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TME, vol10, no.3, p. 621
The Mathematics Enthusiast
, ISSN 1551-3440, Vol. 10, no.3, pp. 621-646
2013©The Author(s) & Dept. of Mathematical Sciences-The University of Montana
DevelopingEffectiveMathematicsTeaching:
AssessingContentandPedagogicalKnowledge,Student‐CenteredTeaching,and
StudentEngagement
SerigneM.Gningue1,RogerPeach1,&BarbaraSchroder2
1LehmanCollege,CityUniversityofNewYork
2CenterforAdvancedStudyinEducation,GraduateCenter,CityUniversityofNewYork
Abstract:TheMathematicsTeacherTransformationInstitutes(MTTI)programattempts
todevelopmathteacherleadersinpartbyprovidingcontent,inquiryandleadership
coursesaimedatmakingthemmoreeffectiveteachers.Weassessedprogressbyobserving
teacherleaders’teachingpractices,andencouragingthemtointroduceorextendstudent‐
centeredpedagogyintheirclassrooms.Wefoundtherewaslittlerelationshipbetweenour
measuresofmathematicscontentknowledgeandstudent‐centeredpedagogy.But
teacherswhoemployedstudent‐centeredpedagogytendedtohavemorehighly‐engaged
mathstudentsintheirclassrooms.
Keywords:effectivemathematicsteaching;mathcontentknowledge;student‐centered
teaching;studentengagement.
Improvingstudentachievementinmathematicsandsciencehasbeenaconcernin
theUnitedStatesofAmericasincetheearly1980swheninternationaltestsbeganshowing
U.S.studentsfallingbehindmostdevelopedcountriesinmathematicsandscienceskills.
ManyU.S.studentsdonotobtaintheknowledgeandskills,particularlyinscience,
technology,engineering,andmathematics(STEM),whicharerequiredforsuccessinthe
globalmarketplaceofthe21stcentury(NationalAcademyofSciences,2006).
1Serigne.gningue@lehman.cuny.edu
Gningue, Peach & Schroder
Educators,educationalresearchers,andpolicymakershavenotalwaysagreed
aboutthereasonsforthefailureofU.S.studentstoperform.Somearguemany
mathematicsteachershaveinadequatemathematicalcontentknowledgethemselves,and
thusareunabletoteachtheirstudentstothehighestlevel(Ahuja,2006;Ginsburg,Cooke,
Leinwand,Noell&Pollock,2005).Others(Darling‐Hammond,2007;NationalCouncilof
SupervisorsofMathematics[NCSM],2008;NationalCouncilofTeachersofMathematics,
2000;OfficeofScienceandTechnologyPolicy,2006;U.S.DepartmentofEducation,2004;
NationalScienceBoard,2006),inpart,relatesuchaneducationalfailurenotonlytothe
lackofqualifiedteacherswithsolidcontentknowledgeinSTEM,butalsotoaprofoundlack
ofunderstandingofteachingandlearningingradesK‐12,whichmayleadtotheuseof
ineffectiveteachingpractices.ForBrownandBorko(1992),andBallandBass(2000),
understandingcontentknowledgeandmethodsofinquiryinmathematicsareatthecoreof
effectiveteachingandlearning.Theuseofinquiry‐basedapproachestoinstruction,in
whichstudentshaveopportunitiestoconstructtheirownunderstandingofbasicconcepts,
isthoughtbymanyeducationaltheoriststobemostappropriateindevelopingstudents’
understandingofmathematicsandscienceconcepts.Suchapproachescallforteachersto
beabletoengagestudentsincritical,in‐depth,higher‐orderthinkingthroughuseof
manipulatives,technology,cooperativelearningandotherpedagogicalapproachesthat
enablestudentstoconstructmathematicsconceptsontheirownthroughreasoning,
verifying,comparing,synthesizing,interpreting,investigatingorsolvingproblems,making
connections,communicatingideasandconstructingarguments(Grouws&Shultz,1996;
NationalCouncilofTeachersofMathematics[NCTM],2000).Theseapproachesare
characteristicofwhatisoftencalledstudent‐centeredteachingasopposedtotheso‐called
TME, vol10, no.3, p. 623
“traditional”approachesinwhichthepredominantviewisthatmathematicsteachingisa
show‐and‐tellaswellasasupervisionofdrillsandpractice(Davis,1988).Inthisview,itis
assumedthatlearningoccurspassivelywhenstudentsabsorbreceivedknowledgefroman
all‐knowingteacherorexpert.Thisapproachisoftenreferredtoas“teacher‐centered.”
TheMathematicsAssociationofAmerica(MAA,2008)arguesthatinordertoprepare
studentsfortheincreasinglycomplexmathematicsofthiscentury,astudent‐centered
approachtoteachingismoreappropriatethanthetraditionalteacher‐centeredapproach.
TheMAA(2008)assertstheneedtodeveloppedagogiesthatcouldbeusedeffectivelyto
facilitatestudents’mathematicalabilities.InessencetheMAA(2008)advocatesforan
increaseinstudent‐centeredteachingandlearningandadecreaseinteacher‐centered
pedagogy.Oneassumptionisthatanincreaseinstudent‐centeredteachingwillresultin
increasedstudentengagementinmathematicsand,byimplication,thisincreased
engagementwilllead,inturn,toincreasedstudentachievement.Forexample,various
researchersarguethatstudentsaremoreengagedandachievemorewhenteachersrelate
newlearningtopriorlearning,modelproblemsandprovidethemwithavarietyof
opportunitiestoapplyanduseknowledgeandskillsindifferentlearningsituations(Kemp
&Hall,1992;Rosenshine,2012;Taylor,Pearson,&Walpole,1999).
LogicModelandTheoryofActionfortheProject
OneoftheaimsoftheMathematicsTeacherTransformationInstitutes(MTTI)isto
encourageparticipantteacherstodevelopboththeirmathematicscontentknowledgeand
astudent‐centeredpedagogy,assumingthatthesedevelopmentswillleadtoincreased
studentengagementinmathematics.Thisresearchaimedtoseewhetherthegoalwas
met,andtheassumptionwasjustified.
Gningue, Peach & Schroder
MTTIisaNationalScienceFoundation(NSF)‐fundedprogramdesignedtosupport
thedevelopmentofteacherleaderstostrengthenmathematicsteachingandlearningin
NewYorkCity,especiallyinBronxmiddleandhighschools.MTTIdevelopedathree‐year
three‐dimensionalprogramthatfocusesondeepeningparticipatingteachers’content
knowledge,broadeningtheirpedagogicalrepertoirethroughtheprocessofinquiry,and
developingtheirleadershipcapacitiesacrossanumberofdomainswithinthecontextofa
professionalcommunity.Themodelengagesteachersinaprocessofinquirythatdoesnot
ceaseinaskingquestionsandunderstandingproblems,continuallyrevisitingcriticalissues
relativetoteachingandlearning,designingplanstoresolvetheissues,implementingthe
plans,andcollectingandanalyzingdatatoassesstheeffectivenessofthedesignedplans.As
teachersimprovetheirpedagogicalskills,theyincreasetheirabilitytoexplaintermsand
conceptstostudents,interpretstudents’statementsandsolutions,engagestudentsin
critical,in‐depth,higherorderthinking(Copeland,2003;Grouws&Shultz,1996;Hill,
Rowan,&Ball,2005;NationalCouncilofTeachersofMathematics[NCTM],2000).
Essentially,theaimistodevelopteachers’student‐centeredpedagogy.
MTTIisfundedtosupporttwocohortsof40teacherswithatleastfouryears
teachingexperienceoverfiveyears.Thefirstcohortcompletedtheprogramafterthree
yearsinJune2011.Thispaperreportsresultsfromthefirstcohort.Theresearch
componentofMTTIseekstobroadentheknowledgebaseonteachingandlearningin
mathematicsthroughnewunderstandingof:1)howthestudyofconceptually‐challenging
mathematics—particularlyinalgebraandgeometry—benefitsteachers;2)howclassroom‐
basedactionresearchcontributestocriticalandanalyticalunderstandingofthe
TME, vol10, no.3, p. 625
relationshipsbetweenteachingpracticesandstudentlearning;and3)howmulti‐levelsof
supportprepareteacherswithatleastfouryearsteachingexperienceforleadershiproles.
MTTI’stheoryofaction,depictedinFigure1,hypothesizesinessencethatteacher
backgroundandcharacteristics,schoolclimate(especiallyasrepresentedthroughteacher‐
teacherinteractions)andMTTIexperienceswillimpactparticipants’teacher‐leader
practices,oneofwhichiseffectiveteaching.ThethreemaincomponentsmakingupMTTI
experiencesaremathcontentcourses,inquiry‐basedactionresearchcourses(pedagogy),
andaleadershipcourse.
MTTIaimstosupplementmathteachers’contentknowledgeandhelpteachers
makeandsustainfundamentalshiftsinpractice.Ourhopeisthatsuchchangeswillresult
inmoreeffectiveteachingandteacherleadership.Inturn,wehopethateffectivemath
teachingwillleadtoincreasedstudentengagementinmath.
Students’
Outcomes
Teacher knowledge
Content/Pedagogy/
Leadership
Teacher-Leader
Practices:
Effective Teaching
Teacher
Background and
Characteristics
Teacher Self-
efficacy
Teacher-teacher
Interactions
MTTI
Experiences
School
Leadership/Climate
Gningue, Peach & Schroder
Figure1.MTTI’stheoryofaction.
MTTIProjectOutline
ImprovingTeachers’MathContentKnowledge
TwocoursesaimedatimprovingMTTIparticipants’mathcontentknowledgewere
runthroughoutthespringandfallsemestersof2009.Oneofthecourseswasinmath
fundamentalsandtheotheringeometry.Themathfundamentalscoursefocusedon
algebraandintegratedmathematics.Thegeometrycoursewasbasedaroundgeometric
proofs,andwasrelatedtotheNewYorkstatestandardsforgeometry.Participantsinthe
geometrycoursewererequiredtoundertakeprojectsrelatedtothetopicstaughtinthe
course.ThecoursesweretaughtbymembersoftheLehmanCollegemathematicsfaculty.
ActionResearchCourses
MTTIparticipantstookatwo‐partcourseseriesinclassroom‐basedinquiry
includingactionresearch.Thecourseseriesranforatotalof90classroomhours.Part1of
thisseriestookplaceduringspring2010,“ClassroomInquiryinMiddleandHighSchool
Mathematics.”Part2,“MathematicsInquiryApplications,”wasofferedduringfall2010.
ThesecoursesfocusedonhelpingMTTIteachersexaminetheeffectivenessoftheir
pedagogicalpracticesbyidentifyinganddescribingtheirstudents’errorsand
misconceptions,reviewingliteratureonresearchandtheoriesaboutmathematicsteaching
andlearning,andusingalternativeassessmentsandtechnology.DuringPart2,MTTI
teachersorteamsofteachersusedmixedmethodstodevelopandcompleteAction
ResearchProjects,toassesstheperformanceoftheirstudents.AsofMay2011,23MTTI
TME, vol10, no.3, p. 627
teachersdeveloped29ActionResearchprojects,involving1,017students:378from
middleschoolsand639fromhighschools.Thecourseserieswastaughtandcoordinated
byamemberoftheLehmanCollegesecondaryeducationdepartment.
StatisticsCourse
Forsummer2010,allMTTIparticipantswereofferedachoiceofmathematics
courses,thelastmathematicscoursetheywouldbetakingaspartoftheprogram.They
couldchooseeitheraStatistics(andProbability)courseorasecondGeometrycourse.
VirtuallyallofthemchosetheStatisticscourseandweofferedtwosectionsofthecourseto
accommodatealltheparticipantswhowantedthecourse(anddidnotoffertheGeometry
course).TheMTTIparticipantswantedastatisticscourseforthreemainreasons:1)they
discoveredduringtheActionResearchcoursesthattheydidnotknowthestatistics
requiredtocompletetheirprojects;2)manyhadtheopportunitytobecomeinvolvedin
theirschool'sself‐evaluationandassessmentandfelttheyneededmorestatistical
knowledgetoanalyzetheoverwhelmingamountofdataavailabletotheminternally,and
theirprincipalswereeagerforthemtoserveontheseteams;and3)severalwerebeing
askedtoteachAdvancedPlacement(AP)Statisticsattheirhighschools.Itappearsthat
mostoftheteachers’preferredthestatisticscourseoverthesecondgeometrycoursefor
professionalreasonsotherthanadesiretoimprovetheirmathematicalknowledgefor
teachingstudents.
LeadershipSeminars1&2
TheLeadershipSeminar1beganinFebruary2011;LeadershipSeminar2beganin
May2011.TheDirectoroftheNewYorkCityMathematicsProject(NYCMP),andtheMTTI
Directorledtheseminars.InFall2010,theymetwiththeparticipantsthreetimesduring
Gningue, Peach & Schroder
theActionResearchcourse.Becauseitwasimportanttolaygroundworkforfurther
explorationoftheCommonCoreStateStandards(2010),thefirstmeetingfocusedonthe
Standards.Theothertwomeetingsfocusedonlevelsofcognitivedemandfor
mathematicaltasksaswellascasestudiesfromImplementingStandards‐Based
MathematicsInstruction(Stein,Smith,Henningsen,&Silver,2009).
MTTITeacher‐Consultants
SixMTTIteacher‐consultantsvisitedparticipantsintheirschoolstoprovide
support.Theteacher‐consultantswereretiredmathematicsteacherswithmanyyears’
experience,andweredrawnfromtheteacher‐consultantswhoprovidedasimilarservice
fortheNYCMP.Theteacher‐consultantsvisitedparticipantstwicepermonthforonehalf‐
dayoneachvisit.Theysupportedparticipantsindealingwithpedagogicalandleadership
issues.
ResearchQuestions
TheMTTIprojectisextremelywide‐rangingandmadeupofseveralcomponents.
However,thispaperconcentratesonourattempttoanswerthefollowingthreeresearch
questions:
1. DidparticipatinginMTTIincreaseparticipants’mathematicaland
pedagogicalknowledge?
2. DidparticipatinginMTTIincreaseparticipants’useofstudent‐centered
pedagogyintheclassroom?
3. Didanyincreaseineithermathematicalcontentknowledgeorstudent‐
centeredpedagogyleadtoanincreaseinstudentengagementin
mathematics?
TME, vol10, no.3, p. 629
Method
MathContentKnowledge
Mathcontentknowledgewasmeasuredbytwosetsofpre‐posttestsdevelopedby
theUniversityofLouisville’sCenterforResearchinMathematicsandScienceTeacher
Education(Bush&Nussbaum,2004).OneofthetestswasforAlgebraandIdeas,andthe
otherwasinGeometryandMeasurement.Bothtestsweresetatthemiddleschoollevel.
ThetestswerepartoftheDiagnosticTeacherAssessmentinMathematicsandScience
(DTAMS)instrumentthatwasvalidatedusingasampleof1,600middle‐schoolteachers
(Saderholm,Ronau,Brown,&Collins,2010).Saderholmandhiscolleaguesdetermined
theequivalencyreliabilityofthepretestsandposttestsbycomputingthePearsonproduct
momentcorrelation.This,theyreport,wasgreaterthan.80.Inter‐scorerreliabilitywas
alsogreaterthan.80.ThetwoLouisvilletestswereadministeredbeforeandafterthe
relevantcontentcourseswerecompleted.
EachUniversityofLouisvilletestcontained20items.Thefirst10itemswere
multiple‐choiceitemsandacorrectanswerscored1point.Items11‐20wereopen‐ended
responseitemseachdividedintotwoparts.Acorrectansweronthefirstpartscored1
point.Amaximumof2pointswereavailableforanswerstothesecondpart,givinga
possiblescoreof40points.ThetestswereblindedandscoredattheUniversityof
LouisvilleCenterforResearchinMathematicsandScienceTeacherEducationbymembers
oftheresearchteamunderthesupervisionoftheCenter’sdirector.
Gningue, Peach & Schroder
ThetwoMTTIcourses,oneinmathfundamentalsandtheotheringeometry,took
placethroughoutthespringandfallsemestersof2009.Twopre‐posttestswereadministered
inassociationwiththesecourses.ThesetestsarereferredtoasMTTItests.TheMTTIAlgebra
andIdeastestdealtwith:patterns,functions,andrelationships;expressionsandformulas;
andequationsandinequalities.TheMTTIGeometryandMeasurementtestdealtwith:
two‐dimensionalgeometry;three‐dimensionalgeometry;transformationalgeometry;and
measurement.
ThesetwoMTTItestsweredesignedbyMTTImathfaculty.Thepossiblescoreonthe
MTTIfundamentalstestwas100,andthepossiblescoreontheMTTIgeometrytestwas90.The
sametestwasusedasboththepretestandtheposttestfortheMTTImathfundamentalsand
geometrytests.TheMTTItestswerescoredbyamemberoftheLehmanCollegemathfacultynot
associatedwiththetwoMTTIcourses,basedonrubricsdevelopedbythemathfacultymembers
whotaughtthecourses.
ThequestionsontheUniversityofLouisvilletestsassessedparticipants’general
contentknowledge.Incontrast,theMTTItestsweredirectlyrelatedtothecontenttaught
inthetwocourses.
MathPedagogicalKnowledge
Accordingtoourtheoryofaction,thesecondcomponentofamathteacher’s
capacityforteacherleadershipconcernstheirmasteryofpedagogicalpractices
appropriatebothfortheirstudentsandforthemathematicsconceptstheyteach.
InformationaboutthiscomponentcomesfromquestionsontheLouisvilleAlgebraand
IdeasandGeometryandMeasurementtests,classroomobservations,andteachers’workin
theclassroom‐basedinquirycourses.
TME, vol10, no.3, p. 631
Asmentionedabove,thesecondpartofitems16‐20ontheLouisvilletests
measuredpedagogicalcontentknowledgeandthemaximumpossiblescoreontheseitems
was10.Anexampleofaquestionmeasuringpedagogicalcontentknowledgeisasfollows:
Q.16Astudentclaimsthatallsquaresarecongruenttoeachotherbecausetheyallhave
fourcongruentsides.
a.Whyisthisclaimincorrect?
b.Explainhowyouwouldhelpthestudentunderstandtheerrorinhis
thinking.
Thepedagogicalcontentscoreswereanalyzedseparatelyfromthescoresontheother
questions.
ClassroomObservations
Threeretiredmatheducatorswhohadpreviousexperienceinobservingteachersin
theirclassroomsweretrainedtobeobserversfortheMTTIproject.Theyweretrainedto
useafive‐minutetime‐samplingsysteminwhichtheywereaskedtoobserveforfive
minuteblocksoftimeandnotewhetherornotanyoneormoreofthepedagogicand/or
managementbehaviors(examplesbelow)wasusedbytheteacher.Attheendoftraining,
inter‐raterreliabilitywas.71.
Beginninginthefall2009term,theobserversvisitedtheMTTIteachers’classrooms
atleastfourtimeseachterm.ThroughJanuaryof2011,265observationshadtakenplace.
Theclassroomobservationprotocol([COP],Lawrenz,Huffman,&Appledoorn,2000)
contains,amongotherthings,informationabouttypesofinstructionalactivities.Someof
theseactivitieswerejudgedaprioritobeindicationsofstudent‐centeredpedagogy,
includingsmallgroupdiscussions,classdiscussions,hands‐onactivities,cooperative
learning,studentpresentations,anduseofalearningcenterorstation.Somewere
Gningue, Peach & Schroder
consideredaprioritoindicateteacher‐centeredpedagogy,includinglecturing,lecturing
withlimitedclassdiscussion,modelingproblemsolving,anddemonstrationsbythe
teacher.Theexactnatureofsomeactivities(e.g.writingworkorreadingseatwork)could
notbedeterminedapriori.Inthesecases,theobserversusedtheirownjudgmentwhether
theactivitywasstudent‐centered,teacher‐centered,orindeterminate.
Onaverage,eachobservationlastedforabout50minutes,withmostobservations
beingfor45or50minutes.Anobservationwascappedat60minutes.Thevastmajorityof
observationsinhighschoolswereconductedinalgebra,integratedmath,orgeometry
classes.Afewobservationswereconductedinadvancedmathclasses,includingseven
observationsinpre‐calculusclassesandeightobservationsincalculusclasses.
StudentEngagement
Oneofthesectionsoftheobservationprotocolmentionedconcernedthelevelof
StudentEngagement(SE)ratedashigh,medium,orlow.Duringeachobservation,SEwas
ratedashighwhen80%ormoreofstudentswereengaged,aslowwhen80%ormoreof
studentswereoff‐task,andasmixedotherwise.Anengagedstudentwasseenasonewho,
duringthetimeoftheobservation,wasinvolvedinthelessoninmeaningfulways;thatis,
he/sheparticipatedinallclassroomactivities,collaboratedeffectivelywiththeteacherand
withotherstudents,andwasreflectiveabouthis/herlearning.
Thefindingsfromtheuseoftheinstrumentsoutlinedaboveforassessingmath
contentknowledge,pedagogicalknowledge,andstudent‐centeredpedagogywererelated
tothoseforstudentengagementoutlinedinthissectiontodetermineiftherewasany
relationshipamongthevariables.
Results
TME, vol10, no.3, p. 633
MathContentKnowledge
Thirty‐twoparticipantstookboththepretestandposttestversionsofthetwoUniversityof
LouisvilletestsandtheMTTIfaculty‐designedtests.MeanscoresontheUniversityofLouisville
testofalgebraandideasincreasedsignificantlyfrom25.8atpretestto29.8atposttest.However,
meanscoresontheUniversityofLouisvilletestofgeometryandmeasurementdidnotdiffer
significantlyfrompretest(22.6)topost‐test(20.7)(Tables1&2).
ScoresontheMTTIfaculty‐designedfundamentalstestincreasedsignificantlyfrom
36.5atpretestto48.0atposttest.ScoresontheMTTIgeometrycoursecontenttestalso
increasedsignificantlyfrom26.6atpretestto36.0atposttest(Tables3&4).
Table1
Pre‐andpost‐testmeansfortheLouisvilleAlgebratest
Mean Std.Deviation N
LouisvilleAlgebraPretestTotal/40 25.75 6.309 32
LouisvilleAlgebraPosttestTotal/40 29.81 5.544 32
Significant:t(30)=4.61,p<.001
Table2
Pre‐andpost‐testmeansfortheLouisvilleGeometrytest
Mean Std.Deviation N
LouisvilleGeometryPretestTotal/40 22.56 7.211 32
LouisvilleGeometryPosttestTotal/40 20.72 6.371 32
Notsignificant:F(1,31)=3.45,p=.073
Table3
Pre‐andpost‐testmeansfortheMTTIFundamentalstest
Mean Std.Deviation N
Gningue, Peach & Schroder
MTTIFundamentalsPretestTotal/100 36.47 6.567 32
MTTIFundamentalsPosttestTotal/100 48.00 5.639 32
Significant:t(29)=5.01,p<.001.
Table4
Pre‐andpost‐testmeansfortheMTTIGeometrytest
Mean Std.Deviation N
MTTIGeometryPretestTotal/90 26.58 6.421 32
MTTIGeometryPosttestTotal/90 36.03 5.894 32
Significant:t(30)=4.61,p<.001
PedagogicalContentKnowledge
TheaveragenumberofcorrectanswersforthefivequestionsoftheLouisville
AlgebraandIdeastestrelatingtopedagogicalcontentknowledgeincreasedsignificantly
from4.44to5.16acrosstestadministrations.ThissuggeststhatMTTIparticipants’
pedagogicalcontentknowledgeforalgebraandideasincreasedfollowingengagementwith
acourseinthefundamentalsofmathematics.Themeanpedagogicalcontentknowledge
scoresfortheLouisvilleGeometryandMeasurementtestdeclinedslightlyfrompretest
(3.90)toposttest(3.55)administrations,butthisdecreasewasnotsignificant(Tables5&
6).
Takentogethertheseresultsindicatethatingeneralparticipants’mathcontentand
pedagogicalcontentknowledgeincreasedfrombeginningtoendoftheMTTIcourse.
Table5
Pre‐andposttestmeansforthepedagogicalitemsontheLouisvilleAlgebratest
TME, vol10, no.3, p. 635
Mean Std.Deviation N
LouisvilleAlgebraPretestTotal/10 4.44 1.722 32
LouisvilleAlgebraPosttestTotal/10 5.16 1.629 32
Significant:t(31)=2.49,p=.018.
Table6
Pre‐andposttestmeansforthepedagogicalitemsontheLouisvilleGeometrytest
Mean Std.Deviation N
LouisvilleGeometryPretestTotal/10 3.90 2.146 32
LouisvilleGeometryPosttestTotal/10 3.55 2.602 32
Notsignificant:t(31)=.706,p=.486.
Asmentionedabove,fromtheclassroomobservationprotocols,instructional
activitieswerecodedasteacher‐centered,student‐centeredorindeterminate,at5‐minute
intervals.Forexample,lecturewasconsideredteacher‐centeredwhilecooperative
learningwasconsideredstudent‐centered.Howeverforsomeactivities(e.g.“writing”),
therewasinsufficientinformationontheobserver’sreporttodeterminethestudent‐
centerednessoftheactivity;theseweregivenacodingof“indeterminate.”Foreachlesson,
thepercentoftimespentineachofthesethreecategorieswasthencalculated.Acrossall
observationsandallteachersandallsemesters,therangeoftimespentwas:inteacher‐
centeredactivities,30.2%;instudent‐centeredactivities,30.4%;andinactivitiesthatcould
notbeclearlyclassifiedaseither,39.4%.Therewasnosignificantchangeacrossthe
semesterforthepercentoftimespentinteacher‐centeredvs.student‐centeredactivities
Gningue, Peach & Schroder
(χ2(10)=5.29,p=.87).Thus,itappearsthatstudent‐centeredpedagogydidnotincrease
overthetimespanoftheMTTIcourseforCohort1.
StudentEngagement
Inthefall2009,spring2010,andfall2010semesters,observersassessedthelevel
ofstudentengagementinmathclassatfive‐minuteintervals.Theyrecordedthreepossible
levelsofengagement:lowengagement(80%ormoreofstudentsoff‐task);medium
engagement(mixedengagement);andhighengagement(80%ormoreofstudents
engaged).Highengagementincreasedfromfall2009tospring2010.Inthespring
semester,highengagementhadincreasedsignificantlyfromabout40%ofobservationsto
63.5%ofobservations.Infall2010highengagementdecreasedto48%.However,across
thethreesemesterslowengagementdecreasedfromninepercentinfall2009tofour
percentinfall2010(Figure2).Thesefindingsprovidesomeevidenceforanincreasein
highstudentengagementoverthetime‐spanoftheMTTIproject,andcertainlyevidenceof
adecreaseinlowstudentengagement.
Figure2.Levelofstudentengagementbysemester.
Semester
Fall 2010Spring 2010Fall 2009
Percent
70
60
50
40
30
20
10
0
Low engagement
Mixed engagement
High engagement
TME, vol10, no.3, p. 637
Student‐Engagement,MathContentandPedagogicalKnowledge,andStudent‐Centered
Teaching
Mathcontentknowledgeandpedagogicalcontentknowledgedidnotsignificantly
predictthepercentageclasstimefeaturingstudent‐centeredpedagogy(Tables7&8)or
percentageofhighstudentengagementinmathclass(Tables9&10).
Table7
MathcontentandpedagogicalcontentknowledgeasmeasuredbytheLouisvilletestsas
predictorsofstudent‐centeredpedagogy.
Sumof
Squares
df
Mean
Square
F Sig.
Regression 205.206 4 51.302 .104 . 980
Residual 8390.215 17 493.542
Total 8595.422 21
a. Predictors:(Constant),GeometryContentKnowledgechange,GeometryPedagogical
KnowledgeChange,AlgebraContentKnowledgechange,AlgebraPedagogicalKnowledge
change
b. DependentVariable:PercentStudentCenteredPedagogy
Table8
MathcontentknowledgeasmeasuredbytheMTTItestsaspredictorsofstudent‐centered
pedagogy.
Sumof
Squares
df
Mean
Square
F Sig.
Regression 619.584 2 309.792 .729 . 497
Residual 7228.263 17 425.192
Total 7847.847 19
a. Predictors:(Constant),MTTIGeometrychange,MTTIAlgebrachange
b. DependentVariable:PercentStudentCenteredPedagogy
Table9
MathcontentandpedagogicalcontentknowledgeasmeasuredbytheLouisvilletestsas
predictorsofhighstudentengagementinmathclass
Sumof
Squares
df
Mean
Square
F Sig.
Gningue, Peach & Schroder
Regression 5659.604 4 1414.901 .837 . 520
Residual 28728.310 17 1689.901
Total 34387.915 21
a. Predictors:(Constant),LouisvilleGeometryContentKnowledgechange,AlgebraContent
Knowledgechange,AlgebraPedagogicalKnowledgechange,GeometryPedagogicalKnowledge
change
b. DependentVariable:Percenthighengagement
Table10
MathcontentknowledgeasmeasuredbytheMTTItestsaspredictorsofhighstudent
engagementinmathclass.
Sumof
Squares
df
Mean
Square
F Sig.
Regression 5772.912 2 2886.456 1.973 . 170
Residual 24873.178 17 1463.128
Total 30646.090 19
a. Predictors:(Constant),MTTIGeometrychange,MTTIAlgebrachange
b. DependentVariable:Percenthighstudentengagement
Todetermineiftherewasarelationshipbetweenstudent‐centeredteaching(SCT)
andstudentengagement,wederivedtwogroupsofparticipants;GroupA(HighSCT)
consistedofthesixparticipantswhowereobservedtodisplaythemoststudent‐centered
teachingtechniquesasassessedbytheclassroomobserversacrossboththefall2009,
spring2010andfall2010semesters;andGroupB(LowStudentCentered)consistedofthe
sixMTTIparticipantswhoexhibitedtheleaststudent‐centeredteachingtechniques
assessedinthesamemanneracrossthesametimeperiod.ForGroupA,themean
percentageoftimespentinstudent‐centeredteachingactivitieswas48.7%(s.d.=9.0)
acrossallsemesters,whileforGroupB,itwasonly15.7%(s.d.=9.2).
Wethenexaminedtherelationshipbetweenstudentcenteredteachingandstudent
engagement.Wecalculatedthelevelsofstudentengagementforthetwogroups(highand
TME, vol10, no.3, p. 639
lowSCT)foreachsemesterandameanvalueacrosssemesters.Wefoundthatstudentsof
GroupA(highSCT)teachersweresignificantlymorelikelytobehighlyengagedintheir
mathclassesthanstudentsofGroupB(lowSCT)teachers:χ2(1)=5.81,p=.02(SeeTable
11).
Table11
LevelofstudentengagementfortheHighandLowSCTgroups
LevelofSCT HighEngagemen
t
MixedEngagemen
t
LowEngagemen
t
High 62.4% 33.4% 4.3%
Low 44.7% 48.7% 6.6%
Discussion
WefoundthatMTTIteachers’contentknowledgeinthefundamentalsof
mathematicsimprovedsignificantlyfollowingtheirparticipationintheprogram.However,
therewasnosignificantrelationshipbetweenteachers’increaseincontentknowledgeand
theiruseofstudent‐centeredteachingortheengagementleveloftheirstudentsinmath
class.Thismayhavebeenbecausethemeasuresweusedtoassesscontentknowledgedid
notadequatelytapintoparticipants’pedagogicalknowledge.Supportforthisviewcomes
fromadditionaldatafromtheobservations,whichshowthattheclassroomobservers
Gningue, Peach & Schroder
ratedteachers’masteryofmathconceptshighly.Theobserversalsoreportedthat
participantsmadeextremelyfewmathematicalerrorswhiletheywereteaching.
ItisalsoworthnotingthattheUniversityofLouisvilletestsweretestsofgeneral
mathematicsconceptsandpedagogy,whiletheMTTImathtestswererelatedtotheMTTI
mathcourses,butnotnecessarilytothespecificconceptsandpedagogythatMTTIteachers
wereusingintheirclassrooms.ThemathcontentoftheMTTIcourseswasdeterminedby
theLehmanCollegemathematicsfacultymemberteachingeachcourse.Ingeneral,the
contentofthemathcourseswasrelatedtotheNewYorkStatemathstandards,butitwas
notrelatedspecificallytothecontentthattheteacherswereteachingintheirclassroom.It
mightnotbesurprising,therefore,thattherewasnosignificantrelationshipbetweenMTTI
teachers’mathconceptknowledgeasmeasuredbytheLouisvilleandMTTItestsandtheir
classroompracticesasreportedbytheobservers.
WesuggestthatthediscrepancybetweentheUniversityofLouisvilleGeometryand
Measurementtestresults(lackofimprovement)andthoseoftheMTTIGeometrytest
results(significantimprovement)mayhavebeenduetothelackoffitbetweentheMTTI
geometrycourse,whichwasdesignedtocorrespondtoNewYorkState’ssecondary
geometrycurriculum,andtheitemsontheLouisvilleexam.
ThecontentoftheLouisvilletestshadbeenestablishedwithreferencetoteamsof
mathematicians,matheducators,andmathteacherswhoconductedliteraturereviewsfor
appropriatecontentasdefinedbynationalrecommendations(Saderholm,Ronau,Brown,&
Collins,2010).Thisresultedinteststhatcontainedcontentthatmathexpertsthoughtthat
mathteachersgenerallyoughttoknowandbeabletoteach,ratherthanitemsthat
TME, vol10, no.3, p. 641
assessedmasteryofspecificcoursecontentorwhatteachersneededtoknowtobeableto
teachparticularstudents.
Inaddition,fewerMTTIteachershadexperienceinorwerecurrentlyteaching
geometrycomparedtoalgebra.Thiswasinpartbecause,untilrelativelyrecently,most
emphasishadbeenplacedonalgebrabyNewYorkState’sBoardofRegents.Sinceteachers
werebeingaskedtofocusmoreonteachingalgebrathangeometry,thismightexplainwhy
theMTTIteachersgenerallyimprovedmoreontheAlgebraandFundamentalstestthanthe
Geometrytests.
Wediscoveredthatteacherswhoemployedahighlevelofstudent‐centered,
inquiry‐basedpedagogytendedtobemoreeffectiveasmathteachersthanthosewhoused
alowlevelofstudent‐centeredteaching,atleastifeffectivenessisassessedbytheextentto
whichtheirstudentswereengagedinthelesson.
Anecdotally,participantsreportedthatasaresultofparticipationintheclassroom‐
basedinquiry(actionresearch)courses,theychangedtheirownteachingpracticesand
sawimprovementsinmotivationtowardparticipatinginmathematicsonthepartoftheir
students.Thesefindingsarebasedonself‐report,andinthefuturewearegoingtoask
teacherstoformallyassesswhetherchangesinstudents’motivationtoengageactually
occur.
Forthisstudy,themainvariableusedforassessingtheeffectivenessofteachingis
levelofstudents’engagementinmathclass.Inpart,thiswasbecausewehaddifficultyin
gatheringpre‐andpost‐testdataforstate‐mandatedstudenttests.Tosomeextentthis
wasbecause,inordertoobtainethicalapprovalfromtheNewYorkCityDepartmentof
Educationforthestudy,wecouldnottrackindividualstudentsduringtheperiodofthe
Gningue, Peach & Schroder
research,norcouldMTTIteachersconductresearchactivitiesusingstudentsintheirown
classesasparticipants.
ForMTTICohort2,weareabletoaskMTTIteacherstocollectdatafromtheir
studentsaslongasthosestudents’identitiesarenotrevealed.Therefore,weareinthe
processofadministeringmathperformancetaskstothestudentsofMTTICohort2.These
performancetasksreflectthenewCommonCoreStateStandardsforMathematics(2010)
whicharebeingintroducedinNewYorkCityschoolsinthefall2012semester.Thisisin
anattempttoobtainstudentachievementdata.Wewillthenbeabletolookatthe
relationship,ifany,betweenstudent‐centeredpedagogy,studentengagement,andstudent
achievement.
TME, vol10, no.3, p. 643
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