KeepingupwiththeTimes:HowAreTeacherPreparationProgramsPreparingAspiring 

ElementaryTeacherstoTeachMathematicsUndertheNewStandardsofToday?    

ShaliniSudarsanan                     

Submittedinpartialfulfillmentofthe 

requirementsforthedegree 

ofDoctorofPhilosophy 

undertheExecutiveCommittee 

oftheGraduateSchoolofArtsandSciences    

COLUMBIAUNIVERSITY 

2015

  

 

                                                          

©2015ShaliniSudarsananAllrightsreserved

  

 

ABSTRACT 

KeepingupwiththeTimes:HowAreTeacherPreparationProgramsPreparingAspiring 

ElementaryTeacherstoTeachMathematicsUndertheNewStandardsofToday?    

ShaliniSudarsanan    

The purpose of this study is to determine how well traditional elementary

preparationprogramsand alternative elementary certificationprogramsareperceivedby

teachercandidates,topreparethemtoteachmathematicstothestandardsrequiredtoday.

Using a qualitative design, this study examines how various ranked elementary teacher

preparation programs and alternative certification programs are preparing elementary

teacherstoteachthemathematicsenvisionedintheCCSSM.

Participants in this study all attended an undergraduate elementary teacher

preparationprogramoranalternateroutecertificationprogram.Topaintaholisticpicture

ofelementaryteacherpreparationinmathematics,eachundergraduateelementaryteacher

preparation programs was ranked in a different level in the NCTQ TeacherPreparation

Review. Qualitative surveys and interviews were used to gather data on participants’

perceptionsoftheirpreparationinmathematics.

Twenty‐five participants agreed to takepart in this study. Participants filled out a

two‐part survey. The first part of the survey asked background questions on their

courseworkandfieldexperiencesinmathematicsalongwithasurveyontheirbeliefsabout

their ability to teach mathematics. The second part was a pedagogy survey that asked

  

participants how they would teach particular mathematics concepts that now require a

conceptualunderstandingintheCCSSM.Sevenparticipantsagreedtoparticipateina

followupinterviewtofurtherinvestigatetheirexperiencesinmathematicspreparationin

theirteachereducationprograms.

The data showed that there is little consistency in the mathematics education of

elementary teachers within a teacher preparation program and across different teacher

preparation programs. There is little standardization in the coursework for participants

from different preparation programs and participants within the same program. The

interviewsrevealedthatthedegreetowhichparticipantswereabletoteachmathematics

in their field experiences also varied within and across teacher preparation programs.

Furthermore,theinterviewsalsounveiledthattheCCSSMwerealsostudiedandutilizedin

different capacities within and across institutions. Lastly, the data from the surveys

disclosed that the majority of participants feel that they have the ability to be effective

teachers of mathematics yet the majority of participants teach mathematics for a

proceduralunderstanding.

i  

TableofContents 

 

ListofTables…………………………………………………………………………………………………………………ivChapterI:INTRODUCTION….………………………………………………………………………………................1 

NeedfortheStudy………………………………………………………………………...…………………....1 

PurposefortheStudy………………………………………………………………………………………….4 

ProceduresfortheStudy………………………………………………………..………...…………………6 

OrganizationoftheReport……………………………………………………………….…..…….……….9 ChapterII:LITERATUREREVIEW…………………………………………………………………...…………….11 

Introduction………………………………………………………………………..........................................11 

TheCurrentStateofTeacherPreparationPrograms…………………………………...………12 

NewDemandsoftheCCSSM&ItsImplicationsonTeacherPreparationPrograms……17 

MathematicalKnowledgeforTeaching………………………………………………..…………….25 

MathematicsEfficacyforPre‐serviceTeachers…………………………..……..….…………….31 ChapterIII:METHODOLOGY……………..……………………………………………………….…………...……37 

Introduction………………………………………………………………………………..……………….…...37 

RationaleforResearchApproach……………………………………………………..………….……..38 

TheResearchSample…………………………………………………………………………..…..….…….39 

OverviewoftheResearchDesign……………………………………………………..……….…….....45 

ProposalDefenseandIRBApproval…………………………………………………...………………46 

LiteratureReview……………………………………………………………………….……..…………...…46 

Data‐CollectionMethods…………………………………………………………….…….….……………48 

PhaseI:TheSurveys…………………………………………………………..…….…………….48 

ii  

TheBackgroundSurvey……………………………………………...…....................49 

YourBeliefsAboutTeachingMathematicsSurvey……………….…..…….49 

TheMathematicsPedagogySurvey……………………………………….……....51 

PilotStudyforSurveys…………………………………………………………..……..57 

PhaseII:Interviews…………………………………………………………………………..…...57 

InterviewScheduleofQuestionsandPilotInterviews………………..….58 

InterviewProcess……………………………………………………………...…………58 

MethodsforDataAnalysisandSynthesis…………………………………………………...………59 

Analysisofthe“YourBeliefsAboutTeachingMathematicsSurvey”…..……..59 

Analysisofthe“MathematicsTeachingPedagogySurvey”……..……….………...59 

AnalysisoftheInterviews………………………………………………..………….………….60 

EthicalConsiderations……………………………………………………………………..….…………….60 

LimitationsoftheStudy……………………………………………………………….…….……………..62 

TheRoleoftheResearcher……………………………………………………………...…………………63 ChapterIV:FINDINGS….……………………………………………………………………………..………….……..65 

Introduction……………………………………………………………………………………...……………...65 

ResearchQuestion1……………………………………………………………………..…………………..66 

ResearchQuestion2…………………………………………………………………….….………………...69 

ResearchQuestion3………………………………………………………………….…….………………..79 

ResearchQuestion4…………………………………………………………………….…….……………..79 ChapterV:SUMMARY,CONCLUSIONS&RECOMMENDATIONS………………….……….………103 

Introduction………………………………………………………………………….……….………………103 

Summary………………………………………………………………………….….….……………………..103

iii  

 Conclusions………………………………………………………………………..….……………………….104ResearchQuestion1…………………………………………………………….….……………………...104

 ResearchQuestion2………………………………………………………………..…………….………..105

 ResearchQuestion3………………………………………………………………..……………………...108

 ResearchQuestion4………………………………………………………………..……………………...109

 Discussion………………………………………………………………………………..…….……………….112

 Recommendations……………………………………………………………………...…………………...115

 SuggestionsforFutureResearch…………………………………………………...…………………118

 Bibliography…………………………………………………………………………………….………..………………122 APPENDIXA:SurveytoParticipants………………………………………………….…………..……………130 APPENDIXB:DrivingQuestionsforSemi‐StructuredInterviews………….…….…….………….135 APPENDIXC:E‐mailtoSupervisorsofPre‐Service/In‐serviceTeachers………….……...…..………..136 APPENDIXD:E‐mailtoParticipants………………………………………………….….………….....………137 APPENDIXE:PMTEFrequenciesofParticipants………………………………….…...…………..……..138 APPENDIXF:PMTOEFrequenciesofParticipants……………………………….…………………..…..139 APPENDIXG:ResponsestoPedagogySurveyQuestion1……………..………..…………….………140 APPENDIXH:ResponsestoPedagogySurveyQuestion2………………………....………...….……142 APPENDIXI:ResponsestoPedagogySurveyQuestion3…………………………….………….…….144 APPENDIXJ:ResponsestoPedagogySurveyQuestion4………………………….…..………………145 APPENDIXK:ResponsestoPedagogySurveyQuestion5……………………….….……..…………..147 APPENDIXL:ResponsestoPedagogySurveyQuestion6…………………………..…….…………...149 APPENDIXM:ResponsestoPedagogySurveyQuestion7………………………………..…………151

iv  

 

 LISTOFTABLES

   

Table1.“YourBeliefsAboutTeachingMathematics”Statements…………………………………….50 Table2.MathematicsPedagogySurveyQuestionsAlignmentwiththeCCSSM………………...53 Table3.Participants’DemographicInformation,Coursework,andSelf‐StudentRating……69 Table4.ConceptualResponseRatestotheMathematicsPedagogySurvey………………………81 Table5.ResponsestothePMTEStatements………………………………………………………………….95

v  

    

ACKNOWLEDGMENTS 

Iamsothankfulforallofthepeoplewhohelpedthroughthislong,rewarding

endeavor.Withoutyourhelpandsupport,Icouldneverhavemadethisdreamofmine

cometrue.FirstIwouldliketothankmyadvisor,Dr.EricaWalkerforallofyouradviceand

guidance.YouwerethereformebeforeIevenenteredthedoctoralprogram,advisingme

onmycareergoalsandaspirations.IwouldalsoliketothankthefacultyintheMathematics

EducationdepartmentinTeachersCollegeforalloftheknowledgethatyouhaveinstilledin

meoverthesepasteightyears.AspecialthankyoutoDr.KarpandDr.Wassermanfor

agreeingtobeonmydefensecommittee.Dr.Weinberg,Iwouldliketothankyouforthe

fabuloustripsthatwehadabroad.Dr.Smith,IwantyoutoknowthatIthinkofyouvery

often,asIcarefullyscaffoldeachbitofmylessontoemulatewhatyoudoinyourclasses.

YoureallyareanartistatyourcraftandIthankyouforservingasamodelforme. 

ImustalsothankDr.MelissaSutherlandforinspiringmetotakethispaththirteen

yearsagowhenIwasinherelementaryschoolmathematicscourse. Withoutthatcourse,

andherflawlessteaching,Iprobablywouldneverhavethoughttogetmydoctorate.IfIcan

havesuchaprofoundinfluenceonjustoneofmystudents,asyoudidforme,Iwouldbeso

lucky.Iameternallygratefultoyou,forIwouldnotbeheretoday,writingthis,ifitwasnot

foryou.

ThankyoutoSethWeitzman,myprincipalandafellowTCdoctoralgraduate,for

tellingmenevertogiveupandallowingmetotakeayearsabbaticaltogetaheadonthis

dream.Yoursupporthasbeeninvaluable.

vi  

ThankyoutoPatriciaSchmidt,DeniseSpangler,MarthaLaTorre,SusanMasterson,

andBrianneWrightforhelpingmerecruitparticipantsforthisstudy.Withoutyourhelp,

time,andpatiencethisneverwouldhavecometofruition.Icannotthankyouenough!

MyTCfriends,Ithankyouforallofthosestudysessions,doingmathematicsfor

hoursinthelibrary.ThankyouShrinaPatel,ToddGrunow,BrianneWright,andNicole

Fletcher.AspecialthankyoutoSusanLicwinko,notonlyforbeingastudypartner,butalso

forsupportingmeeverystepoftheway.Thanksforlettingmeconfideinyouandservingas

mytherapistattimes,whentheroadaheadseemedendless,youwerealwaystheretobring

mebacktolife.

Mostimportantly,Iwouldliketoacknowledgemyfamily.ThankyoutoRema

ThirumulpadandSudarsananThirumulpad,mymomanddad,foralwayspushingmeto

striveforthebest.Youraisedmetobelievethatanythingcanbeaccomplishedaslongas

youtaketheriskandtry.Thankyoutomysister,Dr.RashmiBismark,forbeingarolemodel

formeprofessionallyandpersonally.ThankyoutoAmitMahtani,myhusbandtobe,forall

ofyoursupportandencouragement.Iloveyouall.

vii 

 

 DEDICATION

 Tomyfather,SudarsananThirumulpad,

whotoldmethathisdreamwastohavetwodaughtersthathecouldcall“Doctor”.Youfinallyhaveitdad!

1 

ChapterI

INTRODUCTION

 NeedfortheStudy

 According to the Principles and Standards for School Mathematics, “curriculum

shouldbemathematicallyrich,providingstudentswithopportunitiestolearnimportant

mathematical concepts and procedures with understanding” (NCTM, 2000, p. 3). The

newly developed Common Core State Standards for Mathematics (CCSSM) share this

vision(NationalGovernorsAssociation,2010).

Adopted by forty‐four of the fifty states in the U.S., the CCSSMwere created to

preparestudentstosucceedinaglobaleconomyandsociety.Thesenewstandards

“build on the best of previous state standards” (National Governors Association, 2013,

p.1)andareinformedbytheNCTMStandards(2000),alongwiththestandardsofHong

Kong,Korea,andSingapore,allnationsthathaveoutperformedtheU.S.ineithergrades

four or eight in the Trends in International Mathematical Science Study (TIMSS)

(Gonzales, Miller & Provasnik, 2009). The main design principles behind the CCSS in

mathematicsarefocus,coherence,andrigor.

Rigor in the CCSSM stems from high‐level cognitive demands of reasoning,

justification,synthesis,analysis,andproblemsolving,byaskingstudentstodemonstrate

a deep conceptual understanding through application of content knowledge to new

situations (National Governors Association, 2010). According to the Common Core

Standards for Mathematical Practice, teachers must develop varieties of mathematical

expertise in their students, including the ability to persevere through problems,make

conjecturesandconstruct arguments,model situationswithmathematics, and look for

2 

patterns and structure in mathematics. The CCSSM stress the need for “conceptual

understanding to make sure that students are learning and absorbing the critical

information they need to succeed at higher levels” (National Governors Association

CenterforBestPractices,2010). Ultimatelytheresponsibilityofmakingthisvisioncome

tolifeisinthehandsoftheteacher.Ateachershouldhavetheconceptualknowledgeof

mathematics, along with the knowledge and skills that will enable them to pass this

knowledgeontotheirstudents.

The notion of merging content knowledge and pedagogical knowledge was

emphasizedbyLeeShulman. InShulman’s (1986)view,pedagogicalcontentknowledge

(PCK) “goes beyond knowledge of subject matter per se to the dimension of subject

matterknowledgeforteaching…theparticularformofcontentknowledgethatembodies

theaspectsofcontentmostgermanetoitsteachability”(p.9).Inotherwords,PCKisthe

blending of content and pedagogy, or “the ways of representing and formulating the

subject so that makes it comprehensible to others” (Shulman, 1986, p.9). Therefore, a

teachernotonlyneedsknowledgeofthecontentthatistobetaught,buttheymustalso

haveaknowledgeofhowtoteachthiscontentinawaythattheirstudentscan

understandit.

Ball (2008) utilizes PCK in her framework for the mathematical knowledge for

teaching (MKT). In this framework PCK and subject matter knowledge are two broad

categoriesofknowledgerequiredbyteachersthatarefurtherbrokendownintocommon

content knowledge (CCK), specialized content knowledge (SCK), knowledge of content

and students (KCS), and knowledge of content and teaching (KCT). CCK and SCK fall

under the subjectmatter knowledge category,whileKCS andKCT arewithin the

3 

pedagogical content knowledge umbrella. Together, these four areas encompass the

mathematicalknowledgeforteaching.Afurtherdiscussionofthisframeworkisfoundin

theliteraturereview.

AccordingtoBlackburn(2013),inherbookRigorisNotaFourLetterWord: 

Rigorismorethanwhatyouteachandwhatstandardsyoucover; it'showyou teach and how students show you they understand. True rigor iscreatinganenvironmentinwhicheachstudentisexpectedtolearnathighlevels,eachstudentissupportedsoheorshecanlearnathighlevels,andeachstudentdemonstrateslearningathighlevels(p.10).

 Thereforeaspiringteachersmustbeequippedwiththepedagogicalcontentknowledge

necessarytoenablesuchrigortocometolifeintheirownclassroom.

The National Council on Teacher Quality (NCTQ), a non‐profit, non‐partisan,

research policy group, reports that mathematics coursework should be designed for

elementary school teachers, specifically focusing on: number and operations, algebra,

geometryandmeasurement,andprobabilityanddataanalysis. In their report titledNo

CommonDenominator (2008), after looking at syllabi of institutions with elementary

teacher preparation programs, the NCTQ found that in seventy‐seven accredited

elementaryteacherpreparationprogramsacrossthenation,fewprograms,intheirview,

cover themathematics necessary to build a conceptual understanding in these content

areas.Of the seventy‐sevenprogramsevaluated, only ten required the coursework that

covered these four major areas of mathematics. With vastly different preparation

programs, these institutions displayed little consensus as to what teachers need to be

successfullytrainedtoteachmathematics.SimilarfindingscamefromLynnHart’s(2004)

case study that investigated the beliefs and perspectives of eight first‐year elementary

teachersthatwerepreparedinanalternativecertificationprogram.Shefoundthat

4 

althoughthecertificationprograminthestudymandatedtwelvecreditsinmathematics,

two courses in content and two courses inmethods, themathematics contentwas not

sufficiently developed to create teachers who felt confident in their understanding of

mathematicscontent(Hart,2004).TheCCSSMhaveraisedthelevelofMKTrequiredfor

teachers in the classroom and as a result it is necessary to closely examine how

elementary teacher preparation programs are evolving to keep up with these

requirements.

In June of 2013, the NCTQ released its study examining teacher preparation

program across the nation, titled the TeacherPrepReview. Within this document, the

NCTQ examined elementary teacher preparation programs by subject area, and strictly

looked at documents including syllabi and textbooks of required mathematics courses

and was criticized for not looking into student achievement or speaking with actual

students and faculty of the institutions (Darling‐Hammond, 2013). To add to current

research and to better understand how teacher preparation programs are preparing

elementary teachers to teach themathematics envisioned in theCCSSM, this studywill

addtotheresearchconductedbytheNCTQandexaminethealignmentoftheirstudyof

elementaryteacherpreparationprogramsinmathematicstotheperceptionsofstudents

attendingsomeoftheinstitutionshighlightedinthestudy.

 PurposeoftheStudy

The purpose of this study is to determine how well traditional elementary

preparation programs and alternative elementary certification programs are perceived

by teachercandidates, toprepare themto teachmathematics to thestandardsrequired

today.ItisknownthatthenewlyadoptedCCSSMrequireadeepunderstandingof

5 

mathematics at the elementary level. Less is known about how elementary teacher

preparationprograms are preparing teachers to teachmathematics to the high level of

understandingthatisrequiredatthepresenttime,accordingtheCCSSM.Thisstudywill

examine how various ranked institutions in elementary teacher preparation and

alternative certification programs are preparing elementary teachers to teach the

mathematicsenvisionedintheCCSSMthroughthelensofprospectiveteachersthat

attendtheseinstitutions.

Thespecificresearchquestionstobeansweredinthisstudyare: 

(1)Howdoparticipantsdescribe theirmathematicalexperiencesprior to

enteringtheirteacherpreparationprograms?

(2) How do participants describe the mathematics preparation they 

receive in their respective preparation programs and what types of

mathematics preparation, if any, do theywish they had before beginning

studentteaching?

(3) What kinds of additional professional activities, beyond the official

preparation received in their teacher education program, do participants

engage in outside the classroom to support and enhance their formal

preparationformathematicsteaching?

(4) What are participants’ perceived pedagogical competencies in

mathematics and how do these differ across various preparation

programs?

From this study, it is anticipated thatmore informed decisions can bemade by

policymakersandhigher learning institutions in regards to themathematics

6 

requirements necessary for elementary teacher preparation programs. These decisions

willbemadethroughgainingabetterunderstandingoftheexperiencesparticipantshave

in mathematics prior to entering their preparation programs, the mathematics

courseworkandfieldexperiencesinwhichtheyhaveparticipated,thecompetenciesthey

have gained and can exhibit from their education to teach mathematics, and the

challengestheyfaceinpreparingtoteachmathematics.

 ProceduresoftheStudy

 This study is of qualitative design and is composed of two data collection

methods: one‐on‐one in‐depth interviews and a survey component to gather data on

participants’viewsoftheirabilitytoteachmathematicsandtheirpedagogical

approaches to teaching particular mathematical concepts. Using the NCTQ rankings of

undergraduate elementary teacher education programs inmathematics, the population

for this study is composed of participants from one top‐rated elementary teacher

preparation programs, two average programs, one low ranking program, and one

alternative certification program, totaling twenty‐five participants. Teacher educators

supervising elementary student teachers and supervisors of alternate certification

programswerecontactedtohandoutsurveystotheirstudentteachingcohort.Fromthis

pool, a convenience samplewas obtained for in‐depth interviews to elaborate on their

responsestothesurveys.Triangulationofresultsfromthesurveywasachievedthrough

the use of follow up interview questions. All participants in this study completed the

requiredmathematicscourseworkleadinguptostudentteachingandwereinthemiddle

oftheirstudentteachingexperience,orfirstteachingposition, forthoseparticipantsin

thealternaterouteprograms.

7 

Toexaminethealignmentofcurrentresearchonelementaryteacherpreparation

programs in mathematics and student perceptions of their own experiences in

mathematics preparation, recruitment efforts were directed towards schools of

education listed in the NCTQ Teacher Prep Review along with alternative teacher

preparation programs. Released in June of 2013, the NCTQ reviewed 1400 higher

educationteacherpreparationprogramsacrossthecountry.Forthoseinstitutionswilling

to participate, the NCTQ examined the basic elements of teacher preparation design

including admission standards, syllabi, course descriptions, textbooks, student teaching

manuals, graduate surveys, and other documents as a “blueprint” formolding teachers

(NCTQ, 2013). Using current research, theNCTQ also created a list of standards,

rationales and indicators tohelp identify those institutions thataremost likely tohave

thestrongestoutcomesfortheirstudents.BeingthattheTeacherPrepReviewisthemost

current study of undergraduate elementary teacher preparation programs, and is also

controversial in its findings, this study chose toutilize this study for recruitment

purposes to add to current research on the mathematics education of undergraduate

elementaryteacherpreparationprograms.Participantsforthisstudywereselectedfrom

a top ranked institution, two average institutions, and an institution in need of

improvement at theundergraduate level. Toviewelementary teacherpreparationasa

whole, this study also included participants from an alternative teacher preparation

program.Alternativecertificationprogramsareintendedforgraduateswhodidnotstudy

education in college. After undergoing professional development, graduates in these

programsbeginteachingwhilepursuingamaster’sdegreeineducationafterworkhours,

providingaquickpathwaytoteaching(Foderaro,2010).

8 

This study is comprised of two parts: an interview component and a survey

component. All interviewees in the study were identified by a pseudonym, and all

interviewswereaudiorecordedandtranscribedverbatim.

A two part survey was designed to obtain information to answer research

questions(1)and(4).Thefirstpartofthesurveyasksaseriesofquestionstodetermine

the backgrounds of participants and also investigates their beliefs about teaching

mathematics by using the Mathematics Teaching Efficacy Beliefs Instrument (MTEBI).

Created by Enochs, Smith, and Huinker (2000), the MTEBI is a twenty‐one item self‐

reporting instrument divided into two categories: personal mathematics teaching

efficacy andmathematics teaching outcome efficacy. From the twenty‐one statements,

thirteenpersonalmathematicsteachingefficacystatementswereusedtoobtaindataon

participants’perceivedpedagogical teachingcompetencies.Forthisstudy, thedefinition

of pedagogical teaching competencies is adapted from Dineke, Tigelaar, Dolmans,

Wolfhagen,&Cees(2004)as“…anintegratedsetofpersonalcharacteristics,knowledge,

skills and attitudes that are needed for effective performance in various teaching

contexts”(p.255).Thecurrentstudyutilizesthisdefinitionwithrespecttotheteaching

ofmathematics.

Thesecondpartofthesurveywasdesignedasanothertooltogatherdataon

participants’ pedagogical teaching competencies and investigates how participants

wouldteachparticularelementarymathematicsconceptsthatnowrequireaconceptual

understanding in light of the CCSSM. These topics include: understanding place value,

base ten decomposition, modeling operations, reasoning both abstractly and

quantitativelywith fractions, andconceptuallyunderstandingalgorithms (Universityof

9 

Arizona,2011).Forexample,inthepastitwasenoughthatsixthgradersbeabletodivide

two fractionswith theuse of the algorithm,but as indicated in theCCSSM, students in

sixthgradeshouldnowbeabletoapplytheirknowledgeofmultiplicationanddivisionto

divide fractionsby fractionsandbeable touseavisual fractionmodel torepresent the

quotient of two fractions. The problems used in this survey highlight the areas that

require students to have a conceptual understanding of mathematics rather than

memorizing a procedure to solve an exercise. Survey questionswere distributedwhen

theparticipantagreedtoparticipateinthestudyandwerecollectedbeforetheinterview

wasconducted.Thesurveyswerenottimed.

 OrganizationoftheReport

 Thisdissertationisorganizedintofivechapters.

 Chapter 1 (Introduction) includes a discussion of the need for the study, the

purposeofthestudy,theresearchquestionsforthestudy,proceduresforthestudy,and

finallytheorganizationofthereport.

Chapter2istheliteraturereviewforthestudy.Thissectiondiscussesthecurrent

research pertinent to issues in mathematics education and elementary teacher

preparationtoday.

Chapter 3 is the Methodology chapter of the study. This section describes the

researchdesignofthisstudy,includingthestudy’ssetting,howparticipantswere

selected,demographicsofparticipants,anddatacollectionmethods.

Chapter 4 is the Findings chapter. This section contains the results of the study

accompaniedbytableswhereappropriate.

10 

Chapter 5 is the Summary, Conclusions, and Recommendations chapter and

includes a summary of conclusions thatwere yielded by the results of this study. This

sectionalsoincludesforpolicymakers,teachereducatorsandhigherlearning

institutions, a discussion of recommendations to improve elementary teacher

preparationinmathematicsandsuggestionsforfutureresearch.

11 

Chapter II  

LITERATUREREVIEW 

The purpose of this study is to determine how well traditional elementary

preparation programs and alternative elementary certification programs are perceived

by aspiring teachers, to prepare them to teachmathematics to the standards required

today. Specifically, the researcher sought to understand what elementary teacher

preparationprogramsaredoing in theirpreparationprograms toovercome thediverse

mathematicalbackgroundsthatpre‐serviceteachersbring,sothataspiringteachersfeel

thattheyarepreparedtoteachmathematicstothehigherstandardsthataredemanded

at thecurrent time. Tocarryout thisstudy,acompletereviewof thecurrent literature

was conducted.This reviewof the literaturewas ongoing throughout all phases of this

study.

After completing a critical review of the literature surrounding the topic of

elementaryteacherpreparation inmathematics, fourmajorareasofresearcharose: the

current state of elementary teacher preparation programs in mathematics, the new

demands of the Common Core State Standards for Mathematics (CCSSM) and its

implications on teacher preparation programs, mathematical knowledge for teaching

(MKT), and finally pre‐service teachers’ mathematics efficacy and its relation to

mathematics education coursework. A review of the literature on the current state of

elementary teacher preparation programs in mathematics provides the background

information necessary for one to understand the autonomy of teacher preparation

programsandthevariation inmathematicsrequirements fromone teacherpreparation

program to the next.A reviewof the literature on the CCSSM and its implications on

12 

teacherpreparationprogramsprovidesanunderstandingofthecurrentchangesthatare

takingplaceinK‐12mathematicseducationandwhatteachereducationprogramsmust

dosothatteachersarepreparedtoenterthefieldwiththeknowledgeandskillsthatare

necessarytokeepupwiththechangesincurricula.

A discussion of MKT gives clarity to the kinds of knowledge that teachers of

mathematics should have and develop in their teacher preparation programs to be

successful teachers ofmathematics.With thehigh level ofmathematical understanding

requiredofstudentsinelementaryclassrooms,developingpre‐serviceteachers’MKTisa

crucial piece of elementary teacher preparation programs. Finally, the topic of

mathematicsefficacysheds lighton the research thathasbeenconducted todetermine

the relationships that exist between prospective elementary teachers’ mathematics

preparationandtheirperceptionsoftheirownabilitytoteachmathematics.

 CurrentStateofTeacherPreparationPrograms

 Teacher educationprograms in theU.S. vary across thenation as each state has

their own requirements for coursework and teaching experiences that lead to state

certification (National Council of Teacher Quality, 2008; Kilpatrick, 2001; Center for

Research in Mathematics and Science Education, 2010). Further, individual institution

requirements have been found to have no obvious correlation to state requirements

(National Council of Teacher Quality, 2008). As a result, inconsistencies exist in the

preparationofteacherswithineachstateandthenation.

Teacherscanobtainalicensetoteachviavariouspathways.“Traditional”teacher

preparationprograms are usually housed in a higher education setting and result in a

bachelor’s or master’s degree in education. Alternately, prospective teachers can gain

13 

certification by an “alternative” route, by which a teacher is licensed often without

needingtoenterahighereducationinstitutionfortraditionaleducationcourseworkora

degree in education (Perry, 2011). Alternative route prepared teachers account for

approximatelyoneinfivenewteachersintheUnitedStates(Finn&Petrilli,2007).

Alternativestotraditionalstatecertificationmayresembleavarietyofscenarios.

Alternativeroutecanmeanalternativeways tomeet teachercertificationrequirements,

like a graduate level master’s degree program in lieu of an undergraduate teacher

education program, which may be viewed as maintaining or raising the standards for

entryintoteaching.Anothertakeisalternativecertification,whichadjuststhestandards

for certification, and may include reduced training, or training completed during the

course of a teaching career rather than prior to the start of one’s career. Thismay be

consideredtolowerthestandardsforentryintoteaching.Lastly,alternativecertification

canalsomeanalternativestostatecertificationitself,allowinglocalemployersto

prepare and certify their own candidates, removing state standards for teacher

certificationentirely(Darling‐Hammond,1990).

However, since most alternative certification programs require that candidates

demonstrateknowledgeof subjectmatter toqualify foradmission, the focusof teacher

preparation is not typically on content knowledge but is more on pedagogy, with an

emphasis onhow to teach specific content to a diverse groupof students (US.Dept. of

Education, Office of Innovations and Improvement, Innovations in Education, 2004).

Thereforealthoughsomealternativecertificationprogramsrequireanextensiveamount

ofcoursework, inmanycases, forelementary teachersthiswillnot includecoursework

on elementarymathematics content. In some cases, alternative certified teachersmay

14 

takecoursesinauniversityaspartoftheirprogram;however,courseworkvariesamong

participating universities, and it still remains that these teachers will not have the

necessary coursework in elementary mathematics prior to entering their teaching

position(US.Dept.ofEducation,Officeof Innovationsand Improvement, Innovations in

Education, 2004).

Currently, the amount of mathematics required for traditional pre‐service

elementaryteachingprogramsdiffersvastlyfromoneinstitutiontothenext(Centerfor

Research in Mathematics and Science Education, 2010; National Council on Teacher

Quality,2008;Greenberg,McKee,&Walsh,2013).Insomeprograms,itisthecasethata

student studying to become an elementary school teachermay never even complete a

coursedesignedtoenhancetheirknowledgeonthefundamentalconceptsofelementary

mathematics.In2008,theNationalCouncilonTeacherQuality(NCTQ)publisheda

report titled NoCommonDenominator.This report looked at elementary mathematics

preparation in seventy‐seven randomly selected higher learning institutions across the

nation and unveiled the inconsistencies between the chosen programs. Schools ranged

fromhaving0to12semesterhoursinelementarymathematicscontentand0to6hours

inmathematicsmethodscourses.Outofthe77institutionsinthestudy,14didnotoffer

anycoursesonelementarymathematics.

In its most recent report, the NCTQ found that out of 522 elementary

undergraduate programs evaluated for breadth and depth in elementary school

mathematics, only twenty‐five percent had coursework that addressed essential

elementary mathematics topics in adequate breadth and depth, thereby providing

teacherswiththepreparationnecessarytomeetthedemandsoftheCCSSM.Thisleaves

15 

seventy‐fivepercentof theundergraduateprogramsevaluated in thisstudy, insufficient

toprepareelementaryteacherstomeetthedemandsoftoday’smathematicsclassrooms.

AccordingtotheNCTQ,outof205elementarygraduateprogramsevaluated,ninety‐eight

percenthadprogramcourseworkaddressingessentialmathematicstopicsininadequate

breadthanddepth(Greenberg,etal.,2013).BothreportsbytheNCTQdemonstratethe

notion that elementary teacher preparation is not standardized in theUnitedStates, in

some cases leaving teacher candidates with limited coursework in elementary school

mathematics and unprepared to enter the classroom (NCTQ, 2008; Greenberg, et al.,

2013). 

The Teacher Education and Development Study in Mathematics (TEDS‐M) also

found variation in the type of coursework necessary to prepare elementary teachers

(CenterforResearchinMathematicsandScienceEducation,2010).Thisstudyexamined

coursetakingandpracticalexperiencesprovidedbyteacherpreparationprogramsin16

nations,specifically80institutionsintheUnitedStates.Thisstudyalsoassessedteacher

knowledge of mathematics and mathematics pedagogy. The study revealed that in

elementary teacherpreparationprograms, roughly the sameamountof time isdevoted

to coursework in formal mathematics, mathematics pedagogy (which included both

elementarymathematics content courses andmethods courses), and general pedagogy.

This approximateone‐third split generallyheld true fromone institution to thenext in

theUnitedStatesaswellasincomparisontoinstitutionsinothernations.Thestudyalso

foundthatelementaryteachercandidate’smathematicscontentknowledgevariesacross

institutions.Aftercontrollingfordifferencesrelatedtotheselectivityofstudents,scaled

scoresrepresentingthelevelofaninstitution’smathematicalcontentknowledgeranged

16 

from450to600.TheCenterforResearchinMathematicsandScienceEducation

accounts this variation as a reflection of the inconsistent views ofwhat coursework is

necessarytopreparefutureteachersofmathematics(2010).

Thevaryingviewsofwhatisnecessarytoprepareelementaryteachercandidates

inmathematicsresultsfromtheautonomousstructureofhighereducation.Accordingto

Kilpatrick(2001), inordertobecertifiedtoteachelementaryschool,onlytwelvestates

require a minimum number of credits in mathematics, ranging from six to twelve

semesterhours.Withinthisframework,statesdonotimposeaparticulartypeofcourse

to fulfill thesecredits.Thereforecoursescould includebasicmathematicscourses,such

as college algebra, methods courses that educate students on pedagogical knowledge,

courses focused on elementary mathematics content, or courses that teach a

combinationofcontentandpedagogy.ThiscontradictsthestanceoftheConference

Board of theMathematical Sciences (CBMS), that generalmathematics courses such as

college algebra, calculus and higher‐levelmathematics courses are not a substitute for

elementarymathematicscoursework(CBMS,2013).

To keep teacher preparation programs accountable and provide public

recognitionofthequalityoftheirprograms,accreditingagenciesliketheNational

Council for Accreditation of Teacher Education (NCATE) and the Teacher Accreditation

EducationCouncil(TEAC)exist.Whenevaluatingprograms,NCATElooksforevidenceof

studentlearninganddataforprogramdecisionmakingwhileTEACrequiresprogramsto

define their claims, develop and collect data using reliable instruments that measure

their claims, and demonstrate how findings from data are used for program

improvements(Erickson&Wentworth,2010).AccordingtotheNCATEStandards,with

17 

respect to mathematics, “Elementary teachers know, understand, and use the major

concepts and procedures that identify number and operations, algebra, geometry,

measurement, data analysis, and probability. In doing so they consistently engage in

problem solving, reasoning and proof, communication, connections, and representation

(2008, p. 54).” However, NCATE does not establish a certain number of credits that

teacher preparation programsmust have in elementary schoolmathematics to achieve

this standard. As a result, different accredited teacher preparation programs may not

havecomparablecourseworkinelementaryschoolmathematics.

TheNCTQ(2008)foundauniversityintheMidwesthavingNCATEaccreditation

without requiring any coursework in elementary mathematics. Furthermore, this

institution only offered one two‐credit course on mathematics methods for students,

leaving one to question whether this single course can adequately prepare teacher

candidatestoachievetheNCATEstandard.Thisexamplesurfacesthepointthatalthough

ahigher learning institutionmaybeaccredited, it isnotthecasethat theirteachersare

adequately prepared to teachmathematics, especiallywith respect to the newways of

theCCSSM.

 NewDemandsoftheCCSSManditsImplicationonTeacherEducationPrograms

 According to their developers, the CCSSM were created in a unified effort to

promisechildrenintheUnitedStatesahighqualityeducationthatwillprovidethemthe

knowledge and skills in their K‐12 education necessary to survive in our highly

competitive global economy and society. To obtain this goal, the standards are

internationallybenchmarked,evidence–basedandbuilduponthecurrentstrengthsand

lessonsofstatestandards(NGA,2010).Forthemathematicsstandards,focus,coherence

18 

andrigoraretheevidence‐baseddesignprinciplesthatthestandardswerecreatedfrom.

Focusstemsfromnarrowingthescopeofcontentwithineachgradelevelsothatstudents

canbuildadeeperunderstandingofmathematicsratherthanasurfaceunderstandingof

many topics. Coherence has to dowithmaking connections between topics and seeing

the progression of topics fromone grade level to thenext. Finally, rigor in coursework

comesinthreeequally intensedomains:conceptualunderstanding,proceduralskilland

fluency,andfinallyapplications(NGA,2013).

Forelementarymathematicsinparticular,thestandardsstressnotonly

procedural skill but also conceptualunderstanding inorder to assure that studentsare

learningthecrucialinformationthatisnecessarytosucceedathigherlevels.Inorderto

facilitatetheunderstandingofmathematicsconceptsandprocedureswithaprogression

to more advanced application work, the elementary standards in grades K‐5 provide

studentswithasolidfoundationinwholenumbers,addition,subtraction,multiplication,

division, fractionsanddecimals.Theelementarystandardsset theplatformfor therich

androbuststandardsthatcontinueinmiddleschoolandhighschoollevels (NGA,2010).

To set the foundation for the mathematical progressions that students will

continuetolearnfromkindergartenthroughtwelfthgrade,theCCSSMstressthat

students should understand themathematics they aredoing at every level. The CCSSM

definewhatstudentsshouldunderstand,anddefineunderstandingastheabilityto

justifywhyaparticularmathematicalstatementistrueorwherearulestemsfrom(NGA,

2010).Thereforesincestudentsarerequiredtohavethisdeepconceptualunderstanding

ofmathematics, it is clear that teachersmust alsohaveawell‐versedunderstandingof

mathematics,andhavetheinstructionaltoolsnecessarytoconveythistotheirstudents.

19 

AccordingtoSchmidt(2012),teachersmustdelivercontent,usingtheirmathematicsand

pedagogical knowledge to present content to their students in a way that enables

meaningful learning experiences. It is in the teacher’s hands to create an instructional

experience in which student learning takes place. Thus teachers not only need to be

trainedwithmathematicscontentbutalsothemethodsnecessarytodeliverthecontent

inamannerthattheirstudentscanlearn,understandandbuildon.

Itisclearthatteachersneedtobeeducatedonthenewstandardssothattheyare

able to implement them in theirownclassrooms inamanner thatwillhold learning to

therich,rigorousvisionatwhichthesestandardswere initiallycreated.Althoughmany

states have had standards put in place, The Common Core State Standards for

MathematicalPracticepaintsadifferentviewofmathematicslearningthaniscommonin

many content courses for teachers (Science and Mathematics Teacher Imperative/The

Leadership CollaborativeWorking Group on the Common Core State Standards, 2011).

Therefore teacher educators must alter professional preparation courses for aspiring

teachers to keep up with the changes that are embedded in the CCSSM. Without this

change,thedisparitybetweenteacherpreparationandexpectationsoftheCCSSMcould

result inwhat is described as the “perfect storm”, a scenariowhere thenewly adopted

standardsarenotexecutedtothelevelofrigoratwhichtheywerecreatedbecausenew

teachersarenotequippedtoteachmathematicswiththelevelofunderstandingrequired

at the current time (Center for Research on Science and Mathematics, 2010). Thus if

teachers are notadequatelypreparedwith theknowledge and toolsnecessary to teach

mathematics to the level of understanding required today, the basis of the CCSSM,

coherence,focus,andrigorinmathematicswillnotcometofruition.

20 

The report Gearing up for the Common Core State Standards in Mathematics

(Institute for Mathematics and Education, 2011) suggests that new teachers will need

furtherpreparation inmanyareas.Thisnewwayofpreparing teachers is theonlyway

that theCCSSMwill be able tobe implemented in adherence to itsvision.According to

thisdocument,theemphasisonunderstandingplacevalueingradesK‐2isnewformany

teachers.Studentswillnotonly havetobeabletoidentify the differentplacesofa

number but they will have to understand that a multi‐digit number is comprised of

variousamountofones, tens, thousands,etc(…)andthat thenumber isasumof these

given amounts. Thus it is not only necessary that teachers themselves have this

understanding, they will also have to learn and gain experience in how to teach this

conceptforunderstandingwiththeirstudents.

In grades K‐5, algebra is another domain in which teachers will need further

preparation.Teacherswillhave tobewellversedwith thekindsofactionsrepresented

by each operation, the variety of representations, and the limitations of different

representations.Forgrades3‐5,thedevelopmentoffractionsandoperationsonfractions

inacarefullychosensequencemaybenewtoteachereducators.Studentswillbe

requiredtorepresentfractionsinmultiplewaysaswellasbeabletoviewfractionsona

numberline.Whenteachingstudentstoreasonwithfractionsteachersmustcontinually

emphasizethefractionaspartofawhole.Forexampleone‐halfisonlyguaranteedtobe

greater than one‐third when the fractions are referring to the same whole. Teachers

should use fraction models to help students visualize fractions as parts of a whole.

Furthermore,studentsmustunderstandthelogicbehindthealgorithmsinvolvedin

21 

operating with fractions and teachers will need preparation in how to teach for this

understanding.

In sixth grade, teachers shouldno longer limit proportional reasoning to setting

upaproportionandcrossmultiplyingbut instead teachers should incorporate ideasof

ratios,rates,andscalingtoreasonabouttheseconcepts.Ratiosshouldbemodeledusing

tape diagrams, double number lines, and ratio tables. Teaching in this manner in the

elementarygradesisnewformanyteachers. AccordingtoDingman,Teuscher,Newton,

& Kasmer (2013), teacher education programs will need to consider changes to their

currentpracticesinordertohelppre‐serviceteachersmeetthedemandsoftheCCSSM,

includinganemphasisonparticularmathematical contentatdifferentgradebandsand

thedifferent typesof reasoningexpectedacrossK‐8. Their research suggests that that

the CCSSM require an increased expectation of higher‐order thinking in several areas,

particularlyingeometry.Theyfoundthatinthepast,themajorityofK‐8geometrystate

standardshadanemphasisonlevel1ofvanHielegeometricthinking,whereasthe

CCSSMhas a greater emphasis on level 3 of vanHiele geometric thinking and a lesser

emphasisonlevel1.Teacherpreparationprogramswillhavetochangetheirmethodsof

teachereducationsothatnewgraduatesofteachingcanhaveastronghandleon

teachingelementarymathematicalconceptsforunderstanding.Goingforward,education

programs have to educate prospective teachers on the different models and

representations of elementarymathematical concepts so that they can incorporate this

newwayofteaching,thinking,andreasoningaboutmathematicsintotheirownpractice.

Asa response to theCCSSM, in2012 theConferenceBoardof theMathematical Sciences(CBMS)publishedtheMathematical Education of Teachers II,reflectingthe

22 

current changes inmathematics education in gradesK‐12, as a resource for thosewho

prepare both future and practicing teachers alike. The CBMS recommends that before

beginningtoteach,elementaryteachersshouldstudyindepth,fromtheperspectiveofa

teacher, the majority of K‐5 mathematics and its connections to pre‐kindergarten

mathematicsaswellasitsconnectionstothemathematicsfoundingrades6‐8.Teachers

will need coursework in mathematical pedagogy to complement the content they are

learning. Therefore, it does not only suffice for aspiring teachers to strictly have the

knowledgeof themathematics taught in the elementarygradesbut theyneed to know

howthemathematicstheyteachevolvesfromkindergartentothemiddlegrades.

Teacherpreparationprogramswillhave toadjust their coursework inpreparing

teacherstomakethisunderstandinghappen. Accordingtoasurveyconductedin2006,

of over five thousand teachers, in sixty school districts, the majority of elementary

teachers demonstrate a lack of confidence in the topics taught in the later grades.

According to this report less than a 25%of elementary teachers felt prepared to teach

proportionality concepts, a topic that is the currently a domain in the sixth grade

standardsoftheCCSSM.Inthissurvey,outofallelementarytopics,therewereonlytwo

topics out of twenty‐four that 75%ormore of elementary teachers felt theywerewell

prepared: whole number meaning and operation and properties. The report

demonstrates that for any given topic, there is a substantial percentage of elementary

teachersthatdonotfeel“verywellprepared”toteachtheparticulartopic,andyetthese

teachershaveacertificationtodoso(PROM/SE,2006).Thus,fromthisreportitisclear

that aspiring teachers need to be given the appropriate coursework in their teacher

preparationprogramssothattheyfeeladequatelypreparedtoteachallelementary

23 

mathematicsconceptsandsimultaneouslyhaveknowledgeof themathematics that isa

precursor to elementarymathematics aswell as themathematics taught in themiddle

grades.

The CBMS recommends that teacher preparation programs should require 12

semesterhoursthatfocusonthekindergartenthrougheighthgrademathematicsfound

inthe CCSSM.Sixofthe twelve hours should focus on number and operations with

specialattentiongiventothepropertiesofoperations.Theremainingsixhoursshouldbe

devotedtoadditionalworkwithalgebra,measurementandgeometry.Coursesshouldbe

structuredsothatfutureteachersofmathematicshavetimetothinkabout,discussand

explain mathematical ideas in order to make sense of the mathematics they will

eventuallyteach,therebylearningmathematicscontentthroughtheCommonCoreState

StandardsforMathematicalPractice(CBMS,2013).

To complement coursework, teacher preparation programs should also identify,

nurture and sustain high quality field experiences for future teachers (Science and

Mathematics Teacher Imperative/The Leadership Collaborative Working Group

(SMTI/TLC), 2011). Field experiencesprovide theopportunity forone to learnhow to

apply the mathematical practices in real classrooms with real students. In a study

conducted to find the impact of field experiences on pre‐service elementary teachers,

significant differences in knowledge of content and students as well as pedagogical

contentknowledgewere foundbetweengroups that tookcontentandmethodscourses

whilealsoparticipatinginfieldexperiencesversusnon‐fieldexperiencegroupsthatonly

tookcontentormethodscourses.Thisstudyalsofoundthatwith50hoursof fieldwork

priortomathematicscontentandmethodscoursework,studentsstillperformedlowon

24 

a content knowledge for teaching mathematics assessment, suggesting that past

experienceshave little impactonmathematicalpedagogical contentknowledge forpre‐

service teachers (Strawhecker, 2005). This study demonstrates the importance of

complementingmathematics courseworkwith fieldexperiences rather than completing

eitheroneinisolation.

With the new reform based practices adopted by most states, it will be

particularlyimportantforallaspiringteacherstobegivenopportunitiestoobservehigh

qualityinstructionalignedwiththeCCSSMandlearnalongsidemasterfulteachers.When

donewell,fieldexperiencesensureteacherscanapplytheeducationprogramknowledge

and skill that they have learned in their programs, in the classroom (Perry, 2011).

AccordingtoastudyconductedbytheFordhamInstitute,outof716teachereducatorsin

four‐yearcollegesanduniversities,48%feltthattoomanycooperatingteacherslackthe

dispositionandskillstobeeffectivemodelsfortoday’sstudentteachers(Duffet&Farkas,

2010). Therefore cooperating teachers should be carefully selected, so that prospective

elementary teachers are not undermining their coursework by observing and learning

fromteachersthatneglectreformandcontinuetousetheirstatus‐quopractices. Instead,

fieldexperiences shouldprovideanopportunity forbothprofessionals andprospective

teachers alike, to develop communities of mathematical practice, and for higher

education to simultaneously engage in improving practice in schoolswhile nurturing a

newgenerationofteachers(SMTI/TLC2011).

TheCCSSMalonewillnotrevampthequalityofeducationintheUnitedStates.In

orderforthestandardstobe implementedappropriately,teacherpreparationprograms

must also play their role in this time of transition.Higher learning institutions should

25 

reevaluateandaltertheirpreparationprogramstoreflectthehigherlevelof

mathematicalunderstandingsuggested in thestandards.Teacherpreparation, including

mathematics content courses, methods courses and field experiences need to work

togethertodevelopprospectiveteachers’knowledgeofelementarymathematicscontent

andteaching.

 MathematicalKnowledgeforTeaching

 Research conducted by Liping Ma (1999) comparing the understanding of

elementary mathematics of U.S. teachers versus teachers in China suggests that U.S.

teachers tend to be procedurally focused rather than teaching for a conceptual

understanding.Forexample,whenexaminingsubtractionwithregrouping,Mafoundthat

seventy‐sevenpercentofU.S.teachersdisplayedonlyproceduralknowledgeofthetopic

versusfourteenpercentofChineseteachers.Theunderstandingdisplayedwas

superficial, limited to aspects of the algorithm rather than demonstrating a conceptual

understanding.Shefoundthatthisrigidunderstandingextendedintotheclassroomand

limited the children’s understanding, indicating that elementary teachers in the United

Statesmusthaveabetterunderstandingoftheconceptualfoundationsofthe

mathematicstheyteachinorderfortheirstudentstoalsogainthisdeepunderstanding.

Ma’sresearchexemplifiesthepresentneedforhighereducationtorevisitthe

professional preparation of teachers and revise the mathematical pedagogical content

knowledgethatisnecessaryforteacherstoteachthenewstandards(SMTI/TLC2011).

According to Lee Shulman, content knowledge “refers to the amount and

organizationofknowledgeperseinthemindoftheteacher”whilepedagogicalcontent

knowledgeis“thewaysofrepresentingandformulatingthesubjectmatterthatmakeit

26 

comprehensible to others” (1986, p. 9). Therefore, in Shulman’s framework, content

knowledge in mathematics requires a deep understanding of the structures of

mathematics while pedagogical content knowledge is a particular form of content

knowledgethatateachermusthaveinordertoteachmathematicsforstudentlearning.

This includes knowing the misconceptions that students bring with them about the

content they are learning and knowledge of strategies to successfully reorganize their

pre‐existing understanding of the content (Shulman, 1986). Thus, PCK is the bridge

betweencontentknowledgeandthepracticeofteaching(Ball,Phelps&Thames,2008).

Deborah Ball built on Shulman’s work on PCK and applied it specifically to

mathematics to coin the phrase “mathematical knowledge for teaching” (MKT). This is

the “mathematical knowledge needed to carry out the work of teaching mathematics”

(Ball, Phelps, & Thames, 2008, p. 395). It is the knowledge that enables a teacher to

predict and identify themisconceptions that students have, engage in error analysis of

mathematical tasks, have the ability to consider themultiple approaches to a solution,

andbeabletoidentifyalternateapproachesthatyieldacorrectanswer.MKTalso

includesbeingabletoconceptuallymakesenseofproceduresinmathematics(Ball,etal.,

2008). Hence, MKT is a kind of professional knowledge of mathematics, specific to

teachers of mathematics, which is different from the knowledge demanded by other

mathematicallyintenseoccupations(Ball,Bass&Hill,2005).

The MKT framework refines Shulman’s framework of teacher knowledge and

further delineates two of his original categories, subject matter knowledge and

pedagogicalteacherknowledge,intofourdomains.Thefourdomainsunderitsumbrella

include: common content knowledge (CCK), specialized content knowledge (SCK),

27 

knowledgeofcontentandstudents(KCS),andknowledgeofcontentandteaching(KCT).

Thefirsttwodomainsfallundersubjectmatterknowledgewhilethelattertwodomains

fallunderPCK.

Commoncontentknowledge is themathematicalknowledgeandskill that isnot

unique toteachingwhileSCKis themathematicalknowledgeandskill that isunique to

teaching. KCS is content knowledge combined with knowledge of how students think

about,know,orlearnparticularcontent. Inutilizingthisparticularknowledge,teachers

shoulduse their knowledgeof the topic and their insights on students to further think

aboutwhat studentsaredoing (Ball,Hill,&Schilling,2008).Forexample,beingable to

anticipatewhatconceptsstudentswillmostlikelyfindconfusing(Ball,Phelps&Thames,

2008).KCTcombinesknowledgeabout teachingandknowingaboutmathematics.This knowledge requires an understanding of mathematics content as well as an

understandingofpedagogicalissuesthatwillaffectstudentlearning.

TomeasurethedomainsofCCKandSCK,Ball,Hill,&Bass(2005)developeda250

multiple‐choice questionnaire using research on the teaching and learning of

mathematics,analysisofcurriculummaterials,studentwork,andpersonalexperience.In

measuringCCK,researchersdevelopedquestionsthatweresetinteachingscenarios,but

stillonlyrequiredalevelofmathematicalunderstandingheldbymostadults.Questions

pertaining to SCK required teachers to analyze student work to determine validity of

alternatealgorithms,identifyrepresentationsofnumberoroperationsusingpicturesor

manipulatives, andgive explanations for commonmathematical rules. This instrument

was then used in studies to examine whether teachers’ scores are a predictor of high

qualityinstructionandincreasedstudentlearning(Ball,Hill,&Schilling,2004).

28 

Ball, Hill, & Rowan (2005) conducted a study looking at the effects of teachers’

mathematical knowledge on student achievementwith a sample of 700 first and third

grade teachers and almost 300 students. The study focused on the effectiveness in

teaching mathematics as a predictor of student gains, including what knowledge a

teachershasandhowtheythenusethisknowledgeintheirclassrooms.Although

teachersmaybehighlyproficientintheirknowledgeofmathematics,thisdoesnot

suffice, to help students learn theymust also be able to use their pedagogical content

knowledge to perform the tasks that they perform as teachers. From the MKT

measurement instrument, researchers selected questions dealing with specialized

mathematical knowledge and skills used in teaching mathematics, and found that

teachers’CCKandSCKwassignificantlyrelatedtostudentachievementinfirstandthird

grades, even when controlled for various factors including socioeconomic status,

absences,andteachercredentials.Thesizeoftheeffectofteachers’MKTwascomparable

tothesizeoftheeffectofsocioeconomicstatusonstudentgainscores,therebyameans

ofslowingdownthegrowthofanachievementgap(Ball,Bass&Hill,2005). Thus it is

clearby thisstudythat teachersneedtobepreparedwiththevariousdomainsofMKT,

notonlysotheycanbesuccessfulintheclassroom,butalsotobetterensurethesuccess

oftheirstudents.

Hillandresearchers(2008)conductedastudytofindtherelationshipbetweena

teacher’sMKTandtheirmathematicalqualityof instruction(MQI).MQI isa“composite

ofseveraldimensionsthatcharacterizetherigorandrichnessofthemathematicsofthe

lesson, including the presence or absence of mathematical errors, mathematical

explanationsandjustification,mathematicalrepresentation,andrelatedobservables”(p.

29 

431).Usingasampleoftenteachersfromsecondtosixthgrade,theMKTmeasurement

instrumentandvideo‐tapedteacherlessons,thisstudyattemptedtotesttherelationship

between teachersubjectmatterknowledgeand instructionbycomparing teacherswith

varying levels ofMKT to understand howMKT is expressed in instruction. Tomeasure

MQI researchersdesignedacodingschemecomprisedof12codes for teachers’MKT,8

codesforteachers’useofmathematicswithstudents,and10equitycodes.Eachteacher

submitted9videotapedlessons,andlessonswerebrokendownandcodedinfive‐

minutesegments.Eachteacherwasthengivenanoveralllessonscorebasedonthemean

ofthenineindividuallessonscores.

According to their results, teachers carrying a high level ofMKT provide better

mathematicalinstructionfortheirstudentsbyavoidingmathematicalerrors,usingtheir

knowledge to support rigorous reasoning and explanations, having the ability tobetter

analyze and use mathematical ideas, and using more mathematics overall in the

classroom. Researchers foundmore consistentexamplesofelegantmathematics in the

classroomsof teacherswith ahighMKTcompared to teacherswith lowMKT.Teachers

withalowMKTwerefoundtohavemorevariabilityinthequalityoftheirlessons,with

just asmany lessonsgoingpoorlyasgoingwell (Hill, et al., 2008). Inaneffort togive

students in today’s classroom access to a rigorous understanding of elementary

mathematics, this finding supports the need for teacher preparation programs to build

pre‐serviceteachers’mathematicalknowledgeforteaching.

Sincethetimethat teachereducatorsspendwithprospectiveteachers is limited,

it is impossible foreducators toaddressall themathematical topics thatanelementary

teacher may encounter in their teaching career. The goal of elementary mathematics

30 

courses should be to develop independent, reflective learners that have the ability to

addressnewcontentandnewpedagogiesastheyarepresented,andmakesenseofnew

mathematicalideasontheirown(Browning,Garza‐Kling,Moss,Thanheiser&Watanabe,

2010).AccordingtotheCBMS(2001),teachereducatorsshouldidentifywhat

conceptions pre‐service teachers have when they enter classrooms and then build on

theseconceptions.Thereforestudentsshouldbeengagedintaskscreatedtoaddressthe

conceptsthattheyarefamiliarwithinanefforttomakewayforthedevelopmentofmore

sophisticatedideas.

Courses should aim to develop connections between mathematical content

knowledgeandknowledgeof contentand teaching,knowledgeof contentandchildren,

andknowledgeof thecurriculum.Students shouldbeprovidedwith theopportunity to

grapple with unfamiliar mathematical concepts by using their common content

knowledge tomake new connections for relatedmathematical ideas. This processwill

develop and build on their current knowledge ofmathematics (Browning et al., 2010).

Burton,Daane,andGiesen(2008)foundthatbyinfusing20minutesof5th and6th grade

mathematics content into a mathematics methods course, pre‐service teachers

demonstrated greater improvement in content knowledge for teaching mathematics

compared to the control group that was not exposed to this content. Thus blending

content andpedagogy into courseworkproves very beneficial inbuildingMKT for pre‐

serviceelementaryteachers.

Teachersshouldprovideopportunities formathematical communicationanduse

formative assessments to inform instruction, to not only build their understanding of

mathematics but also to provide amodel for their own teaching. Pre‐service teachers

31 

shouldlookatstudentexamplestobringtosurfacethestrugglesthatchildrenfacewith

certain concepts in an effort to build on their knowledge of student thinking and their

ownunderstandingof these concepts. Only looking intowhatthemathematics content

should be in elementary teacher preparation misses an important part of the picture,

whichishowmathematicscontentshouldbetaught(Browning,etal.,2010).Thereforein

order to build MKT, teacher educators have to incorporate a balance of content and

pedagogy in their teacher preparation classes so that pre‐service teachers can develop

theknowledgeencompassedinallthedomainsofMKT.

 MathematicsEfficacyforPre‐ServiceTeachers

 The term teacherefficacy canbedefinedasa teacher’s “ judgmentof his orher

capabilities to bring about desired outcomes of student engagement and learning”

(Tschannen‐Moran&WoolfolkHowy,2001,p.783). Teacherefficacywasbuiltuponthe

construct of self‐efficacy, first described by Bandura (1977) as the judgment of one’s

capability to accomplish a task. Enochs, Smith,&Huiniker (2000) further separate the

term into a two‐dimensional construct that includes personal teaching efficacy and

teachingoutcomeexpectancy.Personalteachingefficacyisateacher’sbeliefinhisorher

ownteachingeffectiveness,whileteachingoutcomeexpectancy isateacher’sbeliefthat

effective teaching can result in positive student learning outcomes, regardless of any

externalfactors.

Research studies have indicated that teachers with a high teaching efficacy are

morelikelytotrynewstrategies,includingtechniquesthatmaybedifficulttoimplement

and those that involve risks such as sharing control with students (Hami, Czerniak &

Lumpe,1996;Riggs&Enochs,1990).Highlyefficaciousteachershavebeenfoundtobe

32 

more likely to use inquiry and student‐centered teaching strategies, whereas teachers

with a low sense of efficacy aremore likely to use teacher‐directed strategies, such as

lecturingandreadingfromthetext(Czerniak,1990). Briley(2012)foundthatpersonal

mathematics teaching efficacy is found to have a statistically significant positive

relationship to the beliefs about the nature of mathematics, the beliefs about doing,

validating,andlearningmathematics,andtothebeliefsontheusefulnessof

mathematics. Therefore building teachers’ mathematics efficacy is very important,

especiallywiththenewteachingstrategiesandtechniquesinvolvedintheCCSSM.

Numerous studies have found significant increases in mathematics teaching

efficacy after completion of amathematicsmethods course (Huinker&Madison, 1997;

Utley,Mosley,&Bryant,2005 ; Swars,Hart,Smith, Smith,&Tolar, 2009). Swars, et al.

(2009)foundsignificantincreasesinbothoverallpersonalmathematicsteachingefficacy

and mathematics teaching outcome efficacy during methods courses. In teacher

preparationprograms,mathematicsmethodologycoursesactastheprimarymechanism

for influencingteachercandidates’beliefsaboutmathematicspedagogy(Mixel&Cates,

2004).Methodscoursesthatemphasizeadevelopmentofconceptualknowledgebefore

procedural knowledgeandpromotea cultureof understandingand successful problem

solving have found to help alleviate the mathematics anxiety of aspiring elementary

teachers (Swars, et al., 2009), thereby increasing their mathematics teaching efficacy

(Swars,Daane,Giesen,2006).

Swars, et al. (2006) found thatmathematical anxietyhas a negative relationship

with mathematics teaching efficacy. Results from their study suggest that pre‐service

teachers need experiences within mathematics methods courses which address their

33 

past experiences with mathematics. This self‐awareness of negative experiences with

mathematics may be a building block towards reducing mathematics anxiety and

increasingmathematicsteachingefficacy.

Althoughmethodscourses inpreparationprogramshavebeen found to increase

teacher candidates’ mathematics teaching efficacy and influence their beliefs on the

teaching ofmathematics, this has not been found to transfer into their practice in the

classroom (Raymond, 1997;Vacc&Bright, 1999).This canbe attributed to the finding

thatbeliefsaboutmathematicscontentaremoreclosely linked to instructionalpractice

than beliefs about mathematics pedagogy (Raymond, 1997). Briley (2012) found that

mathematical beliefs are a statistically significant predictor of and have a statistically

significant effect on mathematics teaching efficacy. Therefore teacher preparation

programsshouldaimtheirefforts towards influencing teachercandidates’beliefsabout

mathematics content throughcontent courses (Mizell&Kates,2004). According to the

research mentioned, content courses can be a venue to influence teacher candidates’

beliefsaboutmathematicscontent,therebyaffectingtheirmathematicsteachingefficacy

andinstructionalpractices.

Data indicates that as prospective teachers develop their understanding of the

mathematics content knowledge needed to teach at the elementary level, they become

better able to understand and embrace pedagogical beliefs that are more cognitively

orientedandbecomemoreconfidentintheirownskillsandabilitiestoteach

mathematics effectively (Swars et al., 2009). Yet when researching the impact of

additional content courses on teacher candidates’ beliefs on mathematics content and

pedagogy,Cates&Mixell(2004)foundthatalthoughtheexperimentalgroupwith

34 

additional content coursework gained confidence in their own mathematical abilities,

theyalsogrewmoretraditionalintheirpedagogicalbeliefsandmorestronglybelievedin

alotofworkbookpracticeforstudents.Theauthorsattributethisfindingtobearesultof

teacher‐directed instruction in their content courses. Therefore content courses should

be structured to be student centered rather than teacher directed, so that prospective

teachers are exposed to the kind of teaching thatwill be expected of themwhen they

enter the classroom (Browning, et al., 2010). According to Briley (2012), pre‐service

teachers’ reflections on their mathematical beliefs in a constructivist classroom may

emerge lasting change to their mathematical beliefs, mathematics self‐efficacy, and

mathematicsteachingefficacy,whichmaycontinueonintotheirteachingcareers.

Fieldwork is another source for changing teachers’ efficacy beliefs. During fieldwork, pre‐service teachers’ efficacy beliefs in mathematics are subject to change

(Charlambos, Charlambous, Pillipou, & Kyriakides, 2008). After interviewing

participantsfromastudyonthedevelopmentofpre‐serviceteachers’efficacybeliefson

mathematics, Charlambos, et al., (2008), found that being in schools during student

teaching and experimenting with teachingmathematics on nearly a daily basis helped

prospectiveteacherstodevelopasenseoftheircompetence.However,resultsfromtheir

studyrevealed thatpre‐service teachers’ efficacybeliefsdonotevolve inuniformways,

forexamplesometeachersfinishedtheirstudentteachingwithverylowefficacybeliefs.

The authors suggest that in addition to personal mathematics beliefs, mentors play a

pivotalroleininformingefficacybeliefsforprospectiveteachersastheyhavethe

potential to inhibit opportunities to experimentwith reform‐oriented approaches and

35 

evendamageteacherefficacybeliefs.Thusasmentionedearlier,carefulmeasuresshould

betakenwhenrecruitingteachersasmentorsforpre‐serviceteachers.

Studentteachinghasbeenfoundtohavevaryingresultsonmathematicsteaching

outcome efficacy beliefs (Hoy &Woolfolk, 1990 & Swars, et al., 2009). Some studies

indicate that pre‐service teachers’ outcome efficacy beliefs decline during student

teachingbecauseof theunrealisticoptimismthatprospectiveteachershavebeforethey

areimmersedintheirstudentteachingexperience(Hoy&Woolfolk,1990).Swars.etal.

(2009) found thatmathematics teaching outcome efficacy (MTOE) beliefs remained at

thesamelevelduringstudentteaching.Theresearchersattributethispositivefindingto

thenumerous fieldexperiences that teachercandidatesparticipated inprior to student

teaching,whichmayhavetemperedanyunrealisticexpectations,preventingadeclinein

theirMTOE beliefs. Therefore it is beneficial for teacher preparation programs to have

ongoingopportunitiesforfieldexperiences,notonlytopreventunrealisticexpectations,

but so that students candevelopa senseof their competence earlyon and improveon

theirareasofneedbeforeenteringtheworkforce.

 ConcludingRemarks

 This chapter displays a review of the literature that forms the basis for this

proposed study. First the chapter began by displaying the vast disparity between the

currentmathematicspreparationofpre‐serviceelementaryteachersandthepreparation

that is necessary to adhere to the vision of the Common Core State Standards. To

accomplish this the chapter first reviewed the structure and varying coursework

requirementsfound inelementaryteacherpreparationprogramsacrossthenationand

36 

then moved on to discuss the current expectations of elementary school mathematics

teachersunderthenewlydevelopedCommonCoreStateStandardsforMathematics.

As discussed in the previous chapter, for this study, teaching competencies are

definedas“…anintegratedsetofpersonalcharacteristics,knowledge,skillsandattitudes

that are needed for effective performance in various teaching contexts” (Dineke, et al.,

2004,p.255).Sincetheroleofteacherpreparationprogramsistoprepareteacherswith

the teaching competencies necessary to be successful in today’s classroom, MKT and

mathematicsteachingefficacybeliefswerereviewedtoinformthisstudy.

Comparedtothepast,theCCSSMrequiredifferenttypesofreasoningskillsinK‐8

mathematics, including expectations of higher order thinking inmany areas. Therefore

teachers require a deeper understanding of mathematics, beyond that of a procedural

understanding.Thusadiscussionofmathematicalknowledgeforteaching(Ball,Thames,

&Phelps, 2008)presents a theoretical context for this studyas it displays themultiple

dimensionsofknowledgethatteachersofmathematicsneedinordertobesuccessfulin

theclassroom.

Building mathematics teaching efficacy beliefs in pre‐service teachers is also a

crucial piece of teacher preparation programs. Teacher candidates should feel well

preparedandconfidentintheirabilitiestoteachmathematics.Sinceelementaryteacher

candidates enter teacher preparation programs with a variety of mathematical

experiences that influence their own beliefs on their abilities to teach mathematics,

teacherefficacybeliefsarealsopresentedinthisliteraturereviewasanothertheoretical

contextforthisstudy.

37 

ChapterIII

METHODOLOGY

Thepurposeofthisstudyistodeterminehowwelltraditionalelementary

preparation programs and alternative elementary certification programs are perceived

by aspiring teachers, to prepare them to teachmathematics to the standards required

today.Theresearcherbelievedthatfindingsfromthisstudywouldallowpolicymakersto

make more informed decisions on mathematics coursework and field experiences

provided in elementary teacher preparation programs. To understand how we are

currently preparing elementary teacher candidates to teach mathematics, the study

addressedfourresearchquestions:(1)Howdoparticipantsdescribetheirmathematical

experiences prior to entering their teacher preparation programs? (2) How do

participants describe the mathematics preparation they receive in their respective

preparationprogramsandwhat typesofmathematicspreparation, if any,do theywish

they had before beginning student teaching? (3)What kinds of additional professional

activities,beyondtheofficialpreparationreceivedintheirteachereducationprogram,do

participants engage in outside the classroom to support and enhance their formal

preparationformathematicsteaching?(4)Whatareparticipants’perceivedpedagogical

competencies in mathematics and how do these differ across various preparation

programs? This chapter discusses the research methodology behind this study and

addressesthefollowingareas:(1)therationaleforresearchapproach,(2)descriptionof

theresearchsample, (3)summaryof informationneeded,(4)overviewoftheresearch

design,(5)methodsofdatacollection,(6)analysisandsynthesisofthedata,(7)ethical

considerations,and(8)limitationsofthestudy.

38 

 

RationaleforResearchApproach 

Qualitativeresearch isbestutilized topromoteadeepunderstandingofa

social setting or activity as viewed from the perspective of the research participants,

implying an emphasis on exploration, discovery, and description (Bloomberg & Volpe,

2008). Since the current study uses the perspectives of teacher candidates to paint a

pictureofhowelementarypre‐serviceteachersarebeingpreparedtoteachmathematics,

a qualitative approach seemed to be the most suitable research method. Specifically,

Marshall (1985) suggests that qualitative research be used for research that seeks to

explorewhereandwhypolicyand localknowledgeandpracticeareatodds.Througha

review of the literature, it was found that in order for the CCSSM to be instituted in

classrooms appropriately, teachers require a deep understanding of the mathematics

taughtattheelementarylevel,yettheliteraturesuggeststhatthereisawidevariancein

how elementary teacher preparation programs are preparing teachers to teach

mathematics. Qualitative studies are used to understand how people interpret their

experiences, how they construct their worlds, and what means they attribute to their

experiences (Merriam, 2009). This study uses a qualitative approach to describe the

varying experiences of participants in their elementary mathematics preparation

programs in an effort to understand how higher learning institutions are preparing

aspiringteacherstoentertheCommonCoreclassroomsoftoday.

Thisstudyusesphenomenologyastheframeworkforthisqualitativestudy.

Thepurposeofphenomenologicalresearchistoinvestigatethelivedexperienceof

peopletodeveloppatternsandrelationshipsofmeaning(Volpe,2008).Sincethisstudy

uses the perspectives of participants to understand and paint a picture of how

39 

elementary teacher preparation programs are preparing elementary school teachers to

teachmathematicsintoday’sclassrooms,phenomenologyisthemostsuitableapproach

forthisstudy.

 TheResearchSample

 Purposeful sampling was used to select participants for this research study. In

purposefulsamplingsubjectsareselectedbecauseofsomecharacteristic(Patton,1990).

Inparticular,maximumvariation isa typeofpurposefulsampling thatwasused in this

study.Maximumvariationsamplingaimstocaptureanddescribethecentralthemesthat

cutacrossagreatdealofvariation.Toobtainthistypeofsampleonebeginsby

identifyingdiversecharacteristicsorcriteriaforselectingthesample. Thedatacollection

and analysiswill yield two types of findings, first high‐quality, detailed descriptions of

eachcasethatwilldocumenttheuniquenessineachsample,andsecond,anyimportant

sharedpatternsthatemergeacrossthevaryingcases(Patton,1990).

Toobtainasamplewithawidevarietyofpreparationinmathematics,the

principal investigator reached out to institutions that participated in the TeacherPrep

Review2013Report,publishedby theNationalCouncilonTeacherQuality (NCTQ).The

TeacherPrep Review is a report examining the basic elements of teacher preparation

design including admission standards, syllabi, course descriptions, textbooks, student

teachingmanuals, graduate surveys, and other documents as a “blueprint” formolding

teachers (NCTQ, 2013). Based on these elements, elementary teacher preparation

programsweregivenarankingoutoffourstars,fourbeingthebestandzerostarsbeing

themost inneedof improvement.Forthisstudy,recruitmenteffortswerebasedonthe

report’sgradeforstandard5,theCommonCoreElementaryMathematicsStandardand

40 

standard1,SelectionCriteria.Toobtainafullpictureofelementaryteacherpreparation

in mathematics and keep the institutions selectivity standardized, the researcher

recruited participants strictly from institutions earning four stars in the selectivity

standard(meaningtheseprogramsonlyadmitthetophalfofcollegestudentsintotheir

teacherpreparationprogram)andvariedtheamountofmathematicspreparationinthe

sampleby recruitingparticipants from five institutionswith stars ranging fromzero to

four.

The data used to score the mathematics standard included course descriptions

and credit information of elementary mathematics content and methods courses in

institution of higher education catalogs, syllabi of required elementary mathematics

content courses, required primary textbooks in required elementary mathematics

content courses, and integratedPostsecondary EducationData System (IPEDS) data on

meanuniversitySAT/ACTscoresormeanofSAT/ACTscoresself‐reportedtotheCollege

Board.Usingthisdata,onesubjectspecialistthenevaluatedeachprogramusinga

detailedscoringprotocolandtenpercentoftheprogramswererandomlyselectedfora

secondevaluationtoassessanyvariancesinscoring(NCTQStandard5Breakdown,

2013). From this analysis, undergraduate andgraduate elementary teacherpreparation

programswereassignedascorefortheirmathematicspreparationrangingfromzeroto

four stars. Four or three stars designate a preparation program whose coursework

addresses essentialmathematics topics inboth adequatebreadth anddepth.Two stars

designate a program whose coursework addresses essential mathematics topics in

adequatebreadthbutnotdepth.Finally,aoneorzerostardesignatesaprogramwhose

coursework addresses essential mathematics topics in inadequate breadth and depth.

41 

Programs may have earned a few or no stars due to a variety of reasons including:

mathematics coursework that includes courses that are designed to prepare both

elementaryandmiddleschoolteachers,contentcourseworkthatfocusesonelementary

mathematicsmethodsratherthancontent,andlackofahigh‐caliberelementarycontent

mathematicstextbooktosupportinstruction(NCTQFindings,2013).

Althoughthe2013 Teacher Prep Review hasundergonemuchscrutinyinthe

media,beingthatthisreportisthemostcurrentdocumentexaminingelementary

teacher preparation programs at the undergraduate level, the researcher felt that the

current studywould add to the body of research in elementary teacher preparation in

mathematics by examining the alignment of the 2013 Teacher Prep Review to the

perceptions of future teachers that attend the institutions in the report. Thereby

recognizingthecriticismsofthemethodologyoftheNCTQreport,theresearcherfeltthat

using a research sample based on its rankings would provide another means of

investigatingtheaccuracyofthereportwhileadvancingcurrentresearch inelementary

teacher preparation in mathematics. The researcher’s initial intent was to recruit

participants from one four‐star program, one two‐star program, and one zero‐star

program in hopes to obtain ten participants from each. After emailing 32 institutions

listedasfourstarsintheselectivitycategoryoftheReview,fourinstitutionsrespondedin

agreement to participate in the study: one high ranking undergraduate elementary

teacher preparation program inmathematicswith a four‐star designation, two average

programswithatwo‐stardesignation,andoneprograminneedofimprovement,witha

zero‐star designation. The researcher had to obtain IRB approval from one of the

institutionsbeforeprogramsupervisorswereabletosendouttheelectronicsurvey.

42 

Thefour‐starteacherpreparationprogramparticipatinginthisstudyispartofa

stateinstitutionlocatedinthesouth.Thisinstitutionadmitsseventy‐fiveundergraduates

intoitsearlychildhoodeducationprogrameachsemester.StudentsmustobtainaGPAof

2.8 to apply to the program. The coursework requirements for this program in

mathematics includethreemathematicscourses,oneofwhichisrequiredforentry into

theprogram,theothertwosatisfyafoundationsandquantitativereasoningrequirement.

According to the institution’s website, two elementary education mathematics courses

shouldbe substituted to satisfy these tworequirements.One is ageometry course that

examines mathematical topics designed for future elementary teachers discussing

propertiesofangles,circles,spheres,triangles,quadrilaterals,measurement,length,area,

volume, transformations, congruence, and similarity. The other is a numbers, algebra,

andstatisticscourseforelementaryschoolteachersdesignedtoexaminenumbersense,

algebraincludingexpressionsandsolvingequations,andbasicdescriptivestatistics.The

required mathematics course necessary to be admitted into the program is a course

examining the meaning of all arithmetic operations with whole numbers, fractions,

decimals and discussing ratios and proportions. Once students are admitted into the

program, they then must complete two methods courses. One is a course titled

“Children’sMathematicalLearning,”examiningthedevelopmentofmathematical

thinking and learning by including a field component and another titled “Mathematics

Teaching and Curriculum in PreK‐5th Grade,” incorporating content and materials

appropriate for teaching mathematics in these grades by integrating an analysis of

mathematicsteaching,includingtheuseoftechnologyalongwithafieldcomponent.The

program coordinator sent out an email to all elementary student teachers in the

43 

undergraduateprogramand five studentsagreed toparticipate, approximatelya seven

percentresponserate.

Thefirsttwo‐starteachereducationprogram(TEP), letitbeknowninthisstudy

as “TEP A”, is a mid‐size university in located in the Midwest. Students must have a

minimum GPA of 2.5 in order to be accepted into the elementary education major.

Coursework inmathematics education includes two content courses and onemethods

course.Onecourseistitled“MathforElementaryTeachersNumberSenseandAlgebra”,

while the other is titled “Math for Elementary Teachers: Probability, Statistics, and

Geometry”. Theuniversityalsorequiresanadditionalmathematicsmethodscourse.No

description of this course can be found on the institution’s website. Program A has a

partnershipwith localschooldistricts toprovide fieldexperiences forstudents for four

semestersfollowedbystudentteaching.Aftertherecruitmentemailwassenttostudent

teachers, ten out of seventy‐two student teachers participated in the survey, or a 14

percentresponserate.

The second two‐star program, let it be known in this study as “TEP B”, is small

stateschoollocatedintheMidwest.CourseworkincludesFoundationsofMathematicsin

ElementaryEducationCourses I and II, aswell as onemethods course titled “Teaching

Elementary/MiddleSchoolMathematics”thatprovidestenhoursofteachingexperiences

outonthefield.Coursedescriptionscouldnotbefoundonthewebsite.Twoparticipants

fromthisinstitutionagreedtoparticipateinthesurveyandtheresearcherwasnotable

toascertainhowmanystudentteacherswereintheprogram,thusnoresponseratecan

benoted.

44 

Thezero‐starprogramthatagreedtoparticipate in theresearchstudy isasmall

private institution located inNewYork State. The institution requires students to take

one three credit, “TeachingMathematics” course and one three creditmethods course.

Theprogramsupervisornotedthattherewereonlythreestudentteachersinthe

programandallthreeagreedtofilloutthesurvey,withaonehundredpercentresponse

rate.

Theresearchersoughttofindalternateroutecertificationparticipantsthathadall

gone through the same alternate route certification program and thus a snowball

samplingstrategywasused.Snowball,chain,ornetworksamplingisaformofpurposeful

samplingthatinvolvesfindingafewkeyparticipantsthatmeetthecriteriaofyourstudy

who then refer you toother participants (Merriam,2009). In this study the researcher

didnotuseparticipantsto“snowball”,butinsteadreachedouttoacolleaguethatgained

certificationandnetworkswithteachersfromaparticularalternativeroutecertification

program. This colleague reached out to her network of teacher colleagues and they

identifiedfirstyearelementaryteachersintheirschoolsthatgainedcertificationthrough

aspecificalternaterouteprogram.

The researcherwas able to obtain thirteen potential participants and contacted

themdirectlyviaemail.Fromthispoolofpotentialparticipants,fiveagreedtoparticipate

in the study. All participants were part of the same alternative certification program

located inNew Jersey.Thisparticularprogram is designedparticularly fornovice

teacherswithuptotwoyearsofteachingexperienceandprovidesaninety‐hour

program that leads to state teacher certification. During the weekend and weeknight

seminars teachers learn teaching techniques and subjectmatter knowledge.They then

45 

spendpartof theirweekly seminarspracticing these techniqueswith facultyandpeers

and obtain feedback on their teaching. The curriculum is delivered through both in‐

personandonlineplatforms.The seminarsareheld twoweeknightsperweekandone

Saturdayeachmonthduring theacademicyearalongwithsessionsduring thesummer

prior to the start of the school year and a one‐week session following the academic

schoolyear.

Allparticipantsinthisstudyhadcompletedallthemathematicscoursesrequired

intheirpreparationprogramandwere inthemiddleof theirstudent teachingor in the

caseofthealternativeroutecertificationparticipants, theirfirstyearteaching.Withthe

exceptionof oneof the alternative route certificationparticipants, all participantswere

femaleinthisstudy.

Once participants filled out the electronic survey, they were then asked to

participate in a follow up interview. Out of the 25 students participating in the study,

seven agreed to a follow up interview. Thus a convenience sample was used for the

interviewsbasedonresponsestotherecruitmentletter.

 OverviewoftheResearchDesign

 Thefollowing listdescribesthestepsusedtoconducttheresearchforthisstudy

followedbyanin‐depthdiscussionofeachstep.

1.Following the proposal defense, the principal investigator was granted IRB 

approvaltobegintheresearch.

2. Prior tocollectingdata,a reviewof the literaturewasconducted toresearch

existingcontributionsmadetothestudyofelementaryteacherpreparationin

mathematics.

46 

3. Potential elementary teacher preparation programs, adhering to the criteria

mentioned in the previous section were contacted via email, and those that

agreed to participatewere sent a recruitment letter alongwith a link to the

electronic surveys.The surveys collecteddemographic alongwithperceptual

datafromtheparticipants.

4. Oneonone, in‐depth interviewswereconductedwithsevenofthe

participants.

5. Interview data responses analyzed within and between the different

interviewees.

 ProposalDefenseandIRBApproval

 Theresearcherdevelopedaproposalforthisstudythatincluded:theneedforthe

study,thepurposestatement,researchquestions,andprocedures.Oncetheproposalwas

successfullydefended, the researcherappliedandwasgranted IRBapproval toconduct

thestudy.

LiteratureReview 

Anongoingconsultationofcurrentliteraturewasconductedtoinformthisstudy.

Therewerefourmajorareasofresearchthatarosefromthereviewoftheliterature:the

current state of elementary teacher preparation programs in mathematics, the new

demands of the Common Core State Standards for Mathematics (CCSSM) and its

implications on teacher preparation programs, mathematical knowledge for teaching

(MKT), and finally pre‐service teachers’ mathematics efficacy and its relation to

mathematicseducationcoursework.Theliteraturereviewprovidedtheresearcherwith

aholisticunderstandingof theelementary teacherpreparation inmathematicsby

47 

identifying two key elements necessary in preparing teachers, building MKT and

mathematicsefficacy,aswellasprovidingaglimpseintothecurrentstateofelementary

teacherpreparationprogramswithregardstostructure,coursework,andtheCCSSM.

Togainclarityonthecurrentstateofteacherpreparationprogram,theresearcher

reviewed various studies related to elementary teacher preparation in mathematics,

includingtheNCTQTeacherPrepReview(2013),NoCommonDenominator(NCTQ,2008),

anda reporton theTeacherEducationandDevelopmentStudy inMathematics (Center

for Research inMathematics and Science Education, 2010). Recommendations by the

CBMS informed this study on the CCSSM and its implications on the coursework for

teacher preparation programs.This literature was used to inform the interview

questionsonparticipants’ teacherpreparation inmathematics and thequestions asked

in themathematics pedagogy survey.The reportGearingupfortheCommonCoreState

Standards in Mathematics (Institute for Mathematics and Education, 2011) provided

researchonthenewdemandsoftheCCSSMandtheresultingareaswhereteacherswill

need further preparation. These areas were then used to inform the design of the

mathematicspedagogysurveyasconceptswerechosenthatreflecttheseparticular

areas.

Inthepast,itwouldsufficeforstudentstohaveafluentproceduralunderstanding

of elementary schoolmathematics, but theCCSSMhavechanged this, studentsarenow

required to have a deeper understanding ofmathematics concepts. Therefore, teachers

must have a conceptual understanding of mathematics. Teachers must also be able to

make thisknowledgeaccessible for their students.Teacherpreparationprogramsmust

buildthisknowledgeinprospectiveteachers.Therefore,literaturebyDeborahBallwas

48 

reviewed to gain an understanding of the dimensions of mathematical knowledge for

teachingandhowthisknowledgecanbeinstilledinpre‐serviceteachers.Thisliterature

wasusedtodesignthetypesofquestionsaskedinthemathematicspedagogysurvey.

Sincethisstudyutilizedamathematicsteachingefficacysurveytogatherdataon

participants’ perceptions of their pedagogical teaching competencies, literature on

mathematicsteachingefficacyisthelastareaofresearchthatinformedthisstudy.

 Data‐CollectionMethods

 PhaseI:TheSurveys 

Thirty‐twoundergraduateelementaryteacherpreparationprogramswere

contactedviaemailtoparticipateinthestudyalongwiththirteenfirstyearteachers

participatinginanalternativeroutecertificationprogram.Ofthoseundergraduate

institutionsandindividualscontacted,fourprogramsandthreeindividualsagreedto

participate. Participantsweregiventhechoicebetweenpapercopiesofthesurveyoran

electronicmailcontainingfourlinks,participants’rightsandconsentform,abackground

survey,“YourBeliefsAboutTeachingMathematicsSurvey,”anda“MathematicsPedagogy

Survey.”Allbutoneofthefourprogramschosetocompletethesurveyselectronically.

AccordingtoFowler(2013),surveyresearchisaimedat“tappingthesubjectivefeelings

ofthepublic…Thereare,inaddition,numerousfactsaboutpeopleandtheirbehaviors

andsituationsofpeoplethatcanonlybeobtainedbyaskingasampleofpeopleabout

themselves”(p.2).Thesurveyinstrumentutilizedinthecurrentstudyincludedboth

multiplechoiceandopen‐endedresponsestotapintoparticipants’perceptionsofhow

theirteacherpreparationprogramshavepreparedthemtoteachmathematics,including

49 

perceptionsoftheircoursework,mathematicsefficacy,perceivedmathematics

competenciesandtheirviewsonmathematicspedagogy.

 TheBackgroundSurvey 

Thepurposeofthebackgroundsurveywastoobtaindemographicinformationon

participantsandtogaininsightontheirmathematicspreparation,bothingeneral

mathematicsandwithrespecttomathematicseducation,atthehighschooland

universitylevel.Participantswereaskedtoselectthemathematicscoursestheytook

priortoenteringtheirteacherpreparationprogramandindicatethenumberofcourses

theytookinboth,elementarymathematicscontentandmathematicsmethods. Togain

insightonparticipantsasmathematicsstudentspriortoenteringtheirteacher

preparationprogram,informationwasgatheredonparticipants’performancein

mathematics,suchas“IwasanAstudent“,“IwasaBstudent”,andsoon.Withthese

categoriesitwaspossibletomakecomparisonsonthevaryingmathematicsexperience

thatparticipantshadwithinateacherpreparationprogramandacrossthedifferent

programstoanswerthefirstandsecondresearchquestionforthisstudy. Datawas

collected,suchasthegradesthatonestudentteachesin(orhastaughtin),whichwere

notusedinthisstudy.Personalinformationsuchasnameandinstitutionofstudywere

collectedbutwerereplacedwithapseudonymtomaintainconfidentiality.

 YourBeliefsAboutTeachingMathematicsSurvey 

The purpose of the “Your Beliefs About Teaching Mathematics ” survey was to

collect informationonparticipants’beliefsabout teachingmathematics.The instrument

wascreatedbyEnochs,Smith,andHuinker(2000)andisoriginallytitledthe

50 

Mathematics Teaching Efficacy Beliefs Instrument. This twenty‐one question survey is

divided into two categories, personalmathematics teaching efficacy, that is a teacher’s

belief in his or her own teaching effectiveness, and mathematics teaching outcome

expectancy, a teacher’s belief that, regardless of external factors, effective teaching can

result in positive student learning outcomes. Survey choiceswere in the formof a five

choice Likert scale: “strongly agree”, “agree”, “undecided”, “disagree”, and “strongly

disagree”.Thepersonalmathematicsteachingefficacy(PMTE)statementswereanalyzed

forthisstudyandarelistedinthetablebelow.

Table1.“YourBeliefsAboutTeachingMathematics”PMTEstatements Icontinuallyfindbetterwaystoteachmathematics.

EvenifItryveryhard,IdonotteachmathematicsaswellasIteachmostsubjects.

Iknowhowtoteachmathematics conceptseffectively.

Iamnotveryeffectiveinmonitoringmathematicsactivities.

Igenerallyteachmathematicsineffectively.

Iunderstandmathematicsconceptswellenoughtobeeffectiveinteaching elementarymathematics.

Ifinditdifficulttousemanipulativetoexplain tostudents whymathematicsworks.

Itypicallyamabletoanswerstudents’questionsaboutmathematics.

IthinkIdonothavethenecessary skillstoteachmathematics.

Givenachoice,Iwouldnotwantmyprincipaltoevaluate mymathematicsteaching.

Whenastudenthasdifficultyunderstandingamathematicsconcept,Iamusuallyat

alossastohowtohelpthestudentunderstanditbetter.

51 

Whenteachingmathematics,Iusuallywelcomestudents’questions. Idonotknowwhattodototurnstudentsontomathematics.

    

This survey was used to gain insight on the attitudes and perceptions of

participants towards their own teaching of mathematics, informing research question

four, therefore there was no “right” or “wrong” answer. However, with teacher

preparation programs preparing teachers for the Common Core classrooms of today,

buildingmathematicsefficacyisanintegralpartofthisprocess.Theliteraturestatesthat

highly efficacious teachers have been found to use more student‐centered teaching

strategies (Czerniak, 1990). This aligns with the vision of the Common Core State

Standards forMathematicsPractice. Inaddition,mathematicsefficacyhasa statistically

positive relationship to the beliefs about the nature of mathematics, as well as doing,

validating,andlearningmathematics(Briley,2012).

 TheMathematicsPedagogySurvey 

Thissurveywasdesignedtoinvestigatehowparticipantswouldteachelementary

mathematicsconceptsthatnowrequireaconceptualunderstandingforstudentsdueto

theCCSSM.AccordingtotheCommonCoreInitiative,“understandinghowthestandards

differfrompreviousstandards,andthenecessaryshiftstheycallfor‐isessentialto

implementingthem”(Corestandards.org). Thereforetheresearcherusedquestionsthat

alignedwiththekeyinstructionalshiftsoftheCommonCore,includingthecallfor

conceptualunderstandingofkeyconceptsinareasofplacevalueandfractions(Institute

forMathematicsandEducation,2011). Sincearithmeticisakeyfocusinelementary

52 

schoolmathematics,questionsinthissurveywereunderthenumberandoperations

domainoftheCCSSMandspannedacrossgradesonethroughseven.

Thefirstthreequestionsinthissurveywereadaptedfromapedagogysurvey

designedbyLicwinko(2014)toresearchproblematicelementarymathematicstopicsin

areasoffractions,estimation,subtraction,andareaandperimeter.Additionalquestions

onfractionswereaddedtoinvestigatehowpre‐serviceteacherswouldteachparticular

conceptsthatwerenotincludedinthesurvey.Followingtherecommendationsfor

elementaryteacherpreparationintheMathematicalEducationofTeachersII,thissurvey

alsohadaquestionthatdealtwithmathematicsinthemiddleschoollevelbydesigninga

questiononthesubtractionofintegers.Lastly,toinvestigateparticipants’knowledgeof

alternatealgorithmsandplacevalue,thelastquestionofthesurveywasdesignedaftera

questioninthemathematicalknowledgeforteachingmeasures(Hill&Ball,2008).

Thequestionswereopen‐endedsothatparticipantscoulddescribetheirown

approachtoteachingeachtopictotheirstudents.Byleavingthequestionsopen‐ended,

participants’responsescouldbeevaluatedforuseofspecializedcontentknowledge,

knowledgeofcontentandstudents,andknowledgeofcontentandteaching.Thisalso

allowedtheresearchertodeterminewhethertheparticipantwoulduseaproceduralor

conceptualapproachtoteachingthetopic.Thiswasthencomparedtotheteaching

approachsuggestedbytheCCSSM.Eachquestionislistedbelowalongwiththecluster

andstandardsoftheCCSSM.Participantswereaskedtoleavequestionsblankiftheydid

notknowhowtoanswerthequestion.

Table2.MathematicsPedagogySurveyQuestionsAlignmentwiththeCCSSM

53 

 

Question TheStandardsand InstructionalStrategies RecommendedbytheCommonCore

1.Astudentmakesthefollowing

mistake:734‐481=353.Howdoyou

respond?

TheCluster:Useplacevalueunderstanding

andpropertiesofoperationstoaddand

subtract(Grade2).

TheStandards:Addandsubtractwithin 

1000,usingconcretemodelsordrawings

andstrategiesbasedonplacevalue,

propertiesofoperations,and/orthe

relationshipbetweenadditionand

subtraction;relatethestrategytoawritten

method.

Understandthatinaddingorsubtracting

three‐digitnumbers,oneaddsorsubtracts

hundredsandhundreds,tensandtens,

onesandones;andsometimesitis

necessarytocomposeordecomposetens

orhundreds.

2.Imagineyouareintroducingaddition withregroupingtoyourstudents,how

wouldyoushowstudentswhy

27+95=122?

TheCluster:Useplacevalueunderstanding 

andpropertiesofoperationstoaddand

subtract(Grade1).

TheStandards:Addwithin100,using

54 

 

  concretemodelsordrawingsand 

strategiesbasedonplacevalue,properties

ofoperations,and/ortherelationship

betweenadditionandsubtraction;relate

thestrategytoawrittenmethodand

explainthereasoningused.

Understandthatinaddingtwo‐digit

numbers,oneaddstensandtens,onesand

ones;andsometimesitisnecessaryto

composeaten.

3.Createastoryproblemtoillustrate that4¼÷½=8½.

TheCluster:Applyandextendprevious 

understandingsofmultiplicationand

divisiontodividefractionsbyfractions

(Grade6).

TheStandards:Interpretandcompute

quotientsoffractions,andsolveword

problemsinvolvingdivisionoffractionsby

fractions,e.g.,byusingvisualfraction

modelsandequationstorepresentthe

problem.

Forexample,createastorycontextforthe

problem.Howmany½cupservingsarein

55 

 

  4¼cupsofyogurt?

4.Howwouldyouexplainwhywhenyou

dividebyafractionyoumultiplybythe

reciprocalofthefraction?Youcanuse

thefollowingdivisionexercisetoassist youinyourexplanation. 3÷¼=12.

TheCluster:Applyandextendprevious

understandingsofmultiplicationand

division(Grade5).

TheStandards:Interpretdivisionofa

wholenumberbyaunitfraction,and

computesuchquotients.

Createastorycontextfor3÷(¼),anduse

avisualfractionmodeltoshowthe

quotient.Usetherelationshipbetween

multiplicationanddivisiontoexplainthat

3÷(¼)=12because12×(¼) =3.

5.Whenintroducingstudentshowto renamemixednumbersintoimproper

fractions,howwouldyouteachachildto

rename2¾intoanimproperfraction?

TheCluster:Buildfractionsfromunit 

fractions.(Grade4). 

TheStandards:Decomposeafractionintoa

sumoffractionswiththesame

denominatorinmorethanoneway,

recordingeachdecompositionbyan

equation.

Justifydecompositions,e.g.byusingvisual

fractionmodel.Example:2¾=1+1+¼+

¼+¼=4/4+4/4+¼+¼+¼=11/4.

56 

 

   

6.Astudentdoesnotunderstandwhy

whenyousubtractanegativenumber,

thenumberfromwhichyouare

subtractingincreases.

Forexample:7–(‐2)=9. Howwouldyoushoworexplaintoa

studentwhythisistrue?

TheCluster:Applyandextendprevious

understandingofoperationswithfractions.

(Grade7)

TheStandards:Understandp+qasthe

numberlocatedadistance|q|fromp,inthe

positiveornegativedirectiondependingon

whetherqispositiveornegative.

Understandsubtractionofrational

numbersasaddingtheadditiveinverse,p‐

q=p+(‐q).

Showthatthedistancebetweentwo

rationalnumbersonthenumberlineisthe

absolutevalueoftheirdifference,and

applythisprincipleinreal‐worldcontexts.

7.Imaginethatyouareworkingwith yourclassonmultiplyingtwo,twodigit

numbers.Yourstudents’solutionsare

displayedtotheright.Whichofthese

student’smethodcanbeusedtomultiply

anytwowholenumbers?

TheCluster:Useplacevalueunderstanding 

andpropertiesofoperationstoperform

multi‐digitarithmetic.(Grade4)

TheStandard:Multiplyawholenumberof

uptofourdigitsbyaone‐digitwhole

number,andmultiplytwotwo‐digit

numbers,usingstrategiesbasedonplace

valueandthepropertiesofoperations.

57 

                      

PilotStudyforSurveys 

The researcher tested the surveys in a pilot study conducted with three

undergraduateelementarystudentteachers.Thesestudentschosetouseapapercopyof

the survey rather than the electronic version. Upon receiving their responses, the

researcherworkedwith a doctoral colleague to edit thewording ofMathematics

PedagogySurveyquestion5,as thepilotstudyresultsshowedthatparticipantsdidnot

understandthequestionasitwas initiallyworded.Thesurveywasagainhandedoutto

anothergroupof elementary student teachers, responseswereanalyzedandconsistent

responsesmadeitclearthatparticipantsunderstoodthequestions.

PhaseII:Interviews 

Theinterviewwasusedasasecondarymeansofdatacollectionforthisresearch.

According to Siedman (2013), “If the researcher’s goal is to understand the meaning

people involved in education make of their experience, then interviewing provides a

necessary, ifnotalways completely sufficient, avenueof inquiry” (p.10).For this study,

the interviewmethod allowed for clarification to responses in the survey aswell as a

58 

venue for detailed descriptions of the various teacher preparation programs

participating in the study. In particular, semi‐structured interviews were used for this

study. Insemi‐structured interviews the interviewerhasquestionsprepared inadvance

but questionsmust be open, therefore subsequent questions cannot be planned by the

interviewer and must be carefully improvised according to the responses from

participants (Wengraf,2001).Theresearcher felt thatparticipants’answerswouldvary

according to their teacher preparation program and therefore semi‐structured

interviews would be the most suitable choice so that participants would have the

opportunitytounveil,indepth,theirperceptionsoftheirteacherpreparationprograms.

In‐depth interviews,however,dohave limitations. Interviewscanbeextensiveandtime

consuming, furthermore, because of the subjective nature of interviews, analyzing and

interpreting data aremore complex than structured surveys. Another limitation is that

interviewingcanbemorereactivetopersonalitiesand interpersonaldynamicsbetween

theinterviewerandtheparticipant(Kenke,2008).

 InterviewScheduleofQuestionsandPilotInterviews 

Theprincipalinvestigatorusedthestudy’sfourresearchquestionsasa

framework todesign the interviewquestions. Amatrixwas constructed todisplay the

relationshipbetweentheresearchquestionsandtheinterviewquestions.Theresearcher

asked her fellow doctoral colleagues along with her advisor to review and give

suggestionstoimprovetheresearchquestions. Nextapilotinterviewwasconductedto

test the interview questions and make adjustments to allow for more open‐ended

questions enabling the maximum conversation between the researcher and the

participant.

59 

 

InterviewProcess 

Individual emails were sent to all participants who participated in the study.

Initially the researcher chose select participants to interview based on answers to the

survey. However, the response rate was low and therefore the researcher invited all

participants toparticipate inthe interview.Sevenparticipantsagreedtobe interviewed

and a suitable date and time was selected to complete the interview via FaceTime or

Skype.The interviews took place betweenMarch andMay 2014.All interviewswere

taperecordedandtranscribedverbatim.

 MethodsforDataAnalysisandSynthesis

 Analysisofthe“YourBeliefsAboutTeachingMathematicsSurvey” 

Responsestothissurveywerebasedonparticipants’perceptionsofthemselvesas

teachersandonhowtheirteachingaffectstheirstudents’learning.Thereforeresponses

were based on opinion and were not objective, as there were no correct or incorrect

responses. The overall purpose of the research was to understand how well teacher

preparationprograms are perceived in building teacher efficacy in aspiring elementary

teachers inmathematics and lends itself tobe anexploratoryanalysis.Responseswere

analyzed tomake comparisons and distinctions within and across teacher preparation

programs.Thefindingswerethencomparedandcontrastedwiththecurrentresearchin

theareaofteacherefficacyinmathematics.

 Analysisofthe“MathematicsTeachingPedagogySurvey”

Responses to this surveywere based onparticipants’ opinions of best practices,

andalthough thereareno technically “correct”or “incorrect” responses, the responses

60 

could be on parwith research based best practice, particularly those suggested by the

CommonCore. Alphanumeric codeswereassigned todescriptors fromthepedagogical

recommendationsnoted in theCCSSM.Responseswere then categorizedas conceptual,

basedontherecommendationsbytheCCSSM,orprocedural. Responseswereanalyzed

foremergingpatternsand themesbothwithin teacherpreparationprogramandacross

the different groups. The background survey was also consulted to inform why

participantsmayhaverespondedastheydidtocertainquestionsinthepedagogysurvey.

 AnalysisoftheInterviews 

Theresearcherbeganbytranscribinginterviews,verbatim,andcaptured

emotionsandpauseswithinthetranscript.Theneachinterviewwaslistenedtoasecond

timetomakesurethedatawasaccuratelydepictedandtranscribed.Next,theinterviews

werecodedaccordingtoapreviouslyestablishedcodingschemebasedondescriptorsof

categories found in the study’s conceptual framework. The coding process segments

interviewsintoseparatecategories,forcingonetolookateachdetail,whereassynthesis

involvesputting thesesegments together todepictaholisticand integratedexplanation

(Bloomberg& Volpe, 2008). Following this, the researcher decomposed interviews by

placing the codedquotations into their coordinating categories. As the datawas

analyzed, select categories and descriptorswere omitted and added as various themes

andpatterns emerged in participants’ responses. The researcher looked for trends and

patternswithincategories,acrosscategories,andcomparedthesetrendsandpatternsto

the current literature, to synthesize a holistic picture of how elementary teacher

preparationprogramsarepreparing teachers to teachmathematicsto thestandardsof

today.

61 

 

EthicalConsiderations 

To ensure participants were protected, informed consent was maintained

throughoutthestudy.Participantswereaskedtosignaconsentformbeforeundergoing

the survey and/or interview. The informed consent stated the purpose of the study,

followedbynotespertainingtoconfidentiality.Theconsentformwaswrittenatamiddle

schoollevelreadingleveltoensurethatallparticipantscouldreadandcomprehendthe

document.Theconsentformnotedthatallthedatacollectedwouldbeheldconfidential

andthatnonameswouldbeusedinthestudy.Participantswerealsoaskediftheywould

consenttobeingtaperecordedinthepossiblecasethattheywerechosenforafollowup

interview.

Oncethesurveyswerecompletedandreturned,anIDcodewasassignedforeach

participantandthiscodewaswrittenonthetopofthebackgroundpagealongwiththe

subsequent pages of the survey. The background page was ripped off and a list of all

participants’alongwiththeirassignedcodeswaskeptinasafe,securelocationseparate

fromthedata.Afterreviewingthedata,thoseparticipantsthatwereaskedbacktodoan

interviewwereassignedapseudonymto link their interviewto their identity.Once the

interviewswere transcribed, all audiotapeswere destroyed.All coding and data

materialswerestoredinalockedfinalcabinetintheprincipalinvestigator’shomeoffice

Therewereminimal risks toparticipants thatparticipated in thisstudy, suchas confusion in how to answer questions in the Mathematics Pedagogy Survey due to a

possible lackof knowledgeon the subjectmatter. Someparticipantsmayhave also felt

anxiousduetopre‐existinganxietytowardsmathematics.Toensurenonegativeimpacts

ontheparticipants,all identifyinginformationwasremovedfromsurveyandinterview

62 

materials. Furthermore, in all but one teacher preparation program, professors and

supervisors did not see participants’ responses since the surveys were done

electronicallyandsubmitteddirectlytotheresearcher.Theteacherpreparationprogram

thatusedapapercopywastheonlycasewheretheprofessorhadanopportunitytosee

participants’responses,butsincethesestudentswerestudentteachingandnottakinga

coursewiththeprofessor,itisunlikelythatparticipantsfacedanybiasasaresultoftheir

responsestothesurvey.

 

 

LimitationsoftheStudy 

 

Aswithanystudy,thisstudydidhavesomelimitingfactors.Sincethisstudywas

qualitativeinnature,subjectivityintheanalysisofdatawasonelimitingfactor.Asecond

limitation results from participant reactivity to the survey questions in the teacher

efficacy survey. Although responses were based on opinion, participants may have

believed that there was a “correct” response, or a response that they perceived made

them look better by the researcher, rather than choosing the response that accurately

describedtheirfeelingstowardsaparticularstatement.

Finally,a last limitationwasthesizeof thestudy.Theparticipantstruggledwith

recruiting participants since there were time limitations, IRB approval in some

institutions, and a smaller pool to choose from based on the selection criteria for the

study.Therefore,therewasasmallsamplesizefromeachteacherpreparationprogram,

particularlyforthefour‐starinstitutionwhereonlyfiveparticipantsfilledouttheefficacy

survey,andfourparticipantsfilledoutthepedagogysurvey.Sincethesefiveparticipants

arerandomsamplefromaratherlargeelementaryteacherpreparationprogram,theydo

63 

notportraytheteacherpreparationprogramasawhole.Thismade itdifficult tomake

comparisonsacrossthedifferentteacherpreparationprograms.

Tocompensatefortheselimitationstheresearchertookprecautionarymeasures.

To reduce subjectivity and bias in the study the researcher coded data according to a

peer‐reviewed coding scheme. Second, the researcher stated thepurpose clearly in the

consent documents that participants signed prior to the data collection. Next, in the

directionsofeachsurveytheresearchernotedthatthesurveywasnotatestand

thereforetherewerenocorrectorincorrectresponses.Sincethesampleofthestudywas

small, transferability of findings to similarly ranked elementary teacher preparation

programs is questionable. To compensate, the researcher collected data usingmultiple

instrumentstopaintaholisticpictureofthevaryingprogramsbycarefullydescribingthe

experiencesofparticipantsintheirteacherpreparationprogramsbywayofrich,detailed

descriptions.Bydoingthis,theresearcherfeltthattheknowledgegainedfromthisstudy

couldbeconsideredandutilizedwhenitsapplicationappropriatelyfitanothersituation.

   

TheRoleoftheResearcher 

Giventhattheresearcherhasadegreeinelementaryeducation,mathematicsand

mathematicseducation,shehashadpriorexperiencesintheareaofelementaryteacher

educationinmathematics,assheherselfhasoncebeenintheshoesofherparticipants.

Theseexperiencesprovidedherwithsomebackgroundknowledgewhenshecreatedthe

pedagogysurvey,assheknewtheparticulartopicsandconceptsthatelementaryteacher

candidateswouldfinddifficult.Duringthesemi‐structuredinterviews,theresearcher

calleduponherpriorexperiencesinherownpreparationtoquestionparticipantsabout

64 

theircourseworkandexperiencesteachingmathematics.Thisallowedtheresearcherto

probedeeperintotheirpersonalexperiencesandobtainthickdescriptionsoftheir

teacherpreparationprograms.

65 

ChapterIV

FINDINGS

The purpose of this study is to determine how well traditional elementary

preparation programs and alternative elementary certification programs are perceived

by aspiring teachers, to prepare them to teachmathematics to the standards required

today.Theresearcherbelievedthatfindingsfromthisstudywouldallowpolicymakersto

make more informed decisions on mathematics coursework and field experiences

provided in elementary teacher preparation programs. Data was obtained from 25

backgroundsurveys,25personalmathematicsteachingefficacysurveys,20mathematics

pedagogy surveys, and seven interviews. Following the data collection, the data was

analyzedtoanswerthefollowingfourresearchquestions:

(1) How do participants describe their mathematical experiences prior to

enteringtheirteacherpreparationprograms?

(2) How do participants describe the mathematics preparation they receive in

their respective preparation programs and what types of mathematics

preparation,ifany,dotheywishtheyhadbeforebeginningstudentteaching?

(3) What kinds of additional professional activities, beyond the official

preparationreceivedintheirteachereducationprogram,doparticipants

engage in outside the classroom to support and enhance their formal

preparationformathematicsteaching?

(4) What are participants’ perceived pedagogical competencies in mathematics

andhowdothesedifferacrossvariouspreparationprograms?

66 

This chapterwill discuss the findings obtained from the data to answer each research

questionmentionedabove.

   

ResearchQuestion1:Howdoparticipantsdescribetheirmathematical

experiencespriortoenteringtheirteacherpreparationprogram?

 Finding #1: The majority of participants indicated taking a mathematics course

beyond Algebra/Trigonometry prior to entering their Teacher Education Program

(TEP).

Themajority(18outof25)ofparticipantsindicatedtakingamathematicscourse

beyondAlgebraorTrigonometrypriortoenteringtheirTEP.Themajority(12outof18)

oftheseparticipantstookCalculusandsome(6outof18)tookstatistics.Some(7outof

25)participantsnotedthattheirhighestlevelofmathematicspriortogainingentryinto

theirTEPwasAlgebraorTrigonometry. Manyof thoseparticipants that tookCalculus

were able to place out of any additional mathematics coursework aside from those

courses requiredby theirTEP. Twoparticipants from the two‐starTEPAnoted taking

additional developmental mathematics courses in order to obtain entrance into their

teacherpreparationprogram.

 Finding #2: Regardless of their overall perceptions of themselves asmathematics

students,fewparticipantshaveanenthusiasticattitudetowardstheirmathematical

experiencesinelementary,middle,andhighschool.

67 

When asked to rate themselves as “A”, “B”, “C”, or “D”mathematics students, 13

participantsdistinguished themselves as “A” students and12participantsdistinguished

themselvesas “B”students. Yet,when interviewed,evenparticipantsrating themselves

as“A”studentsvoicedmixedperceptionstowardstheirmathematicspreparationpriorto

enteringtheirteacherpreparationprogram.

 

WhenIwasinelementaryschool,Ihatedmath.ButIwouldn’tsayIstruggledwithit. Itjust‐‐Iwas‐‐whatever.TherewerecertainsubjectsthatIwas frustratedwith. But as I got older, I became like, really good atmath.(Bethany)

 

ImeanIdidn’tloveit.ItookAlgebraIandIIwhenIwasingradeschoolandthenwhenIwasinhighschoolIwentallthewayup.Soitwasn’tmyfavoritebutIdidwellinitso(pauses)itwasokay.(Laura)

 

Um….wellIactuallyskippedagradewhenIwasyounger.Ididprettywellinmath.It’snotmyfavoritesubject. (Samantha)

  

 Only a few participants (2 out of 7) that were interviewed had strictly positive

perceptionsofthemselvesaslearnersofmathematics.

 

I thinkIwasdefinitelygood inhighschoolandIamgoingtoassumeIwaspretty goodinelementaryandmiddleschool. (Clarissa).

 

Ilovedmath!Itwasmy favorite subject aside from art. Um…and it wasalwaysasubjectIdidreallywellin.(Carol)

 

  

According toBall (1989), “In thecaseofprospectivemathematics teachers, their

experienceshaveoftenpersuadedthemthatmathematics isafixedbodyofrules,adull

and uninteresting subject best taught throughmemorization and that they themselves

are not good atmath“ (p.12). In a study conductedwith high school students, Fleener

(1996) found that by high school students have established their core beliefs about

mathematicsandthesebeliefs are a result of their own experiences in mathematics

68 

classrooms.Thereforeparticipants’responsesseemtoagreewiththestatementbyBall,

as most participants were not enthusiastic in their attitude towards the subject itself,

even when they did find success in mathematics. Pre‐service teachers come to their

teacherpreparationprogramswith informalbeliefs,embeddedintheirmental imageof

classroom practice and this strongly influenceswhat they learn and how they learn in

teacher preparation programs (Richardson, 1996). Thus, teacher educators delivering

mathematics coursework in teacherpreparationprogramsplayan important role since

theyhaveanopportunity to turnprospective teacherson tomathematics.Ballsuggests

thatexperiencesinamethodscoursecanhelpprospectiveteachersacquireanewsense

ofwhatitmeansto“understand”mathematics.

Thetablebelowshowstheinformationgatheredfromthebackgroundsurvey, includingthenumberofmathematicscontentcoursesandmethodscoursesthat studentshadtotakeintheirteacherpreparationprogram. Allparticipantsarenamedby

pseudonyms.

69 

Table3.Participants’DemographicInformation,Coursework,andSelf‐StudentRating 

  

Participant InstitutionRating

HighestLevelofMathematicsPriortoEnteringTheirTEP

StudentTeachingGrade

NumberofContentCourses

NumberofMethodsCourses

Self‐StudentRating

Megan TwoStars Trigonometry 7 2 1 BMaddy TwoStars AlgebraII* 4 3 1 AKathy TwoStars Calculus 1 1 3 ASamantha TwoStars Trigonometry* 2 2 1 BKathleen TwoStars Statistics 6 2 1 BNatalia TwoStars Calculus(AP) K 3 3 ALaura TwoStars Calculus 1 3 3 ASeana TwoStars Statistics K 2 1 AJackie TwoStars Statistics 4 3 2 BNancy TwoStars Calculus(AP) 2 2 1 ACarol Alt.Route Calculus(AP) 2 1 1 AAaron Alt.Route Calculus(AP) K 1 1 AJessica Alt.Route Calculus(AP) 6 0 0 ADiana Alt.Route Statistics 1/3 1 1 BHillary Alt.Route Calculus(AP) 3 1 1 ABethany FourStars Calculus(AP) 2 1 2 AClarissa FourStars Calculus(AP) K 3 2 AChantel FourStars Trigonometry K 3 2 BAbby FourStars Calculus 2 3 2 BChristy FourStars Statistics 3 3 2 BBrittany TwoStars Trigonometry K 0 0 ALindsay TwoStars Calculus(AP) 1/6 2 1 BTeresa Zerostars n/a n/a n/a n/a BLily Zerostars Statistics 1/4 1 1 BLauren Zerostars Algebra 4 1 1 B

   

*indicatestakingtwoadditionaldevelopmentalmathematicscoursespriortoenteringtheirteacherpreparationprogram.

 

  

ResearchQuestion#2:Howdoparticipantsdescribethemathematicspreparation

theyreceiveintheirrespectivepreparationprogramsandwhattypesof

70 

mathematicspreparation, ifany,do theywish theyhadbeforebeginningstudent

teaching?

 Finding#1:There is little consistency in themathematicseducationof elementary

teachers within a teacher preparation program and across different teacher

preparationprograms.

The interviews and surveys revealed the lackof standardizationacrossdifferent

preparationprograms, andmore surprisingly,within the sameprogram.Participants in

the same teacher preparation programs had varying types of mathematics education

courses that they took, just to fulfill the minimum requirements. There were also

differences in their accounts of the amount of time they spent teaching mathematics

during their field experiences and the capacity towhich the Common Core State

StandardsforMathematicswereintegratedintotheircoursework.Thereforetwopeople

studying mathematics education in the same program could have vastly different

experiences.Thiswasalsothecaseacrossthedifferentteacherpreparationprograms.

Afteranalyzingdatagatheredfromthebackgroundsurveysandinterviews,there

is little agreement in thenumber and typeof elementarymathematics content courses

andelementarymathematicsmethodscoursesrequiredbyteachereducationprograms,

bothacrossdifferentprograms,andmorestrikinglywithinagivenTEP.Afterconducting

interviews, participants gave different accounts of the coursework that they were

requiredtotakeintheirteachereducationprogram.

Twooutof twenty‐fiveparticipants indicatedhavingnocontentandnomethods

courses prior to student teaching. Further investigation revealed that one participant

participated in the alternative certification program and the other participant was an

71 

adult learner, going back to the institution for an early childhood degree after already

teachingintheuppergrades.

The majority of the participants from the four‐star TEP program indicated

completing three content courses and twomethods courses; however, one participant

indicatedonlytakingonecontentcourse.Whenaskedaboutthis,theparticipantclarified

that the requirements had changed and that she only had to take one elementary

mathematicscontentcoursewhilemanyofherpeershadtotakethreecoursespriorto

gainingadmittanceintotheprogram.

Participants from the two‐star TEP A had a wide variance in the amount of

mathematics content andmethods courses that theyhad to take. Oneparticipant took

one content course, four participants indicated taking two content courses, and six

participants indicated taking three content courses. Similarly therewasalsoadisparity

inthenumberofmethodscoursesthatparticipantstook.Sevenoftheelevenparticipants

indicated taking one methods course, one participant indicated taking two, and three

participants indicated taking three. After interviewing participants, the researcher

ascertainedthatonereasonforthisvarianceisduetothefactthatsomeparticipantsare

endorsed in mathematics and are certified to teach mathematics from kindergarten

through grade nine and are therefore required to take more mathematics courses.

Interviews with participants revealed that some of their peers, in the elementary

education program, were able to bypass the elementary mathematics content

coursework but all pre‐service teachers had to take themethods course, therewas no

wayofbypassingthiscourse.Theiraccountsarefoundbelow.

72 

Theywouldhavehadtotakethemethodsoneintheprogram,thereisnowaytogetoutofthatone. Theywouldhavehadtotakethatfullsemester.Um…buttherearewaysthattheycouldnothavetakentheearlierones.(Laura)

 

BecauseweasasectiontakeallofourmethodsclassestogethersoImean I got to knowpeople inmy sectionprettywell. ButwheneverwegottothemathIknowalotofpeoplewerejust like“Whydowehavetodothis?Whyareweexplainingthis?”(…)SoIcouldtell thatsome people for math were a little more like frustrated with theclassesbecauseIdon’tthinkthattheyhadthatbackgroundthatsomeofusdid.(Nancy)

 

Participantsfromthetwo‐starTEPBhaddifferentresponsestotheircoursework.

After interviewing, the researcher found out that one participant was the previously

mentionedadult learner going back for her early childhoodeducation certification and

was not required to take any further coursework inmathematics education, while the

secondparticipantnotedtakingtwocontentcoursesandonemethodscourse.

With the exception of one participant who did not take any coursework in

mathematics,allothersfromthealternativeroutecertificationprogramindicatedtaking

one content course and onemethods course. The only consistent responsewithin an

institutionwasamongthezero‐starprogramparticipants,whomallindicatedtakingone

mathematicscontentcourseandonemathematicsmethodscourse.

Currentresearchhasshownthatthereislittleconsistencyinthenumberof

creditsthatpre‐serviceteacherstakeincourseworkdealingwiththemathematicstaught

in the elementary grades across different teacher preparation programs (Center for

Research in Mathematics and Science Education, 2010; Kilpatrick, 2001; NCTQ, 2008;

NCTQ,2013).Inthecurrentstudy,whatismoresurprisingistheinconsistencyfoundin

coursetakingwithinagiventeacherpreparationprogram.Mathematicscourseworkfor

elementaryteachersisnotonlyatimeforprospectiveteacherstobuildtheirknowledge

73 

in the varying domains of MKT, but it is also another chance to foster positive

mathematics teaching efficacy beliefs and encourage student‐centered beliefs about

pedagogy. Two teachers can have similar knowledge, but use very different teaching

stylesbasedontheirvaryingbeliefs(Ernest,1989).Thereforeifprospectiveelementary

teachers are being prepared to step foot into the Common Core classrooms of today,

wheremathematicslessonsarestudent‐centered,andconceptsaretaughtina

conceptualmanner, building teachers’ beliefs about howmathematics is taught in

additiontotheirpersonalmathematicsteachingefficacyisalsoimportant.

The interviews with participants revealed that the CCSSM were studied and

utilizedindifferentcapacitieswithinandacrossinstitutions.Themajorityofparticipants

(five out of seven) mentioned printing out the standards, perhaps looking at each

standardinisolationandhavingtoincludethemwhentheywrotelessonplansfortheir

coursesbutnotcompletinganindepthanalysisofthestandardsandtheirprogressions

withintheircoursework.Onlytwoparticipantsmadementionofcompletingawell

versed study of the standards and their progression from one grade to the next. One

participant was from the four‐star TEP, and the other was from the alternative route

certification program. The first two quotes below are from students in the same TEP,

emphasizingthevariationofintegratingtheCCSSMintocourseworkwithinaninstitution

and the third quote is representative of how most interviewees described their

experiencewiththestandards.

 

OurlastcoursethatwetookwedidanindepthstudyontheCommonCore(…)We spread it out by like going through each‐‐we got into groups andeachgrouphadagradelevelandeithernumberandoperationsorgeometryor whatever the different subtexts were level wise and thenwe had to gothrough each standard and say whether or not we thought the standard

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wouldbeeasyforourgradelevel,hardforourgradelevelorum…justrightforourgrade levelandthenwedissectedthestandards ingroupsandstufflikethat.Um…andthenwewentthroughthesetcategoriesandsaid“ok,weare learning about fractions, these are all of the standards in first, second,third, fourth, and fifth grade that have to dowith fractions and see how itgrowsandstufflikethat.(Bethany)

 

She the teacher didn’t really putmain focuses on them‐ the Common Corewhentheywereteachingit.ButwhenwehadtomakeourownlessonplansforourassignmentstheywantedustomakesurethatwehadthemintheresoreallyIfeltlikeIgotitfromthewebsitemostly.(Clarissa)

 

WedidcovertheCommonCorestandards.Welearnedhowtowritethem‐sohowtotakethemandcreateobjectivesbasedondifferentones.Thenwehadtowritefulllessonplansandturnthemintobegraded. (Samantha)

 

  

According toHiebert andMorris (2009), teacher education inuniversity teacher

preparationprogramsisa“particularcaseofteachingwheretheinstructorsareteacher

educators, the students are prospective teachers, and the learning goals are beginning

teachercompetencies”(p.475).HiebertandMorrissuggestthatthereisalackofshared

knowledgebase in teachereducation. Withregards toelementary teachereducationin

mathematics,Jansen,Bartell,&Berk(2009)statesobtainingagreementamongeveryone

in the system, on lesson‐level learning goals is crucial if collaborative efforts to build

knowledgebaseforteachereducationaretosucceed.Withoutthis,thesystemreturnsto

individualsworkingalonetoimprovetheirownpracticeandeveryonestartsfrom

scratch.The findingsof the current studydisplay a lackof sharedgoals among teacher

educatorstoeducateprospectiveteachersinthecontextoftheCCSSM. Sinceparticipants

from the same teacher preparation programs had vastly different experienceswith the

Common Core, it can be concluded that there is a lack of collaboration among teacher

educators to realign their mathematics education courses to the new standards. One

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reason for thismaybedue to the recentestablishmentof these standards in theK‐12

curriculum.

The majority of participants (20 out of 25) indicated having some sort of field

experienceprior tobeginning their student teachingyet the followup interviewsmade

clear that the opportunities for teaching mathematics vary within and across the

preparation programs. The participants not completing any field experiences were

teachers in the alternative certification program. With the exception of this group of

teachers, allparticipants thatwere interviewedcompletedweekly fieldvisits.Yet, since

thefieldexperiencewasinanelementaryclassroom,itwasnotnecessarilythecasethat

they had ample opportunities to teach mathematics in a whole group setting. The

majorityofundergraduateparticipantsthatwereinterviewed(4outof6)madenoteof

studentteachingbeingtheirfirstrealexperienceinteachingmathematicsasopposedto

simplyobservingitbeingtaught.

Idon’tthinkthatIeverreallytaughtmathematicsbeforestudentteaching.Ithinkuhh(pause)…Ididalittletutoringwithkindergartnersbutthatwasaboutit.Yeah‐Ididn’treally–Ididn’tteachit–Ihavebeenscaredtoteachit. I ‐ I ‐ because it is the most difficult concept in my opinion to teach.(Clarissa)

 Clarissaaddsthisnoteaboutstudentteaching: 

Mathwasthelastonetoaddonandthefirstonetotakeoff.(laughs)SoIthinkIspentawholetwoweeksteachingmath.SoyeahIhaven’thadalotofexperienceteachingmath.

   

Bethanysimilarlystatesthisaboutherfieldexperience: 

WellIwasinkindergarten.Soourmathwasprettysimple.Itwas“Johnnygaveyoufiveapples‐oryouhavethreeapplesandJohnnygaveyou5more,sohowmanydoyouhavenow?”(Child’svoice) Sothatisprettymuchallwedidmathwise.Wedidshapes,andwedidum(pauses)somethingelse,butwedidn’treallydomuchmath.

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Two participants made note of being able to choose the subject they wanted to teach

duringtheirfieldexperiences.Clarissawasoneparticipantwhomadementionofthisas

didNancy.

LastyearwekindofgottodowhateverwewantedbecauseyouwouldonlygototheschoolfortenweekssomyfirstplacementwastheschoolIamatnowwithmysameteacherssosheactuallyletmeteachmathalotbecausethatwasthearea thatIwasthemost‐themostcomfortablewith,andshehatesmath,soshewasfinewithlettingmetakeover.(Nancy)

 State certification requirements also stifled participants’ ability to obtain high

quality field experiences in mathematics. Due to state requirements, one participating

programrequiredstudentstobe“endorsed”inasubjectareaandfieldplacementswere

directedtowardsthisparticularsubjectarea,leavinglittletimeforteachingexperiences

intheareasthatstudentswerenotendorsedin.

MymiddleschoolplacementItaughtlanguageartsandliteracy,andsocialstudies,butIdidnotteachmathuntiltheonerightbeforeIstudenttaught(…)SoIdowishthat therewasmorepreparation for that,or thatwegotmorepracticedoingitinthefield,likeateveryplacement.(Samantha)

 The four‐star institution provided students with a unique opportunity to gain

experienceinthedevelopmentofmathematicalthinkingbyparticipatingina

partnershipprogramwitha local school. The classwasheld at thepartnership school

itself,andaftereachclassstudentssteppedintheclassroomtoworkwitha“buddy”.The

participantnotedthattheintentofthiscoursewasnottogainexperienceteachingbutto

ratherunderstand the logicofachild’s thinking inmathematics.Theparticipantgavea

child amathematicsproblemandobserved and recorded their strategies for solving it.

Theparticipantexplained that theywerenot supposed to correct the child ifheor she

obtainedanincorrectsolution.

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Thisparticipantwishedthatmethodscoursescouldbetaughtinconjunctionwith,

or closer to student teaching and field experiences that involved practice of teaching

mathematics.Theparticipant found ithard to recall the strategiesand lessons that she

learned in her methods coursework since it was in the beginning of her program

coursework.Bethanysays, “I reallywish thatwewould havewaited until not our ‐‐

becausewehadourentiremathcontentdoneinour firstyearbeforewewereeven inthe

classroom.”Incontrast,thealternativecertificationintervieweenotedthatbeingableto

complete her courseworkwhile she taughtwas a positive for her program. She states,

“Soitisdefinitelynicetohaveitwhileteaching.Youwouldforgetalotofstuffifitwaslike

you do your four years or your two‐ year program orwhatever and then you go into a

classroomand thenyoudon’thave thatcycle feedbackor thatcycleof instruction so it is

reallyhelpfulhavingitthisway.”

Fieldexperiencesgiveprospectiveteachersanopportunitytoapplythe

knowledgeandskillstheyhavelearnedintheirprogramtotheclassroom(Perry,2011).

When done in conjunctionwith content ormethods coursework, field experiences can

increase knowledge of content and students (Strawhecker, 2005). Therefore students

needanadequateamountof timespentteachingmathematics intheirfieldexperiences

anditwouldbeworthwhileforprogramstointegratetheseteachingopportunities

withintheirmethodscoursework.Thefindingsofthisstudysuggestthatthecapacityto

whichpre‐service teachers teachmathematics during their field experiences (including

student teaching) varies according to many factors, including but not limited to state

certification requirements, the cooperating teacher, efficacy beliefs, and beliefs and

attitudestowardsmathematics.

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Finding #2:Participantswould like to havemore preparation in the teaching of

early childhoodeducation concepts, thegeneral scopeand sequenceof elementary

mathematics, and tactics to integrate test preparation into daily mathematics

activities.

Three out of seven participants mentioned wishing that their coursework

involvedmorepreparationwithintroducingearlychildhoodconceptsfrombasicfraction

workinsecondgrade,countinginpre‐KandKindergarten,andlookingatandreceiving

moretrainingintheparticularconceptsthatearlychildhoodstudentsfinddifficultywith.

 

SoIwouldhavelikedtohavemoreclassesoncounting,liketheeasystuff.Ifeel once the kids have a general concept or grasp of mathematics theywill‐itiseasiertoteachthembutstartinginpre‐korkindergartenyouareteaching these kids from scratch so Iwould’ve liked to learnmore aboutthosethings.(Clarissa)

 

What I wish to see in the beginningwere just samples of great class, orgreat lessons.Um…howdo Iphrase this…maybe justaheadsuponwhatwouldbesomeofthetrickiersubjects,andbiggerconceptsinsecondgrade to teach, andhow theywere executedby teacherswhodo it reallywell.(Carol)

 

Butum…rightnow I amdoingpartitioningandbasic fractions in secondgrade, and I really wish I would remember those activities that we did.(Bethany)

 

One participant wished for more work with the general scope and sequence of

elementarymathematicsconceptsacrossthegrades.

ButIguessitishardformetounderstandwhereeachgradeisatIguess.Iknow it is different in each school so I know that it is hard for them toprepare uswith, but Iwish thatwe kind of,well inmymethods class atleast, in kindergarten youknow this, in first grade you should know this,andkindofwentoverthatalittlebitmore.(Nancy)

 Oneparticipantmademention foraneed formorepreparationon strategies for

imbeddingtestpreparationintotheregularcurriculum.

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Idefinitelywishwedidmorepracticewiththestatewideassessments.ThekidsnowtakeaPARCCexamortheISM,likebeingabletobuildthoseintotherebecauseitseemslikealotoftimeswearestuckcrammingtheweekbefore ISManddoing randomthings thataregoing topopupon the testinsteadofworkingthemintothecurriculum.(Samantha)

 ResearchQuestion#3:Whatkindsofadditionalprofessionalactivities,beyondthe

officialpreparationreceivedintheirteachereducationprogram,doparticipants

engageinoutsidetheclassroomtosupportandenhancetheirformalpreparation

formathematicsteaching?

 Finding #1:The overwhelmingmajority of participants, regardlessof their teacher

preparationprogram,didnotengageinanyoutsideactivitiestosupporttheirformal

preparationinmathematicsteaching.

Out of the seven participants that were interviewed, only one participantmade

mention of attending or being provided with professional development workshops or

conferences sponsored by her school district to better her teaching of mathematics. It

should be noted that this participant is part of an alternative certification program and

thereforehasaccesstotheseopportunitiesthroughherschooldistrict.Someparticipants

(three out of seven) mentioned tutoring children in mathematics outside of their

institutionasawaytobettertheirteaching.

 ResearchQuestion#4:Whatareparticipants’perceivedpedagogicalcompetenciesin

mathematicsandhowdothesedifferacrossvariouspreparationprograms?

Finding #1: Participants’ responses to the pedagogy survey varied within each

teacher education program. Therefore, no outright comparisons can be made in

distinguishingthecompetenciesofstudentsfromoneparticularprogramtoanother.

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Ingeneral, participantstendtoteachmoreprocedurallyrather thanfora

conceptualunderstandingofmathematicsconcepts.

Due to thevarying sample sizesrepresentingeach teacherpreparationprogram,

when analyzing and categorizing responses from the pedagogy survey as conceptual,

procedural, incorrect, or did not answer, no clear distinction could bemade across the

varyingrankedteacherpreparationprograms.Thatis,foranygivenquestion,responses

fromtheparticipantsrepresentingeachteachereducationprogramrankingfellintojust

theprocedural categoryorweresplit between the conceptual andprocedural category.

Thusparticipantsrepresentingthefour‐starteacherpreparationprogramdidnotalways

give explanations in the pedagogy survey that stressed a conceptual understanding.

Howeverwhilekeeping inmind the small sample size, for anygivenquestion, ahigher

percentageofparticipantsfromthefour‐starinstitutiontendedtoanswerquestions

fromthepedagogysurveyconceptuallyratherthanprocedurallywhencomparedtothe

other teacher education programs. The pedagogy survey did reveal the low number of

conceptual responses, particularly from those participants attending a zero‐star

institutionandthetwo‐starinstitutions.

Correct conceptual response rates for each level of ranking are displayed in the

table below. For this table, both two‐star teacher preparation programs were grouped

together since theybothhad the same ranking.Mathematically correct responseswere

categorizedasconceptualwhentheymadementionofunderstandingtheconceptbehind

thealgorithmbyuseofmanipulatives,drawings, explanationofmodels, and the likeas

recommended by the CCSSM. Mathematically correct responses were categorized as

proceduralwhenexplanationsinvolvedtheuseofaruleoralgorithmtosolvethe

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problem, with no further explanation. Responses were noted as incorrect when they

were incoherent or used mathematically incorrect explanations and/or procedures to

answer thequestionandblankresponseswerealsonoted. Responses to thePedagogy

Surveymaybefoundintheappendices. Itshouldbenotedthatfiveparticipantsdidnot

filloutthepedagogysurvey.

Table4.ConceptualResponseRatestotheMathematicsPedagogySurvey  

Question ConceptualResponseRates

Astudentmakesthefollowingmistake:734‐481=353.Howdoyourespond?

Withtheexception of the zero‐star teachereducationprogram (TEP), nodistinctionscouldbeestablishedamong thedifferentrankingprogramsas responses from participants attending the same institution fell intoboth procedural and conceptual categories regardless ofwhich TEP theparticipantattends.Thethreeparticipantsfromthezero‐starinstitutionall had responses that lackedmention of place value and/or the use ofmanipulatives.Correctconceptualresponseratesareasfollows:2outof4participants fromthe four‐starprogram,3outof10participants fromthe two‐starprogram,2outof3participants fromthealternativeroutecertification program, and 0 out of 3 responses from the zero‐starprogram.

Imagineyouareintroducingadditionwithregroupingtoyourstudents,howwouldyoushowstudentswhy27+95=122?

All three participants from the zero‐star institution used a proceduralstrategy to answer the question. Responses fromparticipants from theother institutions varied in terms their response being procedural orconceptual in nature. Therefore there was no consistency in allparticipants from one TEP belonging to one category versus the other.Conceptual response rateswere as follows: 3out of 4participants fromthefour‐starprogram,7outof10responsesfromthetwo‐starprograms,2 out of 3 responses from the alternate cert. program, and 0 out of 3responsesfromtheofthezero‐starprogram.

Createastoryproblemtoillustratethat4¼÷½=8½.

CorrectversusincorrectresponsesvariedwithineachTEP,butonly1outof the10participantsattendingatwo‐stardesignatedprogramwasableto come upwith a correct story context for the problem. Other correctresponseswere as follows: 3 out of 4 participants attending a four‐starprogram, 1 out of 3 participants from the alternative certificationprogram,and2outof3participantsattendingazerostarprogram.

Howwouldyouexplainwhywhenyoudividebyafractionyou

Correct conceptual responses were as follows: 3 out of 4 participantsfrom the four‐star TEP had a correct conceptual response, 2 out of 10participantsfromthetwo‐starprograms,1outof3participantsfromthealternativeroute certificationprogramand1outof3participants from

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multiplybythereciprocalofthefraction?Youcanusethefollowingdivisionexercisetoassistyouinyourexplanation.3÷¼=12.

thezerostarTEPprogram

Whenintroducingstudentshowtorenamemixednumbersintoimproperfractions,howwouldyouteachachildtorename2¾ intoanimproperfraction.

Themajority of responses fell into the procedural category therefore nodistinctionscouldbemadeamongstTEPprograms.Conceptualresponserate is as follows:2out of4participants from the four‐starprogram,1out of 10 participants from a two‐star program, 2 out of 3 participantsfrom the alternative route certification program, and 1 out of 3participantsfromthezero‐starprogram.

Astudentdoesnotunderstandwhywhenyousubtractanegativenumber,thenumberfromwhichyouaresubtractingincreases.Forexample:7–(‐2)=9.Howwouldyoushoworexplaintoastudentwhythisistrue?

Themajority of responses fell into the procedural category therefore nodistinctionscouldbemadeamongstTEPprograms.Correctresponserateisasfollows:2outof4participantsfromthefour‐starprogram,3outof10participantsfromatwo‐starprogram,2outof3participantsfromanalternative certification program, and 0out of 3 correct responses fromthezero‐starprogram.

Imaginethatyouareworkingwithyourclassonmultiplyingtwo,twodigitnumbers.Yourstudents’solutionsaredisplayedtotheright.Whichofthesestudent’s

Five out of twenty participants identified all three methods as beingcorrect,sevenoutoftwentyidentifiedtwooutofthreemethodscorrect,seven out of twenty identified one out of the three methods as beingcorrect (the traditionalmethod), andoneparticipantdidnot answer.Ofthe five participants that identified all three methods correct, two arefrom the alternative certificationprogram, one attended a two‐starTEP,andtwoattendedthezero‐starprogram. Surprisinglynotoneparticipantfromthefour‐starTEPwasabletoidentifyallthreemethodsascorrect.

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methodcanbeusedtomultiplyanytwowholenumbers?

 

  

Basedontheseresponses,whenanalyzingthesurveysasawhole, it isclearthat

participants tend to give responses that focus on procedural aspects of mathematics

concepts, and in some cases are incorrect in their explanations of elementary

mathematics concepts. When looking across institutions, since the number of

participantsfillingoutthesurveyfromvaryingrankedinstitutionswasnotconsistent,no

cleardistinctions couldbemadecomparing the competencies ofparticipants.However,

when looking for trends in the conceptual response rate of the different teacher

educationprograms,itisnotedthatinallbutonecase,thezero‐starprogramhad0out

of3participantsor1outof3participantsprovideaconceptualexplanationforanygiven

question. In particular, for the two questions dealing with place value, where more

participants gave a conceptual response, all three participants wrote a procedural

explanation. Incontrast, the four‐starprogramhad2outof4or3outof4 conceptual

responsesforanygivenquestioninthepedagogysurvey.Similarly,withtheexceptionof

one question, 2 out of 3 participants from the alternative route certification program

consistently gave conceptual explanations. The conceptual response rates from the

participantsattendingthe two‐starranked institutionswerealso lowforeachquestion.

Aside fromthe twoquestionsdealingwithplacevalue,eachof theremainingpedagogy

questions had only one or two conceptual explanations from the ten participants that

filled out the survey. From the explanations in themathematics pedagogy survey, it is

clearthatteachingstudentsforunderstandingwillbeachallengefornewteachers.The

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followingisadiscussionofeachquestionfromthepedagogysurveyalongwithsample

responsesfromparticipants.

Question1:Astudentmakesthefollowingmistake:734‐481=353.Howdoyou respond?

    

7 out of 20 participants used an explanation that taught this concept in a

conceptualmanner.Conceptualresponsesmadeuseofplacevalue intheirexplanations

and incorporated manipulatives like base ten blocks, place value mats or diagrams

involvingplacevalue.Thefollowingareexamplesofconceptualresponses.

 

I would have the student draw out a picture and start at the ones andcrossofthenumbers.Atthetens,thestudentwouldhavetoregroupa100tobeabletocrossof8tens.(Aaron)

 

I would sit with them with base‐10 blocks and work with them onexchanging tens and hundreds to make the problem able to solve.(Bethany)

 

Iwoulduseaplacevaluemattodemonstratetheneedtoborrowfromthehundreds.Whenastudentmovesonehundredtothetensitwillbeeasierforthemtovisualizethattheyhave6‐4insteadof7‐4.(Brittany)

 

  

12 out of 20 participants answered with a procedural response, using

explanations that had to dowith the procedure of an algorithm, including phrases like

“borrowing”,“crossout”,and “takeoneaway”.Someresponsesdidnotmakementionof

the procedure but instead used phrases like “check your answer” or “try a different

strategy”.Thefollowingareexamplesthatillustratethesekindsofresponses:

 

Iwouldshowthestudentthatwecanborrowfromthe7tomakeita6andmakethe3a13,andthenwecantake8awayfrom13.(Jackie)

 

Theyforgettoregroupandborrow,Iwouldsuggestthatthestudenttriesadifferentsubtractionstrategy.(Abby)

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IwouldexplainthatwhenImadethe“3”intoa“13”,Imusttakeaway“one”fromtheleft.(Teresa)

 

  

Question2:Imagineyouareintroducingadditionwithregroupingtoyourstudents,howwouldyoushowstudentswhy27+95=122?

    

More participants answered this question conceptually than the subtraction

example.12out of the20participantswere able to use some sort of explanation that

dealt with the concept of regrouping rather than the algorithm. The majority of

participants mademention of using base ten blocks to demonstrate the regrouping of

ones tomake a ten. The following are examples of typical conceptual responses from

participants.

 

Iwouldexplaintothestudentsthat7+5=12,butsincetheonesspotcannothold12,wewrite the2(for2ones)andcarrythe1overintothetensspottorepresentouronesetoften.ThenIwouldshowstudentstoadd1+2+9=12.Iwouldrepeatthisandshowstudentstowritethe2downagainandcarrytheoneintothehundredsspot.Thenaddagain. (Seana).

 

I would use base ten blocks to show that the ones aremore than ten byliningupwithatensstick. Thentradetheonesout.(Abby)

 

Use manipulatives or drawings while reinforcing place value. Draw out 2tensand7ones,thendrawout9tensand5ones.Circle10oftheonesandshowhowitcanbecombinedintoaten. (Carol)

 

  

Fiveparticipantsusedproceduralresponsestoanswerthequestion,andcommon

phrasesincluded“lineuptheproblemvertically”or“carrying”.

 

Iwould explain thatwe can only put onenumber at a time under the lineuntilwearefinishedorrunoutofnumbers,thereforewecanonlyputthe2.Iwouldthenputthe1ontopofthe2. (Jackie)

 

Iwouldstartbyrewritingtheproblemvertically.NextIwillexplainhowtoregroupbycarryingthe(1)tothetensplaceandthenaddthethreenumberstogethertogetthefinalanswer.(Lily)

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Three participants used methods that did not involve regrouping but alternate

methods including grouping numbers tomake 100, partial sums and rounding. One of

thesethreeparticipants(Natalia)notedthatshewas“notsurewhatregrouping is”and

insteadchoseanalternatemethodtoaddthe twonumbersbyregrouping thenumbers

tomake100.

In particular, the pedagogy survey revealed that teacher candidates struggle to

conceptualize fractions. The CCSSM state that students should demonstrate their

interpretationsofoperationsbycreatingincontextstoryproblemsthatcorrespondtoa

givenmathematical exercise.Therefore it is important thatnew teachers alsohave this

understanding. It no longer suffices for students to memorize a procedure to solve a

probleminvolvingthedivisionoffractions.Nowstudentshavetocreatetheirownword

problems that interpret the division of fractions. The following is a discussion of the

threequestionsinthepedagogysurveythatdealtwithfractions.    

Question3:Createastoryproblemtoillustratethat � �ૡ.

    

Whengiventhisexercise,onlysomeparticipants(7outof20)wereabletocreate

an accurate storyproblem thatdepicted thedivisionexercise. Correct responseswere

similar to the following,“MaryandAlicearemakingmuffins.Eachmuffintinneeds½cup

ofbatter.Theyhave4¼cupsofbatterintotal.Howmanymuffintinscantheyfill?“(Lily).

All of these responses started with 4¼ units of some object and involved finding the

numberof½‐sizedunitsintheobject.

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Themajority (13 out of 20) of participants could not create a story problem to

represent thedivisionexercise. Someparticipants (5outof20) created storyproblems

that interpreted the problem as a division by two rather than dividing by one‐half.

Responsesweresimilar to thisstoryproblemwrittenby Jackie,“Ifmomhas4¼cupsof

milkandwantstodivideitinhalfsoyouandherhaveequalparts,howmuchwouldeachof

youhave?”

Other incorrect responses (8outof20)were storyproblems thatdidnotmake

sense (3outof20),participants thatchosenot toanswer(3outof20),orparticipants

indicatingthattheydidnotknowhowtocreateastoryproblemtorepresentthe

situationbutknewhowtodotheproblemusingthetraditionalalgorithm(2outof20).

Sampleresponsesarebelow.

 

Ms.Sweetzerwants tobakecakesforherson’sbirthdayparty.Shehas4¼poundsofsugarathomeandwantstoknowhowmanycakesshecanbake.(Carol)

 

I know how to solve this one but do not know how to go aboutmaking astoryproblemthatexplainswhatneedstobedone.(Brittany)

 

Bakingrecipe formuffins.4½cupsofsugar.Eachrequires½cupof sugar.Howmanycupswillallowtherecipetobake4muffins? (Lauren)

 

  

Two participants “endorsed” in mathematics, meaning that are able to teach

mathematics through 9thgradewere unable towrite a story problem for the exercise.

Oneofthesetwoparticipantsrespondedwithastoryproblemdepictingdividingbytwo,

whiletheotherlefttheproblemblank.Wheninterviewedastowhysheleftitblankshe

respondedwith: “I think therewere two questions that Iwasn’t sure of, I just couldn’t

thinkofhowtoanswerthemandIcouldn’tfigureouthowtoexplainit.”(Nancy)Yetboth

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oftheseparticipantsare certified to teach elementary, middle, and the start of high

school.

ThesefindingsaresimilartotheresearchofLipingMa(1999),whereinasimilar

divisionby½problem, among23 teachers, 6 couldnot create a storyproblemand16

made up stories with misconceptions, many confusing division by one‐half with

multiplicationby½.Thisdivisionexercisewastakenfromasixthgradestandardinthe

CCSSM.Although in some regions this is considered amiddle school problem, as

discussed inthe literaturereview,researchhasdiscussedtheimportanceofelementary

teachersbeingwell versed inbothelementary schoolmathematics andmathematics in

the middle grades. Furthermore, depending on state certification, sixth grade is often

under the elementary certification umbrella, thus it is important that elementary

teachershaveastrongknowledgeoffractionsandbeabletocreateastorycontextfora

givenproblem.

   

Question 4: “Howwould you explainwhywhen you divide by a fraction you

multiplyby the reciprocalof the fraction?You canuse the followingdivision

exercisetoassistyouinyourexplanation:3÷¼=12.”    

AccordingtotheCCSSM,teachersshouldbeabletousevisualmodelsalongwith

story problems to display the quotient of a whole number by a unit fraction, thereby

conceptuallyshowingwhy3÷¼=3x4=12.Whengiventhisquestion,fewparticipants

(7outof20),wereabletocorrectlyuseamodelorastorycontextintheirexplanation.A

typicalresponseisfoundbelow.

competencystressedbytheCCSSM, responses to this question werecategorized as

conceptualiftheparticipant made mention of a visual model or story problem that

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IfIhave3piesandeachpieisdividedinto¼,eachpiewouldyield4pieces.Threepieswouldyield12piecestotal.Wecansolvethisbymultiplying3by4, the reciprocal of ¼. When we divided the three pies by ¼, we couldmultiplybythereciprocal4,becauseweareactuallycountingthenumberoffractions (1/4 pieces of pie) that make up the original number (3 pies).(Carol)

 In addition, one participant (Clarissa) used complex fractions, a higher‐ level

concept, inherexplanation.Mostprocedural responseshaddifferentphraseshaving to

dowithmultiplyingbythereciprocal,suchas, “flipthesecondfractionandmultiply”.A

fewof these responses involved thephrase, “theoppositeofdividing ismultiplying”,or

some equivalent phrasewith the samemeaning. These responseswere very rote and

evidencedthatparticipantshadmemorizedrulestosolvetheproblem.Thefollowingisa

typicalresponseofparticipantsthatisroteinnature:

Whenyoudividebyafractionyoumultiplybythereciprocalbecausedividing istheoppositeofmultiplyingandyouhave to flip the fraction.Forexample,3dividedby¼isthesameis3timestheoppositeof¼whichis4.Therefore,3dividedby¼isthesameas3x4=12.(Kathy)

 Most responses (11 out of 20) were similar to this, using procedures in their

explanationsthatdonotnecessarilyexplaintherootoftheconcept.Twoparticipantsdid

notanswertheexercise.

Question5:When introducing tostudentshowtorenamemixednumbers into 

improperfractions,howwouldyouteachachildtorename2 intoanimproper

 

fraction? 

Thestruggleamongstnovice teachers toconceptualize fractionswasonceagain seen in theirresponses to thisquestionon thesurvey. Sincevisualmodels isa

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describeddecomposingwholesintopartsandusingthesumoftheparts incomparison

to the whole to answer the question. Contrastingly, responses were categorized as

proceduralifparticipantsmadementionofalgorithm,relyingonapreviouslymemorized

rule.

When analyzing responses, the majority of responses (12 out of 20) were

considered procedural. Common phrases were used in explanations, such as “multiply

the4andthe2andadd3andputthisanswerontopof4”or“takethedenominatorin

thefractionandmultiplyitbythewholenumber,thenaddthenumerator.Thisisthenew

numeratorand thedenominator remains the same.” It is clear to see that these

responses are strictly based on the algorithm rather than the concept. Only some

responses (six out of twenty) were conceptual and involved explanations of concrete

models to demonstrate the concept of breaking down two wholes into fourths. The

followingaretypicalexamplesofconceptualresponses:

 

I would draw out two rectangles undivided and a third showing the ¾, Iwouldthenshowhowthewholesneedtobedividedintofour pieces. Wecanthencountthenumberofpiecesintotal.(Abby)

 

IfIhave3piesandeachpieisdividedinto¼,eachpiewouldyield4pieces.Threepieswouldyield12piecestotal.Wecansolvethisbymultiplying3by4, the reciprocal of ¼ . When we divided the three pies by ¼ , we couldmultiplybythereciprocal4,becauseweareactuallycountingthenumberoffractions (1/4 pieces of pie) that make up the original number (3 pies).(Carol)

 

Twoparticipantsdidnotrespond,oneofwhoisendorsedinmathematicsandwas

laterinterviewed.Whenaskedwhysheleftthequestionblank,theparticipantsaidshe

“justcouldn’tfigurehowtoexplaintheanswer”(Nancy).

AccordingtoBall(1989),yearsofmemorization,focusonanswers,inattentionto

meanings, have ledway to algorithmicways of knowing and doingmathematics.

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ResponsestothepedagogysurveyconfirmBall’sviewpoint,asthemajorityofresponses

to thepedagogysurveywereprocedural innature. “Teachersmustknowrationales for

procedures,meaning for terms,andexplanation for concepts” (Ball,Phelps&Thames,

2008,p.8).Whatisclearfromtheresultsofthepedagogysurveyisthatthemajorityof

prospectiveteacherslackthisunderstanding.Mostparticipantsstrictlyreliedontheuse

ofanalgorithmintheirexplanations.

Question6:A studentdoesnotunderstandwhywhenyou subtractanegative

number,thenumberfromwhichyouaresubtractingincreases.Forexample:7‐

(‐2)=9. Howwouldyoushoworexplaintoastudentwhythisistrue?    

Out of the twenty responses, 6 out of 20 participants made mention of using

directiononanumberlinetodemonstratesubtractionasaddingtheoppositeormoving

towardstherightdirectiononthenumberline.Howeveronly,fourofthesesixresponses

explained how the number line could be used to demonstrate the concept. Sample

responsesarebelow.

 

Whensubtractingwithnegativenumbersonanumber line, thesubtractionsign means to go in the opposite direction of the directed distance thatfollows.Thus,tofind+7‐(‐2),startat+7,thendotheoppositeof(‐2)–thatis,go2unitstotherightendingat+9. (Clarissa)

 

Showthestudentsanumberlineandexplainhowthenegativesignchangesthedirectionofthehops.(Laura)

 

Youaretakinganumberfromtheothersideofzerowhichchangestherulesof subtraction.Anegativewillmakea largeranswer insubtractionbecauseofplacementonthenumberline(Abby)

 

Use a number line to show direction of subtraction when combined withnegativenumbers.(Carol)

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The remaining responses mentioned the use of a number line to explain the

conceptbutfailedtogointogreaterdetail.Oneadditionalparticipantsaidthatshewould

“showthestudent thatwhenyousubtractanegativenumber,youareactuallyaddinga

positivenumber”(Jackie),butfailedtomentionhowshewouldshowthis.

Fourparticipantsusedprocedures to explain the concept andphrases like “turn

the two minus symbols into a plus”. Five participants wrote mathematically incorrect

responses,threeoftheseresponsesconfusedsubtractinganegativenumberwithfinding

theproductof twonegativenumbers, usingphrases as “wheneveryouget a (‐) (‐) you

add up because two negatives will always equal a positive.” Five participants left the

questionblank.Examplesofproceduralresponsesarefoundbelow.

 

I would teach the slogan, “A negative minus a negative equals a positive.”(Jessica)

 

Iwoulddothetweakmethodoftryingtoexplainandshowthestudentthatwhen you have twomarks in that position just cross them both into plussignstoseethatthisproblembecomeanadditionproblem. (Maddy)

 

Theminusofaminusisaplusrule.(Lauren) 

  

Although this is a middle school concept, as discussed in the literature review,

current research recognizes the importance of elementary teachers learning the

progressionofelementarymathematicsthroughthemiddlegrades,thereforeelementary

teachersstillneedtohaveanunderstandingofoperatingwithintegers.

   

Question7:Imaginethatyouareworkingwithyourclassonmultiplyingtwo,twodigit

numbers.Yourstudents’solutionsaredisplayedtotheright.Whichofthesestudent’s

methodcanbeusedtomultiplyanytwowholenumbers?

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Outofthe20participants,some(5outof20)participantswereabletoidentifyall

three methods as being correct. Seven participants identified two out of the three

methodsasbeingcorrect,andsevenparticipantsidentifiedoneoutofthethreemethods

as being correct. Of the seven participants that chose two correct methods, five

participantschoseStudentBandStudentC,oneparticipantchoseStudentAandStudent

B, and one chose participant chose Student A and Student C. Of the seven participants

that identified one method as correct, four participants chose Student B and three

participantschoseStudentC.Oneparticipantdidnotanswerthequestion.Themajority

ofparticipants(15outof20)wereabletoidentifythestandardalgorithm(StudentB)as

correct,while(14outof20)wereabletoidentifythepartialproductsalgorithm

(Student C) as correct, and only some, (7 out of 20) were able to identify Student A’s

method as correct. Interestingly, oneparticipantwasunsure if the standard algorithm

worked for all whole numbers and three participants thought this method would not

workall.Thesethreeparticipantschosethepartialproductsmethodastheonlymethod

that could work. Six participants did not think that Student A’s method worked and

another six participants were unsure. Whereas, only two participants were unsure if

StudentC’smethodworksformultiplyinganytwowholenumbers,andthree

participantsdidnotthinkthismethodwouldworkatall. Participantshave likelybeen

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exposed to the alternate algorithmof using partial products formultiplying twowhole

numbers,andthereforewerefamiliarwiththisandrecognizedtheuseof thealgorithm

for Student C’s work. While, Student A’s method required participants to use their

knowledge of place value, number sense, and reasoning to verify the validity of this

approach and therefore struggledwith identifying it as correct. Research suggests that

teachers’ beliefs about mathematics are limited and dualistic in the sense of having a

right or wrong orientation with mostly single procedures to obtain a correct answer

(Thompson, 1992). Therefore, some teachers struggle with identifying more than one

approachoralgorithmtosolveanexercisebecausetheyseemathematicsasafixedsetof

proceduresthatmustbedonetoobtainacorrectanswer.AccordingtoBall(2008),being

abletoengageinmathematicalinnerdialogueandprovidemathematicallyvalidanswers

to alternate student solutions are a crucial foundation for determining what to do in

teaching mathematics. Since teachers face all types of solutions, they should have the

knowledge and ability to decide if the approach is mathematically correct and

differentiate between those methods that work in general for a similar problem and

those that do not. The findings of this study suggest that the majority of pre‐service

teachersarenotabletocorrectly identifyvalidalternateapproachestomultiplyingtwo

wholenumbers.

 Finding#2:Themajorityofparticipants feel that theyhavewhat it takes tobean

effectivemathematicsteacher.

The“Yourbeliefsaboutteachingmathematicssurvey”wasgiventoparticipantsto

assess their personal beliefs in their capabilities of being effective teachers of

mathematics.Thesurveycontainedthirteenquestionshavingtodowithvaryingaspects

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of personal mathematics teaching efficacy revealing participants’ beliefs on aspects of

their own pedagogical competencies. Below is a table listing each statement and

participants’ responserates.After lookingat thesurveys, it is clear that themajorityof

participants feel that they have what it takes to be an effective mathematics teacher.

However, theresultsofsomeof thesestatementswere foundtobecontradictory tothe

data gathered from the interviews and pedagogy surveys suggesting a discrepancy

between participants’ perceived competencies and the actual competencies that they

exhibit.Theseparticularstatementsarehighlightedinthissection.

Table5.ResponsestothePMTEStatementsQ1.Icontinuallyfindbetterwaystoteachmathematics.Responses Count %StronglyAgree 6 24.00%Agree 13 52.00%Undecided 5 20.00%Disagree 0 0%StronglyDisagree 0 0%(Didnotanswer) 1 4.00%TotalResponses 25  

Q2.EvenifItryveryhard,IdonotteachmathematicsaswellasIteachmostsubjects.Responses Count %StronglyAgree 1 4.00%Agree 5 20.00%Undecided 4 16.00%Disagree 12 48.00%StronglyDisagree 2 8.00%(Didnotanswer) 1 4.00%TotalResponses 25  

 

Q3.Iknowhowtoteachmathematicsconceptseffectively.Responses Count %StronglyAgree 1 4.00%Agree 10 40.00%Undecided 11 44.00%Disagree 2 8.00%

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StronglyDisagree 0 0%(Didnotanswer) 1 4.00%TotalResponses 25  

Q4.Iamnotveryeffectiveinmonitoringmathematicsactivities.Responses Count %StronglyAgree 0 0%Agree 0 0%Undecided 7 28.00%Disagree 15 60.00%StronglyDisagree 2 8.00%(Didnotanswer) 1 4.00%TotalResponses 25  

Q5.Igenerallyteachmathematicsconceptsineffectively.Responses Count %StronglyAgree 0 0%Agree 1 4.00%Undecided 3 12.00%Disagree 16 64.00%StronglyDisagree 4 16.00%(Didnotanswer) 1 4.00%TotalResponses 25  

Q6.Iunderstandmathematicsconcept wellenough tobeeffectiveinteachingelementarymathematics.Responses Count %StronglyAgree 8 32.00%Agree 11 44.00%Undecided 4 16.00%Disagree 1 4.00%StronglyDisagree 0 0%(Didnotanswer) 1 4.00%TotalResponses 25  

Q7.Ifinditdifficulttousemanipulativestoexplaintostudentswhymathematicsworks.Responses Count %StronglyAgree 0 0%Agree 0 0%Undecided 1 4.00%Disagree 13 52.00%StronglyDisagree 10 40.00%(Didnotanswer) 1 4.00%

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TotalResponses 25  

Q8.Iamtypicallyabletoanswerstudents'questionsaboutmathematics.Responses Count %StronglyAgree 7 28.00%Agree 14 56.00%Undecided 2 8.00%Disagree 1 4.00%StronglyDisagree 0 0%(Didnotanswer) 1 4.00%TotalResponses 25  

Q9.IthinkIdonothavethenecessarytoolstoteachmathematics.Responses Count %StronglyAgree 0 0%Agree 1 4.00%Undecided 1 4.00%Disagree 17 68.00%StronglyDisagree 5 20.00%(Didnotanswer) 1 4.00%TotalResponses 25  

Q10.Givenachoice,Iwouldnotwant my principal toevaluatemymathematicsteaching.Responses Count %StronglyAgree 0 0%Agree 5 20.00%Undecided 1 4.00%Disagree 15 60.00%StronglyDisagree 3 12.00%(Didnotanswer) 1 4.00%TotalResponses 25  

Q11.Whenastudenthasdifficulty understanding amathematicsconcept,I amusuallyatalossastohowtohelpthestudentunderstanditbetter.Responses Count %StronglyAgree 0 0%Agree 0 0%Undecided 5 20.00%Disagree 15 60.00%StronglyDisagree 4 16.00%(Didnotanswer) 1 4.00%TotalResponses 25  

Q12.Whenteachingmathematics,Iusuallywelcomestudents'questions.

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Responses Count %StronglyAgree 13 52.00%Agree 12 48.00%Undecided 0 0%Disagree 0 0%StronglyDisagree 0 0%(Didnotanswer) 0 0%TotalResponses 25  

Q.13 Idonotknowwhattodototurnstudentsontomathematics.Responses Count %StronglyAgree 0 0%Agree 1 4.00%Undecided 8 32.00%Disagree 14 56.00%StronglyDisagree 2 8.00%(Didnotanswer) 0 0%TotalResponses 25  

 

  

Teachers play a crucial role in creatingmathematics environments that provide

studentswithconcrete representations thatenhanceandbuildon theirknowledgeand

understandingofmathematics. AccordingtoMoyer(2001),“manipulativematerialsare

objects designed to represent explicitly and concretely mathematical ideas that are

abstract”(p.176). Inclassroomsoftoday,withtheCommonCore’semphasisontheuse

of concretemodels to conceptually understand the abstractwrittenmethods that

studentsusetosolveproblems,manipulativeshavebecomeincreasinglyimportant.

According to the mathematics efficacy survey, the overwhelming majority of

participants(23outof25)“stronglydisagree”or“disagree”withthestatement“Ifindit

difficulttousemanipulativestoexplaintostudentswhymathematicsworks”,yetthisdid

nottranslateintotheirexplanationsinthemathematicspedagogysurvey.Whenaskedto

explainhowtheywouldteachparticularmathematicsconcepts,onlysomeparticipants

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actually made use of manipulatives in their explanations on a consistent basis, while

most participants used rules, procedures and aspects related to an algorithm in their

explanationsofelementarymathematicsconcepts.Forexample,thefirstquestionofthe

mathematics pedagogy survey asked participants the following question: “A student

makes the following mistake: 734‐481 = 353. How do you respond?”. Out of the 20

participants that filled out the pedagogy survey, only six participantsmademention of

using some sort of manipulative or drawing to reinforce the concept of place value.

However, one of these six responses failed tomention how themanipulativewould be

used to reinforce the concept.Most other participants used phrases like “cross out the

number andwrite one less” or “regroup and borrow” or “have the student check their

workagain”.

The use of manipulatives improved for the second question (“Imagine you are

introducingadditionwithregroupingtoyourstudents,howwouldyoushowwhy27+95

=122?”),with12outof20participantsmakingmentionofbase‐tenblocksorsomeother

type ofmanipulative to showplace value, still leaving eight participantswhomdidnot

make any mention of manipulatives. Five of these eight participants responded with

phrases like “rewrite the problem vertically”, “regroup and carry”, “circle the extra

number”,or“puttheoneontopofthetwo”.Again,usingphraseslimitedtoaspectsofthe

algorithm rather than using concrete models to demonstrate place value and

conceptually understand the written algorithm. It should be noted that the use of

manipulatives, whether concrete models or drawings, are an emphasis of the CCSSM

whenteachingplacevalue.

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The contradiction between participants’ perceived competency on the use of

manipulatives and their actual use of manipulatives in teaching may be a result of

participants’ “surface beliefs” versus their “deep beliefs”. According to Kaplan (1991),

teachers have deep beliefs and surface beliefs. Surface beliefs can be described as the

beliefs that one thinks one should hold, related to “superficial” practices, rather than

what they actually believe, or their deep beliefs. Therefore participants may have

indicatedthattheydonotfinddifficultyinusingmanipulativesbecausethatisthebelief

theyfeeltheyshouldhaveratherthanwhattheyactuallybelieve.Raymond(1997)found

inherstudywithnewelementaryteachersthatdeeplyheld,traditionalbeliefsaboutthe

nature of mathematics may perpetuate mathematics teaching that is more traditional,

even when teachers hold nontraditional beliefs about mathematics pedagogy. She

suggests that, “early and continued reflection aboutmathematics beliefs and practices,

beginninginteacherpreparation,maybethekeytoimprovingthequalityof

mathematicsinstructionandminimizinginconsistencybetweenbeliefsandpractice”(p.

574). 

The majority of participants, (19 out of 25), strongly agree or agree that they

“continually findbetterways to teachmathematics”. However, during the interviews it

wasfoundthatonlyoneparticipantactuallyengagedinprofessionalactivitiestoenhance

their formal learning inside the classroom. Therefore, the interviews revealed that the

majorityofparticipantsdonotparticipateinprofessionalactivitiestocomplementtheir

formalpreparation.However,thismaychangewhenparticipantsenterthefieldandgain

professionaldevelopmentopportunitiesthroughtheiremployer.Thiscontradictionmay

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beduetoparticipants’inclinationtoanswerthesurveywiththeirsurfacebeliefs,rather

thantheiractualbeliefs.

Therewasahighnumberofparticipants(11outof25)thatremained

“undecided” to the statement “I know how to teachmathematics concepts effectively”,

while10outof25participantsagreedwiththisstatement.Yet, themajority(19outof

25) participants feel that they understand mathematics concepts well enough to be

effective in teaching elementary mathematics. Therefore, although some teachers are

unsure if they can teach mathematics concepts effectively, they still feel that they

understand mathematics well enough to teach mathematics and be effective. To gain

clarity on this inconsistency the researcher interviewed a participant that chose these

tworesponses.Theparticipantstated:

Oh I saidundecidedbecause I think thatmath isoneof those thingsthat is a hit or a miss. And it is like did the children pass theassessment, whatever the assessment was, how did they do on it?(pauses) So I don’t know, I don’t know if I am effective unless themajorityoftheclasspassedtheassessment.(Samantha)

 Basedon this response,perhapssomeparticipantsdidnothave theopportunity to see

concretenumbersthatdeemthemaseffectiveteachersofmathematics,eventhoughthey

feel that they understand mathematics well enough to be an effective mathematics

teacher.Formostparticipants,theonlyexperiencetheyhaveteachingmathematicsona

regularbasisisduringtheirstudentteaching,andthismaynotbeenoughtimeforthem

to assesswhether they teachmathematics conceptseffectively. Theymaynothave the

opportunitiestogivesummativeandformativeassessmentswithchildrenandusethese

toreflectonthemselvesasteachersofmathematics.

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Researchers hypothesize that the discrepancy among teacher candidates

perceivedabilitiestoteachmathematicseffectivelyandwhattheyexhibitintheir

teaching is due to their lack of experience in teaching. Asmentioned in the literature

review, when pre‐service teachers spend more time in the classroom, and have more

opportunities to teach mathematics on a daily basis, they develop a sense of their

competence.Field experiences also provide teacher candidateswith anopportunity to

develop perceptions of their effectiveness of their teaching for their students

(Charalambos, Charalambous, Phillippou, & Kyriakides, 2007). Perhaps these findings

wouldbedifferentafterparticipantsfinishtheirfirstyearofteaching.

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ChapterV 

SUMMARY,CONCLUSIONSANDRECOMMENDATIONS 

Thepurposeofthisstudywastodeterminehowwelltraditionalelementary

preparationprogramsandalternativeelementarycertificationprogramsareperceived

byaspiringteachers,topreparethemtoteachmathematicstothestandardsrequired

today.Thestudywasguidedbythefollowingresearchquestions:

(1)Howdoparticipantsdescribe theirmathematicalexperiencesprior to

enteringtheirteacherpreparationprograms?

(2) How do participants describe the mathematics preparation they

receive in their respective preparation programs and what types of

mathematics preparation, if any, do theywish they had before beginning

studentteaching?

(3) What kinds of additional professional activities, beyond the official

preparation received in their teacher education program, do participants

engage in outside the classroom to support and enhance their formal

preparationformathematicsteaching?

(4)What are participants’ perceived pedagogical competencies in 

mathematics and how do these differ across various preparation

programs?

   

Summary 

This study is of qualitative design with multiple data collection methods.

Participantswere recruited from four different teacher preparation programs and one

104 

alternativerouteprogramtofilloutatwo‐partsurvey.Oneteacherpreparationprogram

is a top ranking program in elementary mathematics education and is located in the

South,twoprogramsareaveragerankingandlocatedintheMidwest,oneundergraduate

program is a low ranking program located in the Northeast and the alternative route

certificationprogramislocatedinNewJersey. Asubsetoftheseparticipantsparticipated

inafollowup,semi‐structuredinterview.

The principal investigator designed a qualitative, online survey and a semi‐

structured interviewprotocol thatwasreviewedbyprofessionalcolleagues.Thesurvey

consistedofthreeparts,abackgroundsurvey,amathematicsefficacysurveytitled“Your

Beliefs about TeachingMathematics”, and a “Mathematics Pedagogy Survey”. The “Your

Beliefs about Teaching Mathematics” survey used a Likert‐scale to gather data on

participants’personalmathematicsteachingefficacywhilethepedagogysurveywas

used to gatherdata onhowparticipantswould teachparticularmathematical concepts

that are now require a conceptual understanding among students due to the Common

Core State Standards for Mathematics. The semi‐structured interviews gathered more

information on teacher candidate’s teacher preparationprograms and also allowed the

participantstoexpandontheirresponsestothesurveys.

Conclusions ResearchQuestion1: Howdoparticipantsdescribetheirmathematicalexperiences

priortoenteringtheirteacherpreparationprogram?

Thisstudyexploredthediversemathematicalexperiencesthatteachercandidates

enter their teacher preparation programs with, including their attitudes towards the

subject,theirperceptionsofthemselvesasstudents,andtheircourseworkpriorto

105 

entering their teacher education program. The answers to research question onewere

based on the background survey. The findings suggest that in selectiveprograms,most

teacher candidates have been exposed to higher levels of mathematics, whether it is

calculusorstatistics.Yet,althoughstudentshavetakensuchclassesasAPCalculusorAP

Statistics, their knowledge of elementary school mathematics content and pedagogy

cannotbe judgedbytheamountofmathematicstheyhavetakenpriortoenteringtheir

teacherpreparationprogram.Thepedagogysurveyunveiledteachercandidates’

difficultywith conceptualizing fractions, evenafter taking themathematics contentand

methods courseworkprovidedby their teacherpreparationprogram.This is consistent

with research conducted by mathematics education researchers and mathematicians,

whom agree that it is not enough for teachers to depend on their past experiences as

learnersofmathematicsastheirpreparationtoteachmathematics(CBMS,2012).

The findings also suggest that evenwhenparticipants rate themselves asdecent

mathematics students (“A” or “B” students), teacher candidates are not enthusiastic

towardsmathematicsorthemselvesaslearnersofmathematics.Whenparticipantswere

interviewed about their perceptions of themselves as learners of mathematics,

participantsusedphraseslike“Itwasokay,”“Ididn’tloveit,”or“Iwasfrustrated.”These

findings are consistent with the literature suggesting that prospective teachers are

persuadedbytheirexperiencesinelementaryandsecondaryschool,seeingmathematics

as uninteresting, a subject where they must follow procedures and memorized rules

(Ball,1989).

ResearchQuestion#2:Howdoparticipantsdescribe themathematicspreparation

theyreceive in theirrespectivepreparationprogramsandwhat typesof

106 

mathematicspreparation, if any, do theywish theyhadbefore beginning student

teaching?

The purpose of research question 2was to determine the types ofmathematics

coursework, fieldexperiences,andexposure to theCCSSMthat teachercandidateshave

in their teacherpreparationprogramandalso explorewithparticipants, the additional

typesofpreparationinmathematicseducationtheywouldfindbeneficial.Thedatafrom

the background survey and interviews were used to answer this research question.

Findingssuggestthattherearenopoliciesinplacethatrequireconsistencyacross

teacher education programs in regards to the amount ofmathematics content courses

required in teacherpreparationprograms.Furthermore, institutions themselvesvary in

their requirements resulting in pre‐service elementary teachers gaining different

experiencesintheirmathematicseducation.ThereforealthoughtheNCTQusedthe

syllabiandvariousotherdocuments torankelementary teachereducationprograms in

their teacher preparation in mathematics, it should not be assumed that all students

graduating fromthesameprogramhave thesameexperiences.Forexample,Bethany,a

participant fromthe four‐star institution tookonecourseon themathematics taught in

elementary school, while Clarissa was required to take two courses. This particular

program’spublishedprogramplansuggeststhatprospectiveteachersshouldtakethree

coursesinmathematicscontent.Thesevariationsincourseworkselectionwerealso

foundinbothtwo‐starinstitutionsaswell.

InregardstotheCCSSM,itwasfoundthatthedegreetowhichteachereducation

programs integrate theCCSSM in their curriculumvaries fromoneprogram to another

andwithin the samepreparationprogram.One reason for thismaybedue to teacher

107 

educators themselves not deeming the standards to be of importance, or seeing the

standards as a temporary curriculum shift that will swing in another direction. As a

result teachereducatorsmaynot spend timeredesigning their coursework to integrate

the CCSSM. Although the standards are meant for K‐12 mathematics students, the

standards apply to everyone engaging in mathematics, including elementary teachers

(CBMS,2012).

This study also found differences in the amount of time spent teaching

mathematics.Thesurveysfoundthatwiththeexceptionofalternativeroutecertification

programs,allteachereducationprogramshadsomesortoffieldexperiencerequirement

forpre‐serviceteachers.Yetduring the followupinterviews, it wasfound thatthe

amount of time that prospective teachers spent teachingmathematics varied from one

participant to another. For example, in the four‐star program participants within the

sameprogramhadvaryingresponsestotheirfieldexperiences.Oneparticipantindicated

having ampleopportunity to observe and teachmathematicsprior to student teaching,

while the other noted that student teachingwas the first experience she had teaching

mathematicsanditwasthelastsubjecttobeaddedonandthefirstsubjecttocomeoff

herteachingload.Thus,thereislittleconsistencyintheamountofmathematicsteaching

that teacher candidates obtain from their field experience, before student teaching and

during their student teaching experience. Therefore elementary teachersmay not have

adequate experiences teaching mathematics prior to entering their first teaching

position.

Through the mathematics efficacy survey, this study found that in general, the

majorityofparticipantsfeelthattheyarepreparedtostepintotheclassroomandteach

108 

mathematics. However, thereareareaswhere theywould likemore support, including

areas of introducing and teaching early childhood concepts, the general scope and

sequenceofelementarymathematics,andtacticstointegratetestpreparationintodaily

mathematicsactivities.Whetheritisduetoalackofresourcesorvaryingpriorities,some

teacher preparation programshave limited time to prepare teacherswithmathematics

preparation, in both content and pedagogy. As found in the background survey,

preparation programsmay only require one or two content courses and onemethods

course,thusthecontentofcoursesmustbecarefullychosentomakeworthwhile

learningexperiencesforprospectiveteachers.

ResearchQuestion#3:Whatkinds of additionalprofessionalactivities,beyond the

official preparation received in their teacher education program, do participants

engage in outside the classroom to supportand enhance their formalpreparation

formathematicsteaching?

This study found that the overwhelmingmajority of pre‐service teachers donot

engageinanypreparationinmathematicsoutsideoftheirformalpreparationtoenhance

their practice. Contrastingly, alternative route certification teachers have ample

opportunitiestoattendprofessionaldevelopment.Sincetheyarealreadyoutinthefield,

these teachersareprovidedwithprofessionaldevelopmentopportunities through their

schooldistrict.The interviewsrevealedthe limitedopportunitiesforteachercandidates

toaccessprofessionaldevelopmentoutsideoftheirinstitution.Afewparticipantsmade

mention of tutoring children as an outside source of supporting their teaching of

mathematics.Therefore,ifpre‐serviceteacherswouldliketoattendadditionalactivities,

theymustbemotivatedtopursuetheseontheirown.

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ResearchQuestion#4:Whatareparticipants’perceivedpedagogicalcompetenciesin

mathematicsandhowdothesedifferacrossvariouspreparationprograms?

Fromthepedagogysurvey, it isclear that ingeneral,prospective teachers

tendtoteachprocedurallyratherthanconceptually.Responses toeachof thequestions

from the pedagogy survey were categorized as conceptual or procedural, and for any

givenquestion,themajorityofresponsesfellintotheproceduralcategory.Thisfindingis

consistent with Licwinko (2014), suggesting that teachers tend to have a procedural

understanding, rather than a conceptual understanding of fractions, subtraction, area

andperimeterandaccording toMa (1999), theirunderstanding is limited toaspects to

the algorithm.Ma found that this also extended into the classroomas teachers’ limited

understanding of mathematics stifled their students’ understanding of the various

concepts.

In particular, it was clear that many elementary teacher candidates

struggle with conceptualizing fractions and creating word problems representing

divisionwith fractions. Participantsmade note that they could do the problemusing a

procedure but could not show why when you divide fractions you multiply by the

reciprocal,orshowhowtochangeamixednumbertoanimproperfractionconceptually.

Whencreatinga storyproblem that representeddividingbyahalf,manyconfused this

conceptwithdividingbytwo.Thismisconception isconsistentwiththe findingsofBall

(1989),wheresheaskedparticipantstocreateastorytorepresenttheexercise,2¼÷½,

andmanyparticipants also confuseddividingbyone‐half asdivisionby two, and some

couldnot create a storyproblemat all. Yetwhen asked to complete the initial taskof

completing the exercise, participants remembered theprocedure ofmultiplying by the

110 

reciprocalofthesecondfraction.Twenty‐fiveyearslater,itcanstillbeconcludedthatthe

majority of prospective teachers lack a conceptual understanding of operating with

fractionsandasaresultwillteachoperatingwithfractionsusingtraditional,algorithmic

methods. Thedifferencebetween todayand twenty‐fiveyears ago is that today,upper

primarystudentshavetobeabletowritetheseincontextstoryproblems,sincethisisa

standardintheCCSSM,whereastwenty‐fiveyearsago,teacherscouldgetbywithsolely

teachingthealgorithm.

In general, from the mathematics efficacy survey it was found that the

majority of participants (76%) feel confident in their understanding of mathematics

concepts to teach mathematics effectively but at the same time the majority of

participants (52%) either are unsure or do not feel that they know how to teach

mathematics concepts effectively. These two statements are similar, yet the first one

stresses an understanding of mathematics concepts while the second statement

emphasizesteachingmathematicsconcepts.Teachercandidatesperceivethattheyhavea

grasp ofmathematics content to be effective but they are not as confident inwhether

their teaching of mathematics will be effective. This finding is consistent with studies

conductedbyBates,Kim,andLatham(2011)andCharalambos,Charalambous,

Phillippou, & Kyriakides (2007), that found that pre‐service teachers with high

mathematicsself‐efficacyknowtheycanteachmathematics,butdonotknowiftheywill

be effective for their students. Thismay be due to the little experience that they have

teachingmathematics.Asnotedearlier, theamountof timespent teachingmathematics

during field experiences and student teaching is very limited in some cases, so

prospective teachersmay not feel as confident in their practice as they do with the

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educational theories andmathematics content they have learned in theirmathematics

educationcoursework.

Although teachers generally feel that they have the ability to be effective

teachersofmathematics,thereisadiscrepancybetweentheirperceivedcompetenciesin

being teachers of mathematics and their exhibited competencies. For example, the

overwhelmingmajority(92%)ofparticipantsperceivethattheydonotfinditdifficultto

usemanipulatives and the overwhelmingmajority (88%) feel that they have the tools

necessary to be teachers of mathematics. Yet the pedagogy survey revealed that the

majority of participants have a procedural understanding and approach to teaching

subtraction, fractions, and alternative algorithms and few participants used

manipulatives,models,ordiagramsintheirexplanationsoftheseconcepts.Inthe

survey, the overwhelming majority (76%) of participants also indicated that they

continually find better ways to teach mathematics, yet when participants were

interviewed,onlyoneoutof thesevenparticipantsactuallyparticipated inprofessional

developmentopportunitiestobettertheirteaching.Thesefindingsareanalogoustothe

researchconductedbyBates,Kim,&Latham(2011),inwhichtheyfoundthatpre‐service

teacherswhoscoredlowonabasicmathematicsskillstestfeltjustasconfidentintheir

abilities to teachmathematics as thosewho scored high on the same assessment. This

highlights the discrepancy between pre‐service teachers’ perceptions of their ability to

beeffective teachersofmathematicsandtheiractualcompetencies.Onereason for this

may be due to their lack of teaching experience. Charlambos et al., (2011) note that

teaching mathematics on nearly a daily basis for three months helped prospective

teachersgainasenseoftheircompetence.

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Discussion 

At the conclusion of this study,what became apparent is the danger in treating

conceptual knowledge and procedural knowledge as a rigid dichotomy. While coding

responses from the pedagogy surveys, it became clear that the two constructs are not

mutuallyexclusive,butratherthereisacontinuumofunderstandingfromproceduralto

conceptual with an integration of both between. Take for example, the following

responses to question 2 in the pedagogy surveywhich gave participants the following

scenario: “Imagineyouare introducingadditionwith regrouping toyour students, how

wouldyoushowstudentswhy27+95=122?”

Whenregroupingcircleyourextranumbersoyoudon’t forget itstheretoadd.(Lauren)

 I amwriting theproblemvertically onto theboard. Iwouldmention thatweknowitis12butwecan’tjustwrite12underneathbecause2+9isnot1.Whenweadd#togetherthathavemultipledigitswehavetobepreparedtocarryoneofthedigitsovertotheleft.Wecanwritedownthe2underneaththe7andthe5,buthavetocarrytheoneandadditafterwecalculatethe#’sinthetensplace.(Jessica)

 Iwouldstartbyrewritingtheproblemvertically.NextIwillexplainhowtoregroupbycarrying(1)tothetensplaceandthenaddthethreenumberstogethertogetthefinalanswer.(Lily)

 I would explain to the students that 7 + 5 = 12, but since the ones spotcannothold12,wewrite the2 (for2ones)andcarry the1over into thetens spot to representourone set of ten.Then Iwould showstudents toadd1+2+9=12. Iwould repeat thisandshowstudents towrite the2down again and carry the one into the hundreds spot. Then add again.(Seana)

 Usemanipulatives or drawingswhile reinforcing place value. Draw out 2tensand7ones,thendrawout9tensand5ones.Circle10oftheonesandshow how it can be combined into a ten. Circle 10 of the tens and showhow it can be combined into a single hundred. Count out the hundreds,tens,andonestoget122.Nexttothedrawingstackthenumbersverticallywhilecarrying1tenand1hundred.Showstudentsthatbothwayshelpusarriveatthesameanswer.(Carol)

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Before looking at these responses, it should be mentioned that although a participant

may choose to teach a concept procedurally, it cannot be assumed that the teacher

themselves has a procedural understanding. How one chooses to teach a particular

concept is not necessarily indicative of one’s understanding of the concept. These

responsesdemonstratearangeofunderstanding,fromproceduraltoconceptual.

Lauren’sresponse lacksanymentionofplacevalueandsheonlymakesmention

ofthealgorithmicprocedurewithhercommentof“circleyourextranumbersoyoudon’t

forget it is there to add”. Jessica’s response shows someevidence that sheunderstands

theconcept:althoughshedoesn’texplicitlyuseplacevalue throughoutherexplanation,

she does make connections between the algorithm and the concept. She notes that

because2+9isnot1,itdoesn’tmakesensetoput12underneath,andgoesontomake

thisthereasonforcarryingthe1,butdoesn’tmakementionofthe1representingaten.

Lilysaysshewillexplainhowtoregroupby carrying theone to the tensplace,making

the reader infer that sheknows the1 is a ten, butwhether shewill explain this toher

students is still in question.Thus there are somehints of teaching the concept here as

well, but as written her explanation still does not explain the concept behind the

procedure.

Seana’sresponseevidencesuseofplacevaluetoexplainthealgorithm,addingthe

ones togetherandcarrying the1because it is “asetof ten”,andadding the tensplaces

togetherandcarryingtheonetothehundredsplace.Thusthisexplanationexplainsthe

conceptusingplacevaluemore than theotherresponses,andmaybeconsideredmore

conceptual,butherexplanationdoesnotfullyexplaintheprocedure,aswhenshegetsto

thetensplace,andadds1+2+9toget12,shedoesnotmakementionofaddingtens,to

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get 12 tens, and regrouping 10 tens for 1 hundred. Carol’s explanation makes use of

manipulatives to reinforce place value and explain the concept behind the algorithm.

Fromher explanation it is clear that she is teaching for a conceptual understanding as

eachstepexplainstheconceptbehindthealgorithm.Shemakesmentionofregrouping

10onesforaten,andregrouping10tensforahundred.Carolalsousesmanipulativesto

maketheabstractprocedureinthealgorithmmoreconcreteforchildren.

Proceduralknowledgemaybedefinedasfamiliaritywith“rulesorproceduresfor

solvingmathematical problems” or “chains of prescriptions formanipulating symbols”

while conceptual knowledge can be described as a connected web of knowledge,

“knowledge that is rich in relationships” and that cannot exist as “an isolated piece of

information”(Hiebert&Lefevre,1986,p.7‐8,p.3‐4).

Star(2005)arguesthatconceptualknowledgeshouldnotbedefinedas

knowledge of concept or principals but rather in terms of the quality of a person’s

knowledge of concepts. In his view, “the term concept does not imply connected

knowledge”as“connectionsinaconceptmaybeonlylimitedandsuperficial,ortheymay

beextensiveanddeep”(Star,2005,p.407).Thesameappliesforproceduralknowledge,

asonemayhavesuperficialproceduralknowledgeordeepproceduralknowledge.“Deep

procedural knowledgewould be knowledge of the procedures that are associatedwith

comprehension, flexibility, and critical judgment that is distinct from knowledge of

concepts”(Star,2005,p.408).Separatingthetwotypesofknowledge,allowsonetolook

atproceduralknowledgeaspotentiallydeep.

Baroody, Feil, and Johnson (2007) state that most mathematics education

researchersattempttodistinguishbetweenknowledgetypeandqualitybyviewing

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proceduralandconceptualknowledgeasacontinuumfromsparselytorichlyconnected

knowledge. These researchers challenge Star and argue that although conceptual

knowledge is not necessary for a superficial procedural knowledge, it is unclear how

substantially deep comprehension of a procedure can exist without understanding the

conceptualbasis for eachof its steps. In theory, separating conceptual andprocedural

knowledgemakes sense, but psychologically speaking, the two cannot be separated. In

theirview,“itmakespsychologicalandpedagogicalsensetoviewmeaningfulknowledge

ofmathematicalproceduresandconceptsasintricatelyandnecessarilyinterrelated,not

asdistinctcategoriesofmathematics” (Baroodyetal.,2007,p.127).

Theresponsestothepedagogysurveyprovideadditionalevidencethat

procedural and conceptual knowledge cannot be viewed in isolation of one another. A

teacher may choose to teach concepts using a procedure, but when the procedure is

taught for a deep understanding, the teacher relates the procedures to the concept, as

seenisCarol’sresponse.Itshouldbenotedthatteachingproceduresshouldnotbe

looked down upon, as procedures can be the most efficient way to solve an exercise,

however students who rely on procedures too heavily may lack an understanding of

mathematical concepts,and thereforecannotengage in themathematicalpractices that

theCCSSMencourage.Alistofthesepracticescanbefoundinthefollowingsection.

   

Recommendations 

Teachereducatorsshouldreflectontheircoursesandinsomecasesrevamptheir

curriculum to reflect the changes resulting from the CCSSM. Content courses and

methods courses provide an opportunity for teacher educators to change teacher

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candidates’ attitudes towards mathematics and turn them on to the subject. Classes

should be taught in the context of the CommonCore State Standards forMathematical

Practice. With limited time and resources, it is unreasonable to think that teacher

preparationprogramscanteachallofthemathematicstaughtinelementaryschools,but

thegoalofteachereducatorsshouldbetoadvanceindependent,reflective learnersthat

canaddressnewcontentandnewpedagogiesastheyarepresented,andmakesenseof

these new ideas on their own (Moss, Browning,Moss, Thanheiser&Watanabe, 2012).

Prospectiveteachersshoulddevelopthetypesofmathematicalexpertisedescribedinthe

CommonCore State Standards forMathematics Practice as they learn themathematics

content that they soon will teach (CBMS, 2012). This includes persevering in solving

problems,theabilitytoreasonabstractlyandquantitatively,constructingargumentsand

critiquingthereasoningofothers,andrecognizingpatternsandmakinguseofstructure

inmathematics (NGA,2010). Inmethods classes itwouldbeworthwhile to look at the

standardsandseetheprogressionsof thedomainsfromonegradetothenextandgive

teachercandidatesasenseofthescopeandsequenceofelementaryschoolmathematics.

Teacher educators should pay close attention when teaching students about fractions.

Theymay considermaking it a requirement for students to create a unit plan dealing

with fractionstodisplaytheirunderstandingofhowtheywouldteachvariousconcepts

dealing with fractions, forcing teacher candidates to grapple and understand the

conceptsthemselves.

With the little consistency in thenumberofmathematicseducationcourses that

pre‐serviceteachersarerequiredtotake,policymakersmayconsiderestablishingsome

sortofstandardinordertounifythequalityofteachereducationprograms.The

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autonomousstateofteacherpreparationprogramshasledtotheinconsistent

preparation among different programs, and as this study found, sometimes within a

program.Both the zero‐star and the four‐star institutionswere twoNCATEaccredited

institutions,yettheirteacherpreparationinmathematicseducationwasvastlydifferent.

Accreditation agencies may consider standardized, stronger requirements to obtain

accreditation.

To build teachers’ confidence in their effectiveness in teaching mathematics,

teacher preparation programs should provide high quality field experiences for

prospective teachers, partnering them with model teachers that allow them the

opportunity to teachmathematics using the pedagogies that they are learning in their

coursework,bridgingcontentandtheoryfromtheircourseworktopracticalapplications.

Cooperating teachers and prospective teachers should be given guidelines for field

experiences,includingstudentteaching,thatincorporatetimespentobserving,reflecting

on, and teachingmathematics. School districts should not assume that novice teachers

have adequate exposure to teaching mathematics. They should strongly consider

providingnewteacherswithprofessionaldevelopmentopportunities inmathematics to

supportthemintheirteaching.

Teacher preparation programs may consider creating a development of

mathematical thinking course, if theyhavenot already, so that teacher candidateshave

the opportunity to learn how young children think about mathematics, in an effort to

enlighten them about the differentmisconceptions that students have and the abstract

concepts thatmay be simple to an adult but difficult for young children to grasp. If a

separatecourseisnotpossible,thenperhapsitwouldbeworthwhiletoimbedthistopic

118 

intomethods coursework through readingsandvideosofyoungchildren learning

mathematicsthatstudentscanaccessonline.

Preparation programs may consider advertising professional development

opportunities for students for local conferences and organize transportation to these

events. Professors may also consider educating teacher candidates on professional

organizations like NCTM and encourage them to join and take advantage of their

discountedstudentrates.

SuggestionsforFutureResearch 

Futureresearchersshouldbeawareofthelimitationsofthisstudy.Thelargestof

these is the sample size. Since response rates from teacher preparation programwere

rather low,only fiveteacherpreparationprogramswereused inthisstudy,and inmost

casestheresponserateinagiventeacherpreparationprogramwasalsolow,resultingin

a limited amount of participants from each teacher preparation program. Some of the

participating elementary teacher preparation programs were rather small, with only a

fewstudents in themajor. Inaddition,aconveniencesamplehad tobeused toconduct

the interviews since the response rates were so low. This made it difficult to make

comparisons across different teacher preparation programs. One might consider

conducting a similar study but on a larger scale, recruiting larger elementary teacher

preparation programs and obtaining a larger sample from each, particularly for the

pedagogysurveysand interviews. Itwouldbe interesting to seehow the responsesare

categorizedwithalargersamplesize.Forexample,dothemajorityofparticipantsfrom

the four‐star institutionalwaysanswer thequestionsconceptually?Thiswasdifficult to

ascertainwithonlyfourparticipantsbutwithalargersampleitwouldbeeasiertosee.

119 

Similarly in the interviews, the researcherwasnotable to interviewaparticipant from

thezero‐starteacherpreparationprogram.Howwouldtheydescribetheirexperiencein

their teacher preparation program? The pedagogy surveys could also be designed

differentlyandaskparticipantshowtheywouldsolveaprobleminmultipleways.This

would provide a second explanation and more evidence to determine whether

participants have a procedural or conceptual understanding of the mathematics they

teach.Withmore timeandresources thesequestionscouldbeansweredwitha similar

study.

Asecondlimitationisthatthisstudystrictlyfocusedontherequiredmathematics

coursework that prospective teachers and alternative route teachers had to take, it did

not look into additional mathematics education coursework that elementary teacher

preparation programsmay ormay not have. Itwas clear that as the ratings decreased

among the elementary teacher preparation programs, so did the requiredmathematics

coursework. This study, however, did not look into whether additional electives in

mathematicseducationwereofferedintheseelementaryteacherpreparationprograms.

One may be interested in studying the additional courses that are offered in various

rankedprogramsandincorporateresearchoncourseselection.Forexample,ifadditional

coursework is provided as electives,will students take advantage of these courses and

howdoesmathematicsefficacytieintotheirdecision?

The purpose of this study was to determine how well traditional elementary

preparation programs and alternative elementary certification programs are perceived

by aspiring teachers, to prepare them to teachmathematics to the standards required

today.Thefindingsfromthisstudycanbecategorizedintothreemajorareas:thelackof

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standardization among andwithinmathematics education coursework requirements in

elementary teacher preparation programs, a procedural approach to teaching

mathematics for themajorityofpre‐serviceelementary teachers, and the incongruence

between participants’ perceived confidence in their ability to teach mathematics and

their actual approach to being teachers ofmathematics. The findings of this study are

consistentwiththecurrentresearchintheseareas.

Since the requirements for teacher education programs vary state to state, the

required coursework inmathematics education also varies fromone teacher education

programtothenext(NationalCouncilofTeacherQuality,2008;Kilpatrick,2001;Center

for Research in Mathematics and Science Education, 2010). Although there is little

agreement in the amount of mathematics education coursework necessary to prepare

elementary teachers, the CBMS recommends 12 semester hours of preparation in the

mathematics taught at the elementary level. In this study, the minimum amount of

content coursework inundergraduate teacherpreparationprogramsranged fromthree

tosixcredithours.Thesamewasfoundformethodscoursework.Althoughaccreditation

agencies exist to hold programs accountable, this study found that there was a wide

variationintheamountofrequiredmathematicseducationcourseworkintwo

accredited institutionsparticipating in thisstudy(twocontentandtwomethodsversus

onecontentandonemethodcourse),neitheronefollowingtherecommendationsofthe

CBMS.

Numerous studies have found that elementary teachers have a procedural

understanding of many mathematics topics taught at the elementary level (Licwinko,

2014;Ma,1999;Ball1989).Thereforeitisimportanttobuildprospectiveteachers’MKT

121 

through a carefully structured methods course, one that reflects the CCSSM and the

standards for mathematical practice. With the exception of the alternative route

certification program, all participating preparation programs had at least onemethods

courseduringtheircourseofstudy,yetthemajorityofparticipantsstillusedprocedural

methodstoteachconceptsoffractionsandsubtraction.

Highquality fieldexperiences shouldalsobe imbedded into teacherpreparation

programs (CBMS, 2012; Perry, 2011; Science andMathematicsTeacher Imperative/The

LeadershipCollaborativeWorkingGroup(SMTI/TLC),2011).Fieldexperienceshave

been found to improve mathematics teaching outcome efficacy beliefs by giving

prospective teachers an opportunity to realize their competencies and develop

perceptions on how effective their teaching of mathematics is for their students

(Charlambosetal.,2008). This study foundadiscrepancybetween teachers’perceived

and exhibited competencies in being teachers of mathematics. Field experiences help

teachersdevelopasenseoftheircompetencyandcanminimizethisinconsistency.

The quality of teacher preparationprogramshas been in the spotlight in recent

years, especially with the changing standards in classrooms of today. It is clear that

preparationprogramshavetoevolvewiththechangingtimes,sothatnoviceelementary

teachersareready tocopewiththenewdemandsofmathematicalknowledgerequired

at thecurrent time.The findingsfromthisstudyhighlight theneedfor:standardization

inmathematics educationwithin and across elementary teacherpreparationprograms,

revampingofcontentandmethodscoursestoreflectthechangesoftheCCSSM,andhigh

qualityfieldexperiencesthatallowstudentstospendmoretimeteachingmathematics.

122 

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130 

APPENDIXASurveyto

Participants

 

131 

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133 

Mathematics Pedagogy Survey  This is NOT a test. These are typical mathematics questions that people struggle with. I am interested in your thoughts about how best to teach these topics to students. Please answer each question to the best of your knowledge. You may use drawings in your explanations.  1.) A student makes the following mistake: 734 – 481 = 353. How do you respond?

          

 2) Imagine you are introducing addition with regrouping to your students, how would you

show students why 27 + 95 = 122?  

            3.) Create a story problem to illustrate that 4¼ ÷ ½ = 8 ½ .

              4.) How would you explain why when you divide by a fraction you multiply by the reciprocal of the fraction? You can use the following division exercise to assist you in your explanation.

3÷ ଵ= 12

ସ       

5.) When introducing students how to rename mixed numbers into improper fractions, how would you teach a child to rename 2 ¾ into an improper fraction

134 

  

6.) A student does not understand why when you subtract a negative number, the number from which you are subtracting increases. For example: 7- (-2) = 9. How would you show or explain to a student why this is true?

              7) Imagine that you are working with your class on multiplying two, two digit numbers. Your students’ solutions are displayed to the right. Which of these student’s method can be used to multiply any two whole numbers?

135 

APPENDIXB 

DRIVINGQUESTIONSFORSEMI‐STRUCTUREDINTERVIEWS   

 

136 

APPENDIXC 

EMAILTOSUPERVISORSOFELEMENTARYPRE‐SERVICE/IN‐SERVICETEACHERS

Dear ,

Iamwritingtoaskforyourhelpinconductingaresearchstudyatyourinstitution. IamcurrentlyenrolledintheMathematicsEducationPh.D.programatTeachersCollege,ColumbiaUniversityandamintheprocessofwritingmydissertation. ThestudyisentitledKeepingupwiththetimes:howareteacherpreparationprogramspreparingaspiringelementaryteacherstoteachmathematicsunderthenewstandardsoftoday? Thepurposeofthisstudyistodeterminehowwelltraditionalelementarypreparationprogramsandalternativeelementarycertificationprogramsareperceivedbyaspiringteachers,topreparethemtoteachmathematicstothelevelofrigorrequiredtoday. 

Ihopethatyoucanhelpmebyrecruitingparticipantsfromyourelementarystudentteachingcohorttoanonymouslycompletea5‐pagequestionnaire. Interestedstudents,whovolunteertoparticipate,willbegivenanelectronicconsentformtobesignedandreturnedtotheprimaryresearcheratthebeginningofthesurveyprocess. 

Participantswillalsobegivenalinkforanonlinesurveythattheywillcompleteattheirownconvenience.Thesurveyprocessshouldtakenolongerthanthirtyminutes. Studentsthatparticipatewillbecompensatedwithafive‐dollarI‐tunesgiftcard.Thesurveyresultswillbepooledforthestudyandindividualresultsofthisstudywillremainabsolutelyconfidentialandatnotimewillyourinstitutionbeidentifiedinthisstudy.Basedonsurveyresponses,someparticipantsmaybecontactedtocompleteafollowupinterview. Yourapprovaltoconductthisstudywillbegreatlyappreciated. Iamhappytoansweranyquestionsorconcernsthatyouhave.Youmaycontactmeatmyemailaddress:ssudarsanan@gmail.com Ifyouagree,pleasereplyandIwillfollowupwithaformalinstitutionapprovaldocumentthatyoucansignandemailbacktomeatyourearliestconvenience.Thankyouforyourtime.

   

Sincerely, ShaliniSudarsanan

137 

APPENDIXD 

EMAILTOPARTICIPANTS ToWhomItMayConcern, 

IamwritingtoaskforyourhelpinmydissertationresearchstudyentitledKeepingupwiththetimes:howareteacherpreparationprogramspreparingaspiringelementaryteacherstoteachmathematicsunderthenewstandardsoftoday?.Thepurposeofthisstudyistodeterminehowwelltraditionalelementarypreparationprogramsandalternativeelementarycertificationprogramsareperceivedbyaspiringteachers,topreparethemtoteachmathematicstothenewstandardsoftoday. 

Ihopethatyoucanhelpmebycompletinga5‐pagequestionnaireaboutyourmathematicspreparation,viewsofyourownteachingofmathematics,andyourapproachestoteachingcertainelementarymathematicsconcepts. Ifyouvolunteertoparticipate,youwillbegivenanelectronicconsentformtobesignedandreturnedtotheprimaryresearcheratthebeginningofthesurveyprocess. 

Youwillalsobegivenalinkforanonlinesurveythatyoucancompleteatyourownconvenience.Thesurveyprocessshouldtakenolongerthanthirtyminutes. Studentsthatparticipatemaychoosetobecompensatedwitheitherafive‐dollarI‐tunesgiftcardoranamazongiftcard.Thesurveyresultswillbepooledforthestudyandindividualresultsofthisstudywillremainabsolutelyconfidentialandatnotimewillyouoryourinstitutionbeidentifiedinthisstudy.Basedonsurveyresponses,someparticipantsmaybecontactedtocompleteafollowupinterview. Yourparticipationinthisstudywillbegreatlyappreciated.Iamhappytoansweranyquestionsorconcernsthatyouhave.Youmaycontactmeatmyemailaddress:ssudarsanan@gmail.com Ifyouagree,pleasereplyandIwillfollowupanelectronicconsentformfollowedbythesurveylink.Thankyousomuchforyourtime. Sincerely, ShaliniSudarsanan

138 

APPENDIXE 

PMTEFREQUENCIESOFPARTICIPANTS 

n=25 StronglyAgree

Agree Undecided Disagree StronglyDisagree

Question2Icontinuallyfindbetterwaystoteach

mathematics.

6 13 5 0 0

Question3EvenifItryveryhard,IdonotteachmathematicsaswellasIteachmost

subjects.

1 5 4 12 2

Question5Iknowhowtoteachmathematics

conceptseffectively.

1 10 11 2 0

Question6Iamnotveryeffectiveinmonitoring

mathematicsactivities.

0 0 7 15 2

Question8Igenerallyteachmathematicsconcepts

ineffectively.

0 1 3 16 4

Question11Iunderstandmathematicsconceptswellenoughtobeeffectiveinteaching

elementarymathematics.

8 11 4 1 0

Question15Ifinditdifficulttousemanipulativestoexplaintostudentswhymathematics

works.

0 0 1 13 10

Question16Itypicallyamabletoanswerstudents’

questionsaboutmathematics.

7 14 2 1 0

Question17IthinkIdonothavethenecessary

skillstoteachmathematics.

0 1 1 17 5

Question18Givenachoice,Iwouldnotwantmyprincipaltoevaluatemymathematics

teaching.

0 5 1 15 3

Question19Whenastudenthasdifficulty

understandingamathematicsconcept,Iamusuallyatalossastohowtohelpthestudentunderstanditbetter.

0 0 5 15 4

Question20Whenteachingmathematics,Iusually

welcomestudents’questions.

13 12 0 0 0

Question21Idonotknowwhattodototurnstudentsontomathematics.

0 1 8 14 2

139 

APPENDIXF 

PMTOEFREQUENCIESOFPARTICIPANTS 

n=25 StronglyAgree

Agree Undecided Disagree StronglyDisagree

Question1Whenastudentdoesbetterthanusualinmathematicsitisoftenbecausetheteacherexertedalittleextraeffort.

3 12 4 4 1

Question4Whenthemathematicsgradesof

studentsimprove,itisoftenduetotheirteacherhavingfoundamoreeffective

teachingapproach.

5 13 5 1 0

Question7Ifstudentsareunderachievinginmathematics,itismostlikelyduetoineffectivemathematicsteaching.

2 8 8 5 1

Question9Theinadequacyofastudent’smathematicsbackgroundcanbeovercomebygoodteaching.

6 15 3 0 0

Question10Whenalow‐achievingchildprogressesinmathematics,itisusuallyduetoextraattentiongivenbytheteacher.

6 14 3 1 0

Question12Theteacherisgenerallyresponsiblefor

theachievementofstudentsinmathematics.

1 12 6 5 0

Question13Students’achievementinmathematicsisdirectlyrelatedtotheirteacher’s

effectivenessinmathematicsteaching.

1 12 7 4 0

Question14Ifparentscommentthattheirchildisshowingmoreinterestinmathematicsatschool,itisprobablyduetotheperformanceoftheirchild’steacher.

6 9 7 2 0

140 

APPENDIXG 

RESPONSESTOPEDAGOGYSURVEYQUESTION1 

Question 1: A student makes the following mistake: 734-481 = 353. How do you respond?  

Procedural Conceptual I would tell this student that when carrying

from any place value, to cross out that number and subtract a ten from the number. So for example, if I was to borrow a group of ten from the 7, I would cross it out and put a 6 because I borrowed it. (Kathy)

Lets look in the tens place. If I have 3 tens can I take 8 tens away from that? (Nancy)

I would show the student that we can borrow from the 7 to make it a 6 and make the 3 a 13, and then we can take 8 away from 13. (Jackie)

Remind the student what borrowing means…manipulatives such as base 10 blocks can be used to reinforce the concept. (Laura)

When you borrowed from the 7 in the hundreds place, that reduces the 7 to a six. Take a look at the problem again and see what you can change. (Natalia)

“…Use manipulatives if necessary… hold up three fingers and say is it possible to take 8 away from 3?”… We can borrow!!...We know 1 hundred is the same as ten tens, instead of 7 hundreds, I now have 6. An instead of 3 tens, I have 13 tens. Now we can do 13-8...” (Carol)

Practice borrowing and give them graph paper (Maddy)

I would have the student draw out a picture and start at the ones and cross off the numbers. At the tens, the student would have to regroup a 100 to be able to cross off 8 tens. (Abby)

You are subtracting the lower number from the higher number for each of your number places when this isn’t the case for all. (Clarissa)

I would then sit with them with base-10 blocks and work with them on exchanging tens and hundreds to make the problem able to solve. (Bethany)

They forget to regroup and borrow, I would suggest that the student tries a different subtraction strategy. (Christy)

I would use a place value mat to demonstrate the need to borrow from the hundreds. When a student moves one hundred to the tens it will be easier for them to visualize that they have 6-4 instead of 7-4. (Brittany)

Remember when you borrow a set of ten from a higher place value, cross the number out and write one number less above it. Now borrow from the written number. (Lindsay)

Solve problem myself using multiple strategies, in order to discover likely mistakes and misconceptions that may arise…look at student’s work to see if this provides clues to method…remediate based on the error diagnosed. (Aaron)

Be careful while borrowing. Check your answer. (Lauren)

I look at the number pattern to figure out the logic of the mistake, it appears the student chose to use partial sums, but may need more help using base-10 blocks to illustrate what happens to groups of ten when you need more ones than you were given in order to compute the problem. (Samantha)

I would explain that when I made the “3” into a “13” , I must take away “one” from the left. (Teresa)

 

I would ensure that the students is lining up the correct place values and regroup properly. (Lily)

 

141 

 

The student subtracted 4 from 7, resulting in three hundreds in the answer, I would discuss this with the student. (Seana)

 

I would ask them to look at their work again and say is 6-4 = 3? I would expect them to quickly notice their own error and change the 3 to a 2. (Jessica)

 

142 

APPENDIXH 

RESPONSESTOPEDAGOGYSURVEYQUESTION2 

Question 2: Imagine you are introducing addition with regrouping to your students, how would you show students why

27 + 95 = 122?  

Procedural Conceptual Alternate … I would explain that we can only put one number at a time under the line until we are finished or run out of numbers, therefore we can only put the 2. I would then put the 1 on top of the 2… (Jackie)

.…Because 7+5=12, you bring the 2 down and carry the group of ten over to the tens column…A great way to show this is by using manipulatives, especially base ten blocks. (Kathy)

I am not sure what regrouping is…I would minus 5 from 27 to get 22 and add the 5 to 95 to make it 100…Then I would add… (Natalia)

.…I am writing the problem vertically onto the board. I would mention that we know it is 12 but we can’t just write 12 underneath because 2 + 9 is not 1. When we add # together that have multiple digits we have to be prepared to carry one of the digits over to the left. We can write down the 2 underneath the 7 and the 5, but we have to carry the one and add it after we calculate the #’s in the tens place. (Jessica)

I would first show them that they need to look in their ones column first and do 7+5…and ask if we can put 12 in our ones column. I would then explain how you need to carry the 1 from 12, because that is your tens, to the tens column… (Nancy)

One way to do this problem would be to round the numbers to 100 + 30 and get an answer of 130 then subtract 5 + 3 which is 8 to get 122…Depending on the students each one will have his/her own technique let them pick which one works out best for them. In math it is all about what makes a student more comfortable, not what is most efficient. (Maddy)

I would start by rewriting the problem vertically. Next I will explain how to regroup by carrying the (1) to the tens place and then add the three numbers together to get the final answer. (Lily)

… I would explain to the students that 7 + 5 = 12, but since the ones spot cannot hold 12, we write the 2 (for 2 ones) and carry the 1 over into the tens spot to represent our one set of ten. Then I would show students to add 1 + 2 + 9 = 12. I would repeat this and show students to write the 2 down again and carry the one into the hundreds spot. Then add again. (Seana)

… I would then work with partial sums and break down the numbers to make them “easier” to solve. 27 + 95 = 122, 20 + 90 = 110, 7 + 5 = 12, 110 + 12 = 122. (Christy)

I would explain to students that the order does not matter in addition, to be able to get the answer you can regroup. (Teresa)

Explain the concept of place value and what each digit stands for…Use base 10 blocks or another manipulative to show why we need to regroup into groups of 10 and how. (Laura)

 

When regrouping circle your extra number so you don’t forget its there to add. (Lauren)

I would get out base-10 blocks and set out 2 longs and 7 cubes. Then add 9 longs and 5 cubes… (Samantha)

 

  … 27 = 2 tens and 7 ones; 95 = 9 tens and 5 ones… (Aaron)

 

  Use manipulatives or drawings while reinforcing place value. Draw out 2 tens and 7 ones, then

 

143 

 

  draw out 9 tens and 5 ones. Circle 10 of the ones and show how it can be combined into a ten… (Carol)

 

  I would use base ten blocks to show that the ones are more than ten by lining up with a tens stick. Then trade he ones out. (Abby)

 

  I would use base-10 blocks to work it out…if that is not really working I would use expanded form to help them see each step a little clearer. (Bethany)

 

  A place value mat with manipulatives would visually demonstrate this to students. Unifix cubes can be exchanged for rods, straws can be bundled with bands… (Brittany)

 

  Make groups of ten. Emphasize base ten. (Lindsay)

 

  You need to add the ones place together to get 12 ones. You need to add the tens place together to get 11. You will then need to move the ten from the ones place to the tens place to get 12 and then you have 2 ones left. 12 tens and 2 ones gives you 122. (Clarissa)

 

 

                              

APPENDIXI

144 

 

RESPONSESTOPEDAGOGYSURVEYQUESTION3 

Question 3: Create a story problem to illustrate that 4 ¼ ÷½ = 8 ½ .  

Correct Story Problem Incorrect Story Problem Alternate Responses   If mom has 4 ¼ cups of milk and wants to divide it in

half so you and her have equal parts, how much would each of you have? (Jackie)

4.25/4.5 does not equal 8.5 (Kathy)

Dan has 4 ¼ pounds of pizza dough. He knows that it takes ½ pound of dough to make one pizza. How many pizzas can Dan make? (Aaron)

You and a friend have 4 pies, each cut into 4 pieces. You and your friend want to divide the pieces equally so you each get the same amount. How many pieces would you and your friend each get? (Laura)

Did not answer (Nancy) (Christy) (Lindsay)

I want to make 4 ¼ recipes of salsa. Each recipe calls for ½ a cup of tomatoes. How many recipes can I make? (Abby)

Mike had 4 ¼ pizzas which all needed to be divided into halves. After dividing each pizza in half, how many pieces of pizza would Mike have? (Natalia)

Ms. Sweetzer wants to bake cakes for her son’s birthday party. She has 4 ¼ pounds of sugar at home and wants to know how many cakes she can bake. (Carol)

Every cupcake needs ½ cup of sugar. You have 4 and ¼ cups of sugar. How many cupcakes can you make? (Bethany)

I would say that Sarah has 5 pies. Each pie has 4 slices total. Her friend Amanda comes and eats 3 pieces of pie. What fraction of pies is left? The student would say 4 ¼ then I would say that Mike wants to split the pies EVENLY between himself and Sarah. How many pieces of pie would each of them get if they split them in half EVENLY? The student would then answer 8 ½ pieces of pie. (Samantha)

Not sure how to create a story problem without making the operation be multiplying (multiply by the reciprocal) rather than dividing. (Jessica)

Mary has 4 ¼ yards of material. If each apron takes ½ yard of material, how many aprons can she make? (Clarissa)

I would use cooking and food to explain with actual measuring cups. If John and Sue are trying to make cookies how can they share the ingredients if each one wants to help make the cookies. They must divide the ingredients to equal 8.5. Use multiplication and division to arrive at a solution to this problem. (Maddy)

 

Sam is making big carved gingerbreads. Each gingerbread requires ½ foot of dough. Sam has 4 ¼ feet of dough. How many gingebreads can Sam make? (Teresa)

Baking recipe for muffins. 4 ½ cups of sugar. Each requires ½ cup of sugar. How many cups will allow the recipe to bake 4 muffins? (Lauren)

I know how to solve this one but do not know how to go about making a story problem that explains what needs to be done. (Brittany)

Mary and Alice are making muffins. Each muffin tin needs ½ cup of batter. They have 4 ¼ cups of batter in total. How many muffin tins can they fill? (Lily)

   

APPENDIXJ

145 

 

RESPONSESTOPEDAGOGYSURVEYQUESTION4 

Question4:Howwouldyouexplainwhywhenyoudividebyafractionyoumultiplybythereciprocalofthefraction?Youcanusethefollowingdivisionexercisetoassistyouin

yourexplanation.3÷¼=12.

 Procedural / Incorrect Conceptual Did Not

Answer When you are dividing, you are cutting something into equal shares. (Teresa)

You can use a tape diagram to show how 3 ÷ ¼ = 12…” (Lily)

 (participantdrewapropertapediagramdemonstratingthreewholes,withfourpartsineachwhole)

Lindsay Christy

Divide 3 into ¼ (Lauren) If we are dividing fractions and we set those fractions up in a fraction such as 3/1 / ¼ then how would we get rid of the bottom fraction? We would multiply it by the reciprocal 4/1, and then because we multiplied the bottom by 4/1, we must also multiply the top by 4/1. So now we have 3/1 x 4/1 = 12. (Clarissa)

 

  “I would start by drawing 3 rectangles. I would then say you want to divide the into fourths while splitting the 3 rectangles into 4 equal pieces. I would then say well look at that do we have more pieces or less pieces than we started? I would then saw because we are dividing them into smaller pieces we have more to share with everyone else. (Bethany)

 

I would teach kids that to make dividing fractions easier, you just multiply by the reciprocal (the opposite of ¼). (Jessica)

“Using a picture, 3 rectangles. Divide each into four pieces to represent fourths. This could also be seen as an area model which is how multiplication is often taught now. (Abby)

 

Division is the process of splitting into equal groups. A/B = C is “A (total amount) split into B (number of equal groups) = C (amount in each group). 12/3 = 4 is 12 (total) with 3 in each group = 4 equal groups. However, if B is a fraction less than 1 (i.e. ¼) then we are “splitting up” the total quantity into groups less than 1 (groups of ¼ = 4 groups per 1 unit), hence we will have more than B groups. (Aaron)

If I have 3 pies and each pie is divided into ¼, each pie would yield 4 pieces. Three pies would yield 12 pieces total. We can solve this by multiplying 3 by 4, the reciprocal of ¼. When we divided the three pies by ¼, we could multiply by the reciprocal 4, because we are actually counting the number of fractions (1/4 pieces of pie) that make up the original number (3 pies). (Carol)

 

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I would explain the flip it and multiply method by putting a one under the three and flipping the four on top of the one to then see four times three… (Maddy)

If you had 3 cookies and wanted to break them into fourths, you would end up with 12 pieces. (Laura)

 

I would say because the opposite of dividing is multiplication. (Samantha)

Take 3 candy bars and break them into 4 pieces each. 3 wholes divided into fourths: 4/4 (1) + 4/4 (1) + 4/4 (1) = 3 x 4 = 12. How many ¼ do you have? (Brittany)

 

I would teach the students to remember the phrase “Change the sign flip the second fraction and multiply!” I would then show the students to change the sign to a multiplication sign, change the second fraction to 4/1 or 4, then multiply straight across. (Seana)

   

I would explain to students that any whole number always has a 1 under it. We don’t write whole numbers that way, though. When you need to find the answer to a division problem involving fractions, you need to multiply. The opposite of dividing is multiplying. Therefore, the opposite of ¼ is 4/1. Then you multiply the numerator and denominator. (Jackie)

   

I would say to my students, “When we have 3 divided by ¼ we would first change our whole number to a fraction, like 3/1 or 9/3. Then since we will be dividing fractions we need to find the reciprocal of the second fraction. The reciprocal would be 4/1, you basically flip your fraction around. Next we multiply our fractions, so we would have 3/1 times 4/1 which gives us…12/1 or 12! (Nancy)

   

When you divide by a fraction you multiply by the reciprocal because dividing is he opposite of multiplying and you have to flip the fraction. For example, 3 divided by ¼ is the same is 3 times the opposite of ¼ which is 4. Therefore, 3 divided by ¼ is the same as 3 x 4 = 12. (Kathy)

   

When dividing you multiply the number by the inverse of the fraction. Believe it of not, when you divide whole numbers you do the same thing. For instance 4 divided by 2 can be seen as 4/2, which is essentially the same as 3(1/4). (Natalia)

   

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APPENDIXK 

RESPONSESTOPEDAOGOGYSURVEYQUESTION5 

Question5: Whenintroducingstudentshowtorenamemixednumbersintoimproperfractions,howwouldyouteachachildtorename2¾intoanimproperfraction?

 

  

Procedural Conceptual DidNotAnswer

TheonlywayIrememberismultiplyingthewholenumberbythedenominatorandthenaddingthenumeratortoit.(2x4)+3=11andthenplacethedenominatorbackonthebottom=11/4.(Christy)

4/4=1Demonstratewith a pizza cutinto4pieces.Namethepiecesize¼or1of4pieces.2pizzascutinto4pieces=8/4.Thethirdpizzaisalsocutintofourthsbutonepieceismissingleaving3pieces.Nowcountupthetotalnumberofpieces11alsonamed11fourthsorwritten11/4.Thisisavisualwaytodemonstratewhythefollowingmathworks.Sonow1pizzax4pieces=4,so2pizzas=8.Wholenumbertimesthedenominator.However,westillhavethe3piecesofthethird.(Brittany)

NancyLindsay

Multiplythe4(thedenominator),andthe 2(thewholenumber).Tothisproduct,addthe3(thenumerator)giving11,toformthenewnumerator,andusethe4asthenewdenominator.So,2¾asanimproperfractionis11/4.(Clarissa)

Iwouldstartby drawingout3rectanglessayingthatwedon’thavequiteallofthe3rdoneIwouldthenaskthemhowIcoulddrawthelastrectangletoshowthatweonlyhave¾ofit.Oncewehavepartitionedthethirdrectangleinto¼Iwouldshadein3ofthem.Thenshadeintheotherrectangles.Wewouldthensplitthoserectanglesinto¼aswell.AfterthatIwouldaskhowmanypiecesdowehaveshadednow.(Bethany)

 

2¾=2x4+3=8+3=11/4.(Lauren) Iwoulddrawout tworectangles undividedandathirdshowingthe¾,Iwouldthenshowhowthewholesneedtobedividedintofourpieces.Wecanthencountthenumberofpiecesintotal.(Abby)

 

Iwouldtellthestudenttofirsttimesthe“4”andthe“2”,thenaddthe“3”.Whenyouget“11”,youmustputitoverthe4.(Teresa)

Participant drew acircle diagram withtwocircles,withfourpartsineachwhole,andanothercirclewith3outof4fourpartsandwrote11/4.(Lily)

 

Step1:Multiplythedenominatorbythewholenumberontheleft.Step2:AddthenumeratornumbertoansweryougotinStep1.Step3:Placethefinalanswerontopoftheoriginaldenominator.Youshouldbeleftwithalargernumberontopoftheoriginaldenominator.(Jessica)

Usemanipulatives to break“2”into 8quarters(4quarterseach).Countallquarterpieces(11)andshowthat2¾isequalto11/4.(Carol)

 

Iwouldtellthestudentstowritetheproblemdownfirstandthentakefourtimestwoandgeteight.ThenIwouldhavethemaddthreeontoeightandgetelevenwhichthenwouldgoon

I’dstart witha concrete(or, if difficulttoobtain/create),pictorialrepresentationof2¾,emphasizingthat1=4/4and2=8/4.Forexample,Iwouldshowthat2=8/4bydrawing

 

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topof4tohaveelevenfourths.Iwouldnotonlywritethestepsoutsothateveryonecouldcopythemintotheirnotes,Iwouldthenpracticethistechniquemanytimes,untiltheclassunderstoodwhattodo.(Maddy)

two circlesor squares,dividedinto 4equalpieces,withallpiecesshaded.Teachingkidshowtocreatepictorialrepresentationsofmixednumberswouldallowthemastartingdoingforthistypeofproblem,anditwouldallowthemtochecktheirworkwhentheyattemptedtouseotherstrategies.(Aaron)

 

Iwouldtellthestudentstomultiplythedenominatorbythewholenumberandthenaddthenumerator.(Samantha)

   

Youmultiply2timesthedenominator,which is4andget8,thenyouadd8tothenumeratortoget11,8+3=11.Thereforethenewfractionis11/4.(Natalia)

   

Iwouldteachthemtomultiplythedenominatorandthewholenumberandthenaddthenumerator.Thenlookattheanswerandshowwhyitisthesameasthemixedfraction.(Laura)

   

Iwouldshowstudentsthatthewholenumberismultipliedbythedenominatorandthethenthenumeratorisadded.Thisisthenewnumeratorandthedenominatorremainsthesame.(Seana)

   

Iwouldtellstudentsthatyoutakethedenominatorinthefractionandmultiplyitbythewholenumber.Youaddthenumeratortotheansweryouhadbefore.Youtakethatanswerandputitoverthedenominatorintheoriginalnumber.(Jackie)

   

Firstofall,animproperfractionisafractionwhenthenumeratorisgreaterthanthedenominator.Iwouldtellstudentsthattomakeanimproperfractionwith2¾youfirstmultiplythedenominatorandthewholenumber(4x2)whichis8.Addthenumerator(3)andyouget11/4,thedenominatorstaysthesame.Toturnitbackintoamixednumberyouwoulddivide.Youwoulddo11/4whichis2¾.(Kathy)

   

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APPENDIXL 

RESPONSESTOPEDAGOGYSURVEYQUESTION6 Question7: Astudentdoesnotunderstandwhywhenyousubtractanegativenumber,thenumberfromwhichyouaresubtractingincreases.Forexample:7–(‐2)=9.Howwouldyoushoworexplaintoastudentwhythisistrue?

 ConceptualResponse Procedural Response MathematicallyIncorrect

Response/DidNotAnswerquestion

Whensubtractingwithnegativenumbersonanumberline,thesubtractionsignmeanstogointheoppositedirectionofthedirecteddistancethatfollows.Thus,tofind+7‐(‐2),startat+7,thendotheoppositeof(‐2)–thatis,go2unitstotherightendingat+9.(Clarissa)

Iwoulddo the tweak method oftryingtoexplainandshowthestudentsthatwhenyouhavetwomarksinthatpositionjustcrossthembothintoplussignstoseethatthisproblembecomesanadditionproblem,notasubtraction.Iwouldalsogetoutanumberlineandshowthisproblem,withthehelpofacalculatortoprovemypointasoneotherexample.(Maddy)

I would teach themtheslogan, “Anegativeminusanegativeequalsapositive.(Jessica)

Youaretakinganumberfromtheothersideofzerowhichchangestherulesofsubtraction.Anegativewillmakealargeranswerinsubtractionbecauseoftheplacementonthenumberline.(Abby)

Iwouldshow the student thatwhenyouhave2negativesinarowlikethis,theycometogetherandformapositive,oraplus.Youcanthenaddthetwonumbers.(Seana)

Participant drewadiagram of aballoonandwrote‐“weight‐takeawayweight‐balloongoesup…positive.(Lauren)

Useanumberlinetoshowdirectionofsubtractionwhencombinedwithnegativenumbers.(Carol)

Theminus of a minus isaplusrule”(Lily)

I would tell thestudenttolook at itthisway:Soimaginewebought5goodplayers(+5),sold2goodplayers(+2),bought3badplayers(‐3)andsold7badplayers(‐7).Inthisproblem,goodplayerswouldbepositivebecauseitwouldhelptheteamoutwhilebadplayerswouldbenegative.Wecanwritethefollowingcalculationtofindouttheoveralleffect:+(+5)–(+2)+(‐3)–(‐7).Buying(+5)andselling(subtract)(+2)buying3bad(adding‐3)andselling7bad(subtracting(‐7)equals7.(Kathy)

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Numberline(Aaron)   Whenever you geta(‐)(‐)you addupbecausetwonegativeswillalwaysequalapositive.(Teresa)

Iwoulduseanumberlineandphysicallyshowthechildthemovesbeingillustratedintheproblem.(Samantha)

  ‐(‐2) is actuallypositivebecausetwonegativescanceleachotherout.(Natalia)

Showthestudentsanumberlineandexplainhowthenegativesignchangesthedirectionofthehops.(Laura)

  Did not answer:(Brittany)(Lindsay)(Christy)(Bethany)(Nancy)

Iwouldwritetheproblemthewayitis,butIwouldshowthestudentthatwhenyousubtractanegativenumber,youareactuallyaddingapositivenumber.Hence,minusanegativeisplusapositive.Turntheminussymbolsintoplussymbols.(Jackie)

   

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APPENDIXM 

RESPONSESTOPEDAGOGYSURVEYQUESTION7 7.Imaginethatyouareworkingwithyourclassonmultiplyingtwo,twodigitnumbers.Yourstudents’solutionsaredisplayedtotheright.Whichofthesestudent’smethodcan

beusedtomultiplyanytwowholenumbers? 

             

SelectedAllThreeMethodsasCorrect

SelectedTwoMethodsasCorrect

SelectedOneMethodasCorrect

DidNotAnswer

BrittanyCarolAaronLaurenLily

ClarissaBethanyAbbyMaddyLauraJackieKathy

ChristyJessica

SamanthaSierraNataliaNancyTeresa

Lindsay