

madeandwhy,considerlimitationsofthestudy,andmakesuggestionsaboutfurther

researchonthistopic.

61

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).






62




Thestudyhypothesizedthattreatmentssuchasworkedexamplesandanemphasis

onteachingstudentstheimportanceofmetacognitionnotonlyhelpstudentsknowwhat

thelearningtargetisbutalsohowtoreachit,thushavingapositiveeffectonstudent

successpredictingtheirexpectedsuccessonalearningtarget,and,infact,confirmedthat

workedexamplesandmetacognitivestrategiesdocontributetohowwellstudentspredict

theirexpectedsuccess.ThroughquestionsforastudentGoogleformandateacher

interview,theresearcheralsosoughttodeterminestudentandteacherperceptionsaround

usinglearningtargetstoinformstudentprogressinlearning.Inconsideringhowwell

studentspredictedthelevelatwhichtheyexpectedtoperformforeachlearningtargetand

whattreatmentsmightimprovepredictedsuccess,aseriesofrelatedquestionswas

examined.








performonatest?

63








performancelevel?









students.






64




progressinlearning?


progressinlearning?

CONCLUSIONS

Thestudyconfirmsthehypothesisthattreatmentssuchasworkedexamplesand

metacognitivetreatmentcanhaveapositiveimpactonstudentsuccesspredictingtheir

progresstowardsalearningtarget.

Question1



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


onatest?

Theresearcherconsiderednextwhetherprovidingstudentswithworkedexamplesmight

improvetheirpredictions.

Question2.1


performancelevel?



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?

67

Question2.2



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




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




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


learning?



Overhalfofthestudentsreportedthattheyfrequentlyoralwayspayattentiontothe

learningtargetsineachunit(seeFigure4.5).

Whatisgoingtobeonthetestshouldnotbeasurprisetostudents.Learningtargets

shouldinformteacherswhatcontent-aligneditemstoputonthetestandshouldinform

studentswhatcontent-aligneditemswillbeonthetest.Almostallstudentshavenoticed

thatthelearningtargetsareincludedonthetest,andabouthalfreportedthathavingthe

learningtargetsonthetestfrequentlyoralwayshelpsthemwhiletheyaretakingthetest

(seeFigure4.5).



idealwaytoimprovestudentunderstandingofthelearningtarget.Sharinganovice

workedexamplealongsideaproficientworkedexamplecanalsoilluminatestudent

understandingofthelearningtarget(Popham,2008).Ms.Bairdsharedleveledworked

examplesforeachlearningtargetduringUnit8andUnit9(seeAppendixH).

ForUnit8,Ms.BairdpostedtheleveledworkedexamplesthroughCanvas,theclass

learningmanagementsystem,butshedidnottalkwithstudentsabouthowtheleveled

workedexamplesmightbehelpfultothemintheirlearning.Shereportedduringthe

teacherinterviewthatshedidnotthinkmanystudentsusedthemforUnit8.However,

72

whenstudentsworkedoncorrectingtheUnit8test,sheencouragedstudentstousethe

workedexamples.Thatencouragementandapurposefulmetacognitivetreatmentwith

studentsforUnit9,causedmorestudentstomakeuseoftheworkedexamplesduringUnit

9.Bytheendofthestudy,three-fourthsofstudentsreportedthattheyusedtheworked

examplesinUnits8and9,andthree-fourthsofstudentsreportedthattheworked

exampleswerehelpfulinpreparingforUnits8and9.Nine-tenthsofstudentsreportedthat

havingworkedexamplesforallunitswouldbehelpful.


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


learning?

73

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

74

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.

75

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

76

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.

77



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

78

ordertosaywithconfidencethatworkedexamplesandmetacognitivestrategiesimprove

howwellallstudentspredicttheirprogresstowardsalearningtarget.

CONCLUSIONS




Thisresearchstudybuiltontheimportanceofestablishinglearninggoalsand



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



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.

85

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.

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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

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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

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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

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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.

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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.

103

LISTOFREFERENCES

104

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ThousandOaks,CA:Corwin.

Bailey,K.,&Jakicic,C.(2012).Commonformativeassessment:atoolkitforprofessional

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Baron,L.M.(2016).FormativeAssessmentatWorkintheClassroom.TheMathematics

Teacher,110(1),46-52.doi:10.5951/mathteacher.110.1.0046

Black,P.&Wiliam,D.(1998).Insidetheblackbox:Raisingstandardsthroughclassroom

assessment.PhiDeltaKappan,80(2),139-148.

Boston,M.,Dillon,F.,Smith,M.S.,&Miller,S.(2017).Implementingeffectivemathematics

teachingpracticesingrades9-12.Reston,VA:NationalCouncilofTeachersof

Mathematics.

Bransford,J.D.,Brown,A.L.,&Cocking,R.R.(2001).Howpeoplelearn:Brain,mind,

experienceandschool.Washington,D.C.:NationalAcademyPress.

Ebby,C.B.&Petit,M.(2017).UsingLearningTrajectoriestoEnhanceFormative

Assessment.MathematicsTeachingintheMiddleSchool,22(6),368-372.

doi:10.5951/mathteacmiddscho.22.6.0368

Hattie,J.[CognitionEducation].(2012,May2).SelfreportedgradeswithJohnHattie(New

Zealand)[Videofile].Retrievedfromhttps://vimeo.com/41465488

105

Hattie,J.(2012).Visiblelearningforteachers:Maximizingimpactonlearning.NewYork,NY:

Routledge.

Hattie,J.,Fisher,D.,&Frey,N.(2017).Visiblelearningformathematics,gradesK-12:what

worksbesttooptimizestudentlearning.ThousandOaks,CA:CorwinMathematics.

Heritage,M.(2018).Assessmentforlearningassupportforstudentself-regulation.The

AustralianEducationalResearcher,45(1),51-63.

Hiebert,J.&Stigler,J.(2017).Teacherversusteachersasaleverforchange:Comparinga

JapaneseandaU.S.perspectiveonimprovinginstruction.EducationalResearchers

46(4),169-176.doi:10.3102/0013189X1771189

Horn,I.S.(2011).Strengthinnumbers:Collaborativelearninginsecondarymathematics.

Reston,VA.:NCTM.

Kanold,T.D.,&Larson,M.R.(2012).CommoncoremathematicsinaPLCatwork.

Bloomington,IN:SolutionTreePress.

NationalCouncilofTeachersofMathematics(NCTM).(2014).Principlestoactions:ensuring

mathematicalsuccessforall.Reston,VA.:NCTM.

NationalGovernorsAssociation(NGA)CenterforBestPractices,CouncilofChiefState

SchoolOfficers.(2010).Commoncorestatestandardsmath.WashingtonD.C.:National

GovernorsAssociationCenterforBestPractices,CouncilofChiefStateSchoolOfficers.

OpenUpResources(2017a).Grade8courseguide.Retrievedfrom

https://im.openupresources.org/8/teachers/teacher_course_guide.html

OpenUpResources(2017b).Congruentpolygons.Retrievedfrom

https://im.openupresources.org/8/teachers/1/12.html

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Popham,W.J.(2011).Transformativeassessmentinaction:aninsidelookatapplyingthe

process.Alexandria,VA:ASCD.

Popham,W.J.(2008).Transformativeassessment.Alexandria,VA:Associationfor

SupervisionandCurriculumDevelopment.

Renkl,A.(2014).Learningfromworkedexamples:Howtopreparestudentsformeaningful

problemsolving.InV.A.Benassi,C.E.Overson,&C.M.Hakala(Eds.).Applyingscienceof

learningineducation:Infusingpsychologicalscienceintothecurriculum.Retrievedfrom

theSocietyfortheTeachingofPsychologywebsite:

http://teachpsych.org/ebooks/asle2014/index.php

Smith,M.S.,&Stein,M.K.(2011).5practicesfororchestratingproductivemathematics

discussions.Reston,Va.:NationalCouncilofTeachersofMathematics.

Sousa,D.(2015).Brain-friendlyassessments:Whattheyareandhowtousethem.WestPalm

Beach,FL:LearningSciencesInternational.

Sousa,D.&Tomlinson,C.(2011).Differentiationandthebrain:Howneurosciencesupports

thelearner-friendlyclassroom.Bloomington,IN:SolutionTreePress.

Wiliam,D.(2011).Embeddedformativeassessment.Bloomington,IN:SolutionTreePress.

Wiliam,D.,&Leahy,S.(2015).Embeddingformativeassessment:practicaltechniquesforK-

12classrooms.WestPalmBeach,FL:LearningSciencesInternational.

Wiliam,Dylan;Thompson,M;(2007)Integratingassessmentwithlearning:whatwillit

taketomakeitwork?In:Dwyer,C.A.,(ed.)Thefutureofassessment:Shapingteaching

andlearning.(pp.53-82).Mahwah,NJ:LawrenceErlbaumAssociates.

107

APPENDICES

108

APPENDIXA

InformationSheet

109

APPENDIXB

RecruitmentScript

110

APPENDIXC

ParentalConsent

111

112

APPENDIXD

SchoolDistrictConsent

113

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

⎧⎨⎪

⎩⎪

119

APPENDIXG

LeveledLearningProgressionRatingExample

120

APPENDIXH

Unit8LearningTargetswithLeveledWorkedExamples

121

122

123

APPENDIXI

AssessmentInstrumentQuality-EvaluationTool

124

APPENDIXJ

Unit7LearningTargetsSelf-AssessmentForm

125

126

APPENDIXK

StudentFeedbackonLearningTargetsForm

127

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

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APPENDIXM

InterviewQuestionsforTeacher

• Whatdoyouthinkabouttheimpactofstudentsratingandpredictingtheirprogress

onlearningtargetsbeforeatest?• Whatdoyouthinkabouttheimpactofleveledlearningprogressionswithworked

examplesonstudentlearning?• Whatdoyouthinkabouttheimpactofemphasizingmetacognitionwithstudentson

studentlearning?• Whatplansdoyouhave,ifany,forcontinuinganyofwhatyouhavedonethis

quartero Nextyear?o Forotherunits?o Inotherclasses?

• Whatwouldyoutelland/orrecommendtootherteachersaboutyourexperiencethisquarter?

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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.

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APPENDIXO

Addendum:Unit3_1Pre-testQuestions

134

135

136

137

138

139

APPENDIXP

Addendum:Unit3_1Pre-testForm

140

141

142

143

144

APPENDIXQ

Addendum:Unit3_1LearningGoalsSelf-AssessmentForm

145

146

APPENDIXR

Addendum:LearningGoalsReflectionForm

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.

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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.


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


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.


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.


I got help from a student, and it helped to have someone else’s explanation as well. Watch the videos.


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.


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.


I think it was but I was tired so I didn’t fully absorb the information review with others


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.


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.


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


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


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.



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.



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


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.


It helped me understand my mistakes more Exactly the same thing


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.


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


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


Yes it did. I got a really good score on the test.. Watch some calculus videos


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




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


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


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


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

Documents

Leading Learners to Level Up in a High School Mathematics