Development and Validation of an Improvised Manipulative
Material as Remediation Device in Teaching
Addition of Integers
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Nelvin R. Nool
College of iTeacher Education, Tarlac State University, Philippines
Abstract
The iuse iof imanipulatives ihas ipositive ieffects ion ithe iachievement iof istudents iin
imathematics. iThis istudy iaimed ito idevelop iand ivalidate ian iimprovised imanipulative
imaterial
ias
iremediation
idevice
iin
iteaching iaddition
iof
iintegers.
iUsing
ithe
idevelopmental iresearch idesign, ithe iimprovised imanipulative iwas ideveloped. iTo
ivalidate ithe iimprovised imanipulative, ione igroup ipretest iposttest ipre-experimental
idesign iwas iemployed iamong ilow iperforming ifirst iyear icollege istudents. Findings
showed that the use of the ideveloped imanipulative imaterial significantly improved
istudents‟
performance iand iretention iin iadding iintegers iand ideveloped itheir
confidence in learning mathematics. iTherefore, ithe ideveloped iimprovised
imanipulative
iwas
ian
ieffective
iinstructional
imaterial
iin
iimproving
istudents‟
iperformance
iin
mathematics. iFurthermore, iimplications iof ithe ifindings iand
irecommendations ifor ifuture iresearch iare idiscussed.
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Keywords: addition of integers, imanipulative imaterial, iremediation idevice
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INTRODUCTION
iWhat iI ihear, iI iforget. iWhat iI isee, iI iremember. iWhat iI ido, iI iunderstand. iThis
Chinese imaxim iabove iemphasizes ithat ieffective iand ilasting ilearning icomes ifrom
iexperience iand iactive iinvolvement iby ithe ilearner. iTherefore, iall istudents imust iactively
iengage iin imeaningful, ihands-on, iminds-and ion ilearning iexperiences iin ithe ilearning
iprocess, ispecifically iin imathematics isince iMath ican ionly ibe ilearned iby idoing iit.
i
John Dewey, early in the 20th century, stressed that learning comes from
experience and active involvement of the learner. Jerome Bruner i(1966) advocated
the use of physical manipulatives as providing scaffolding for abstract concepts. Even
Jean Piaget seemed to be aware of the significance of manipulatives in a child‟s
construction of logical-mathematical knowledge. Recently, Goldstone and Son i(2005)
contended that “abstract understanding is most effectively achieved through
experience with perpetually rich, concrete representations…. Abstractions are
effectively learned by exposure to pictures, movies, interactive simulations and realworld physical experiences that embody the abstractions.”
Manipulatives are physical models that represent concretely abstract concepts
and appeal to the senses, can be touched or moved. Notable examples of concrete
representations are Dienes blocks, spinners, number lines, geoboards (boards with a
lattice of pegs and loose rubber bands to wrap around the pegs), Cuisenaire rods
(colored wooden bars cut to integer lengths), algebra tiles, and balance beams.
If la lpicture lis lworth la lthousand lwords, ithen ihow imuch ido iconcrete
iinstructional imaterials iworth? i“When imanipulatives iare iused, ithe isenses iare ibrought
iinto ilearning; istudents ican itouch iand imove iobjects ito imake ivisual irepresentations iof
imathematical
iconcepts”
i(Hoffman,
i2007).
i“Manipulatives
ican
ihelp
ichildren
iunderstand iand idevelop imental iimages iof imath iconcepts” i(Dunlap i& iBrennan, i1979);
ithey iprovide istudents ia iconcrete ibasis ifrom iwhich iabstract ithinking idevelops. iThus,
iinstruction ishould istart ifrom iconcrete iexperiences ito ihelp istudents iin iunderstanding
ian iabstract iconcept, itransition ito isemi-concrete imaterials i(pictures), iand ithen ito
iabstract isymbols i(numbers iand iletters) i (Dunlap i& iBrennan, i1979; iMercer, i1992;
iMiller i& iMercer, i1993).
l
In an effort to de-abstract mathematics, the use of manipulatives brings it to
the concrete level. Recognizing the benefits of manipulative materials in learning
Math, the National Council of Teachers of Mathematics in their 2000 Principles and
2
Standards states that “concrete models can help students represent numbers and
develop number sense; they can also help bring meaning to students‟ use of written
symbols…”i i(NCTM, i2000)
The iStandards irecommends ia ilist iof imanipulatives ifor ikindergarten ithrough
igrade ieight, ibut ino ilist iis irecommended ifor igrades inine ithrough itwelve i(NCTM,
i2000). iStudents iin ithese ilevels ialso ihave ito isee iideas irepresented iat ithe iconcrete ilevel.
iEven icollege istudents ineed ieither iconcrete ior ipictorial irepresentation iof iabstract
iconcepts ito iacquire ideep iunderstanding. iHence, ithis istudy iemerged ito iinvestigate ithe
ipossible ipositive ieffects iof imanipulative idevice ion icollege ifreshmen‟s icomprehension
iof iaddition iof iintegers.
i
Researches, such as those conducted by Freudental i(1983), Janvier i(1985),
Resnick i(1989), among others, have found out that students experienced extreme
difficulties in conceptualizing and performing operations with negative numbers in the
pre-algebraic and algebraic scope. Betaño i(1983) and Gamido i(2001) noted that
college freshmen had difficulties in operations on signed numbers, particularly on
addition and subtraction. iThis isituation iis ialso iexperienced iby ithe iresearcher. iAmong
i98 icollege ifreshmen iwho itook ia iten-item imultiple ichoice itest ion iaddition iof itwo-digit
iintegers, itheir imastery ilevel iwas i56.67%, iwhere i37 ior i38% iobtained iscores iless ithan i5.
iLikewise, iamong i53 icollege isophomores iwho itook ithe isame itest, itheir imastery ilevel iin
iaddition iof iintegers iwas i65.20%, iwhere i16 ior i30% ifailed. iThis ishows ithat ione-third
i(35%) iof ithe ifirst iand isecond iyear istudents idid inot imaster iaddition iof iintegers.
Their difficulty in this operation is quite alarming considering that they had
studied this topic since first year high school. This can be attributed to their inability
to comprehend the rules applied to this operation. They may need physical
representations to see connections between the abstract and concrete which will help
them to understand and develop mental images of this process.
Thus, istudy iaimed ito idevelop iand ivalidate ia imanipulative imaterial ithat iwould
ihelp icollege istudents igain ithorough iunderstanding iof iaddition iof iintegers. iThe
ispecific iobjectives iof ithe istudy iwere ithe ifollowing:
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1. iTo idevelop ia imanipulative idevice ifor iteaching iaddition iof iintegers
2. iTo idetermine ithe ieffectiveness iof ithe imanipulative iin iterms iof icorrected
istudents‟ ierrors, ilevel iof iimprovement iit icaused, iand iretention
3. iTo idraw iimplications iof ithe istudy ito imathematics iteaching
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THEORETICAL iFRAMEWORK
1i
One itheory iof ilearning istates ithat isensory ilearning iis ithe ifoundation iof iall
iexperiences. iStudent‟s ilearning iis ienhanced iif ihe iis iactively iinvolved iin ithe ilearning
iprocess i(Hartshorn i& iBoren, i1990). iExperiential ilearning iprovides istudents iwith
iopportunities ito iengage iin imeaningful ihands-on, iminds-on iand iauthentic ilearning
iexperiences. iThe iuse iof imanipulative imaterials iallows istudents ito ilearn iby idoing.
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Conventionally, imanipulatives irefer ito iphysical iobjects ithat iappeal ito isenses
iand ican ibe itouched ior imoved i(Clements, i1999). iMost ieducators iand iresearchers
ibelieve ithat imanipulatives iare ieffective ibecause ithey irepresent iconcretely iabstract
iconcepts. i“Using irepresentation i– iwhether idrawings, imental iimages, iconcrete
imaterials, ior iequations i– ihelps istudents iorganize itheir ithinking iand itry ivarious
iapproaches ithat imay ilead ito ia iclearer iunderstanding iand ia isolution” i(Fennell i&
iRowan, i2001). iClements i(1999) istressed ithat i“good imanipulatives iare ithose ithat iaid
istudents iin ibuilding, istrengthening, iand iconnecting ivarious irepresentations iof
imathematical iideas.”
i
iGoldstone iand iSon i(2005) ipresented ifour ibenefits iof iusing imanipulatives.
Concrete iinformation iis ieasier ito iremember ithan iabstract iinformation. iIt iis ioften
ieasier ito ireason iwith iconcrete irepresentations iusing imental imodels ithan iabstract
isymbols. iConcrete idetails iare inot ialways i“superficial,” ibut irather iprovide icritical
iinformation iabout ilikely ibehavior iand irelevant iprinciples. iConcrete imaterials iare
ioften imore iengaging iand ientertaining iand iless iintimidating.
i
Piaget‟s iwork ihas ihad ia igreat iimpact ion iAmerican ieducation isince ithe i1960‟s.
iBy iobserving ihis iown itwo ichildren iextensively, ihe ifound ithat ithey imoved ithrough
ithree
i(3)
idevelopmental
ilearning
istages:
ithe
iconcrete
ior
imanipulative,
ithe
irepresentational ior itransitional, iand ithe iabstract. iPiaget iseemed ito ibe iaware iof ithe
iimportance iof imanipulatives ilong ibefore ithe iword ibecame iestablished iin ithe
ieducational ifield.
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Logical-mathematical iknowledge ican idevelop ionly iif ia ichild iacts
i(mentally ior iphysically) ion iobjects. iThe ichild iinvents ilogicalmathematical iknowledge; iit iis inot iinherent iin iobjects ibut iis iconstructed
ifrom ithe iactions iof ithe ichild ion ithe iobjects. iThe iobjects iserve imerely
ias ia imedium ifor ipermitting ithe iconstruction ito ioccur. i Logicali
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mathematical iknowledge iis iconstructed ifrom iexploratory iactions ion
iobjects iwhere ithe imost iimportant icomponent iis ithe ichild‟s iaction, inot
ithe iparticular iobject(s). iNumber, ilength, iand iarea iconcepts icannot ibe
iconstructed ionly ifrom ihearing iabout ithem ior ireading iabout ithem.
i(Wadsworth, i1996, ip. i149)
Indeed, ithe iuse iof iphysical iobjects iaids ichildren ito icreate imental ipicture iof
imathematical iconcepts iwhich ipromotes iconceptual iunderstanding i(Clements, i1999;
iDunlap i& iBrennan, i1979). iIn ithe isame imanner, iBassock i(1996), icited iby iGoldstone iand
iSon i(2005), istressed ithat iabstract ireasoning ican ibe isupported iby iconcrete imaterials
ibecause ithey ipromote itrue iinferences ifrom iphysical irepresentations ito iabstract
iprinciples. iThus, iinstruction ishould istart iwith iconcrete iexperiences ito ihelp istudents iin
icomprehending iabstractions i(Dunlap i& iBrennan, i1979), itransition ito isemi-concrete
imaterials ilike ipictures, iand ithen ito isymbols, ie.g. inumbers iand iletters i(Dunlap i&
iBrennan, i1979; iMercer, i1992; iMiller i& iMercer, i1993). iDunlap iand iBrennan i(1979)
ideveloped ia isequence iof ithree isteps ithat ienable istudents ito itransition ifrom iconcrete ito
iabstract iinstructional iprocedures iand iassist iteachers iin ideciding iwhether istudents iare
iready ito imove ito ithe inext ilevel.
i
At ithe ienactive ior iconcrete ilevel, ithe istudent imanipulates iobjects ito
icorrespond iwith imathematics isymbols. iThe istudent iinterprets imathematical isymbols
iand imanipulates iobjects ito iillustrate ithe iproblem. iThe iteacher iworks iwith ithe istudent
ito icreate ia imental ipicture iof ia igiven iconcept, iusing iblocks ior iother icounters ito isolve ia
inumber isentence igiven iorally. iNext, ithe istudent isolves inumber isentences iwith
imanipulative iobjects. iAt ithis ipoint, itextbook iexercises ican ibe isolved iwith ithe iuse iof
imanipulatives.
i
Once ithe istudent idevelops ian iunderstanding iat ithis ilevel, iinstruction imoves ito
ithe isemi-concrete ilevel iwhere ithe istudent iuses ipictures iof isets ito isolve imathematical
iproblems. iThen, ithe istudent ican itake imathematical isentences iand idraw ipictures ito
iillustrate ithe isets. iIn iother iwords, ithe istudent iinterprets ipictures iof isets iand irelates ito
imathematical isymbols. iThe istudent iinterprets imathematical isymbols iand idraws
ipictures iof isets.
i
After ithe istudent ihas imastered ithe iiconic ilevel, ihe ior ishe iis iready ito imove ito ithe
ihighest ilevel iof irepresentation, ithe iabstract ior isymbolic ilevel, iwhere ithe istudent
iunderstands imathematical isymbols ipresented ialone. iHere ithe istudent ican iread ia
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mathematical isentence, iform ia imental ipicture, iand iobtain ithe icorrect ianswer. iIf ia istudent
iis ihaving idifficulty iwith ia iconcept, iit imay ibe inecessary ito ireteach ihim ior iher i– imoving ithe
istudent ithrough ihis ior iher ipresent ilevel iof ilearning ito iensure iconcept iand iskill
icomprehension.
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The ithree ilevels iof ilearning iwere iemployed iin ithe ipresent istudy iwith ithe iend
iview iof isupporting istudents‟ itransition ifrom iconcrete ito ipictorial ito iabstract ithinking
i(visualization) iabout iaddition iof iintegers.
i
In ispite iof ithe ibenefits iof iusing imanipulative, iClements i(1999) icited isome
idisadvantages iof iutilizing iconcrete imaterials iin iteaching imathematical iconcepts.
iRelying isolely ion imanipulatives ias ia imeans iof iinstruction ican ialso ibe iineffective
ibecause istudents imay ilose ithe iopportunity ifor ideeper iconceptual iunderstanding
iwithout ifurther iformal idiscussion, iabstraction iand imathematical iconnection. iUsing
iconcrete imaterials idoes inot ialways imean ithe imathematical iideas iembedded iin ithose
iobjects iare ifully igrasped iby ithe istudents. iThey imay iinterpret ithe ipurpose iof ithe
iconcrete iobjects ifrom ithe iteacher‟s iintention. iClements inoted ithat isome iresearchers
ifound ithat istudents‟ iuse iof iabacus iand inumber iline idoes inot ialways imatch ithe imental
iactivity iintended iby ithe iteacher. iMoreover, iconcrete imaterials ican ieven ibe iused iby
istudents iin ia irote imanner. iThis idefeats ithe iinstructional ifunction iof imanipulatives
iwhich iis ito ifacilitate ithe iunderstanding iof ian iabstract imathematical iconcept. iLi
i(1999) iwarned ithat i“children ineed ito ihave ia ibalance ibetween imanipulative ipractice
iand iconcept iformation.”
i
Since imanipulatives ido inot iautomatically ilead ito iunderstanding, iteachers imust
i“reflect ion itheir istudents‟ irepresentations ifor imath iideas iand ihelp ithem idevelop
iincreasing
isophisticated
iand
imathematical
irepresentations”
i(Clements,
i1999).
iMoreover, imanipulatives ishould inot ibe iwithdrawn itoo iquickly ibecause istudents iwith
ilearning idisability imay ihave idifficulty iin imaking iconnections ito isemi-concrete iand
iabstract ilevels i(Kelly iet. ial., i1990). i Furthermore, iconcrete imaterials imust ibe icarefully
iselected ito iensure itheir iappropriateness iand iprovide ia isuccessful iand imeaningful
ilearning iexperience ifor ithe istudents i(Baroody, i1989).
i
Manipulative iuse ihas iindeed ibeen ifound ito iyield ipositive ioutcomes ifor
istudent iunderstanding iin ielementary iand imiddle ischool imathematics. iChester, iDavis,
iand iReglin i(1991) ifound ithat ithird igrade istudents iwho iwere ipresented igeometry
iconcepts iwith imanipulatives iscored isignificantly ihigher ion ithe iposttest ithan ithe igroup
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that iwas ipresented iconcepts iusing ionly idrawings iand idiagrams. iHer iresearch iwas
ilimited ibecause iit iwas ionly ia itwo-week istudy iof itwo i(2) ithird igrade iclasses.
i
In ia itwo-year icollaborative iaction iresearch istudy ithat icompared ithe iuse iof
imanipulatives iin isolving iequations iwith ithe iuse iof ia itextbook, iRaymond iand
i Leinenbach i(2000) ifound ipositive ioutcomes iof iusing imanipulatives ias ipart iof ithe
iinstruction ito iteach ialgebraic iequations. iFive iclasses iof ieighth-grade istudents,
iapproximately i120 istudents iunderwent ia imanipulative iprogram iconsisting iof i26
ilessons ithat iintroduced istudents ito ia imanipulative iapproach ito isolving ialgebraic
iequations, iand iguided ithem ithrough ian iintermediate ipictorial iapproach, iculminating
iin iengaging istudents iin iactivities ithat irelate ithe imanipulative ito ithe imore iformal i„high
ischool‟ ialgebra. iThe idata iwere icollected iusing iweekly istudent ireflections, ian iend-ofyear isurvey, istudent iwork isamples iand itest iscores, ia iwhole-class iinterview, iand
iteacher iobservations. iThe iresults iindicated ithat istudents iperformed iat ia ihigher ilevel
iwhen iusing imanipulatives. iAfter ifinishing ithe imanipulative ilessons, istudents ireported
ithat itheir iinterest iand iconfidence ion idoing ialgebra iimproved; ithey iliked iusing
imanipulatives ibecause i“it iwas ifun, ivery ihelpful iand ivery icomfortable, iand ilessons igot
ieasier.”
i
Sharp i(1995) iexamined ithe ieffects iof ialgebra itiles ion istudents‟ iabilities ito
iunderstand iequations. iThe ifindings irevealed ithat ithere iwas ino isignificant idifference
iin iunderstanding ibetween ithe igroup ithat iused ialgebra itiles iand ithe iones iwho idid inot.
iThe istudents iwrote iin ia idiary ithat ithey icould ivisualize ithe ialgebraic isituation ibetter
iwhen imanipulatives iwere iused.
i
Garrity i(1998) iinvestigated iwhether ithe iuse iof ihands-on ilearning, iwith
imanipulatives, iimprove ithe itest iscores iof isecondary ieducation igeometry istudents. iShe
idiscovered ithat istudents ihad ibetter iunderstanding iof igeometric iconcepts, ia imore
ipositive iattitude, iand ipreferred iusing ihands-on iactivities isuch ias imanipulatives iover
iusing imore itraditional ilearning imethods. iSteele i(1993) ireported isimilar ifindings iafter
istudying istudent iengagement iwith ithe imathematics icurriculum. iShe ifound ithat
istudents iwere imore iengaged iand imotivated iwhen iworking iin igroups iand iusing
imanipulatives ithan iwith icompleting idrill iand ipractice isheets.
i
The iuse iof iconcrete-to-representational-to-abstract i(CRA) iinvolves ithe iuse iof
imanipulatives, ipictorial irepresentations, iand iabstract ithinking. iMercer, iMiller, iand
iWitzel i(2003) istudied ithe ieffects iof iCRA ion isixth iand iseventh igraders iand iexamined
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the idata iusing irepeated imeasures iof ianalysis iof ivariance ion ia ipre-test, ipost-test, iand
ifollow-up. iThe iresults ishowed ithat ithe istudents ireceiving iCRA iinstruction idid ibetter
ithan ithose ireceiving ionly iabstract iinstruction.
i
Other ihands-on iactivities ihave ibeen ishown ito ihave ipositive ieffects ion
istudents‟ iunderstanding iof iequations. iThese iactivities iinclude iusing ithe ibalance
imodel, igraphing icalculators, iand imodels. iVlassis i(2002) istudied ithe ieffects iof ithe
ibalance imodel ion istudents‟ iunderstanding iof iequations iand ifound ithat ithe ibalance
imodel ican iincrease istudents‟ iunderstanding iof iapplying ithe isame ioperation iin inonarithmetical iequations.
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Manipulative idevices ior iconcrete imaterials ifacilitate iand ienhance ilearners‟
iunderstanding iof imathematical iconcepts, iand imotivate iand iactively iengage ithem iin
ithe ilearning iprocess i(Chester, i1991; iGarrity, i1998; iMercer, iMiller i& iWitzel, i2003;
iNalipay, i1995; iRaymond i& i Leinenbach, i2000; iSteele, i1993).
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METHODOLOGY
i
Research iDesign
i
This istudy imade iuse iof ithe ione igroup ipretest iposttest ipre-experimental idesign.
iIn isymbol, ithe ipre-experimental idesign iis:
i
R
O1
X
O2
O3
where: iR iis ithe itreatment igroup, iO1 iis ithe ipretest, iX iis ithe iremedial iteaching iwith ithe
iuse iof ithe iimprovised imanipulative, iO2 iis ithe iposttest, iand iO3 iis ithe idelayed iposttest.
i
Sample iiand iiSampling iiProcedure
i
The subjects of this study were the BEEd freshman istudents iat ithe iTarlac
iState iUniversity iCollege iof iEducation iduring ithe ifirst isemester iof ischool iyear i20082009. iPretest iwas iadministered ito i250 iBEEd ifreshmen. iSeventy i(70) iof ithem igot
iscores ithat ifell ibelow i–1 iz-score iand ithey iwere iconsidered ias ithe ilowest iperforming
istudents iin iaddition iof iintegers isince iz-scores ifrom i–1 ito i+1 ibelong ito ithe iaverage
ilevel. iOut iof ithese i70 istudents, i50 iwere irandomly iselected ias ithe isubjects iof ithe
istudy.
8
Data-Gathering iProcedure
i
After the administration of the pretest, the results were analyzed to determine
students‟ common errors and difficulties, which were addressed by the remedial
teaching with the use of the manipulative. iThe iresearcher itaught ithem iwith ithe iuse iof
imanipulative ifor ithree ihours. iHe iguided ithem iin icompleting ithe ilearning iactivities
because low performing students need their teacher‟s assistance to understand the
lesson i(Timbol, i2006). They were allowed to use the manipulative material even
when answering formative tests if they have not yet reached the abstract learning
stage proposed by Piaget. They were also given ample exercises to master addition of
integers. Their answers were immediately checked by their peers with the guidance of
the researcher in order to provide them feedback on their knowledge of the lesson.
The next day after the teaching period, the posttest was administered to
determine the effectiveness of the manipulative as a remediation device. Then after a
week, the delayed posttest was conducted to measure the skills retained among the
students. The manipulative was no longer used in the posttest and delayed posttest to
determine whether the students can visualize the processes in addition of integers.
Data-Gathering Instrument
A iteacher-made itest iwas iused ifor ithe ipretest iand ifor ithe iposttest. iThe itest iwas
ipresented ito ithree iexperienced imathematics iinstructors/professors iin iTSU ito ijudge
ithe iface ivalidity iof ithe itest. iTheir icorrections iand isuggestions iwere iconsidered. iThe
idelayed iposttest iis ia i25-item icompletion itest ion itwo-digit iand ithree-digit iaddition iof
iintegers. iAfter ithe iface ivalidation, ithe ipretest/posttest iwas ifurther ivalidated ithrough
ian iitem ianalysis iafter ia idry-run iof ithe isame, ito i50 isophomore iBEEd istudents. iThe
iresults iof ithe itest iwere isubjected ito iitem ianalysis ito idetermine ithe idifficulty iindices,
idiscrimination iindices iand ireliability icoefficient iusing iKuder-Richardson i20. The
pretest in addition of integers has a moderate difficulty level of 0.61, a discrimination
index of 0.64, and a reliability index of 0.90.
i
i
Statistical iTreatment
Frequency icount iand ipercentage iwas iused ito idetermine ithe ierrors iand
idifficulties iof istudents iin iadding iintegers. iThe imean iscores iof ithe ipretest iand iposttest
iwere icompared ifor isignificant idifference. iThe it-test ifor ithe idifference ibetween imeans
iof icorrelated idata iwas iused ito itest ithe inull ihypotheses. iAll icomputations iwere idone
iusing iMicrosoft iExcel.
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RESULTS iAND iDISCUSSION
Development iof ithe iImprovised iManipulative iMaterial
iThe imain imaterials iin iconstructing ithe imanipulative ibesides ihammer, isaw,
paintbrush iand i10 ione-inch inails iare ithe ifollowing:
i
Quantity
i2 ipieces
i2 ipieces
i10 ipieces
i50 ipieces
i50 ipieces
i20 imL
i
1)
2)
3)
4)
5)
6)
i
Description
i
18 icm i i1.5 icm i i1.5 icm iwood
I24 icm i i1.5 icm i i1.5 icm iwood
i25 icm iwire
iwhite ibeads iof idiameter i1 icm
iblack ibeads iof idiameter i1 icm
ired iwood ipaint
i
Total
i
Estimated iCost
P25.00
P25.00
P10.00
P60.00
The iprocedure iin iconstructing ithe imaterial iis ias ifollows:
1) iIn ithe imiddle iof ithe itwo ipieces iof i18 icm i i1.5 icm i i1.5 icm iwood,
imeasure iand imark i2.7 icm ifrom iboth iends. iThen imeasure iand imark i1.4 icm
ieach iin ithe iremaining ilength.
2.7
1.4 1.4 1.4 1.4 1.4 1.4 1.4 1.4 1.4
e
2.7
2) iUsing ia ione-inch inail, imake ia ihole i0.5 icm ideep iin ithe i10 imarks.
3) iWith ifour ipieces iof iwood, imeasure iand imark i1.5 icm ifrom iboth ilower
iends. iConnect ithis ipoint ito ithe iupper icorner imaking ia idiagonal iline. iThen
icut ialong ithis iline iusing ia isaw.
1.5
1.5
i
18 icm i i1.5 icm i i1.5 icm iwood
1.5
1.5
i
24 icm i i1.5 icm i i1.5 icm iwood
10
4) iUse ione-inch inails ito iattach ithe ifour ipieces iof iwood ito iform ia irectangular
iframe. iThen icoat ithe iframe iwith ired ipaint.
5) iInsert ifive iblack iand ifive iwhite ibeads iin ieach iwire. iThen iinsert iboth
iends iof ieach iwire iin ithe iopposite ihole i of ithe itwo ipieces iof iwood. iThe
ifive iupper irows ihave ifive iblack ibeads ifollowed iby ifive iwhite ibeads.
iThe ifive ilower irows iare ireversed i– ifive iwhite ibeads ifollowed iby i five
iblack ibeads.
Description iof ithe iDevice
i
The imanipulative idevice iused iin ithis istudy ito irepresent iaddition iof iintegers iis
imade iup iof iten iwires, ieach iwith itwo igroups iof ifive ibeads iin icontrasting icolors. iThe
ifirst ifive irows ihave ifive iblack ibeads, iwhich irepresent inegative iintegers, ifollowed iby
ifive iwhite ibeads, iwhich istand ifor ipositive iintegers. iThe itwo icolors iallow iimmediate
irecognition, iso icounting iis inot ineeded. iQuantities iare iconsidered i“entered” iwhen ithey
iare imoved ito ithe ileft iside. iThe ilast ifive irows iare ireversed: ifive iwhite ibeads ifollowed
iby ifive iblack ibeads, ipermitting iinstant irecognition iof imore ithan ifive itens ior ififty.
i
i
Figure i1 ishows ithe imodel iof ithe imanipulative imaterial, iwhich ishows i(–5) i+ i2.
i
Figure i1. iThe iManipulative iDevice
Using ithe iImprovised iManipulative iin iAdding iIntegers
i
The iimprovised imanipulative imaterial ihas ififty iblack iand ififty iwhite
ibeads. iBlack ibeads irepresent inegative iintegers iand iwhite ibeads irepresent ipositive
iintegers. iQuantities iare ientered iwhen ithe ibeads iare imoved ito ithe ileft.
i
11
i
Adding iTwo iNegative iIntegers
Example i1. iTo iadd itwo inegative iintegers, ifor iexample i(–4) i+ i(–7), i–4 iis
ientered ifirst iby imoving i4 iblack ibeads ito ithe ileft ias ishown iin i Figure i2. iThen i(–7) iis
ientered igiving ithe isum iof i(–11) ias iillustrated iin iFigure i3. iThis irepresentation iwill iaid
ithe istudents ito igeneralize ithe isum iof itwo inegative iintegers; iit iis ifound iby iadding itheir
iabsolute ivalues iand iaffixing itheir icommon isign.
i
i
Figure i2. iRepresentation iof i(–4)
i
Figure i3. iRepresentation iof i(–4) i+ i(–7)
iresulting ito i(–11)
Adding ia iPositive iInteger iand ia iNegative iInteger iand iVice-Versa
i
When iadding ia ipositive iand ia inegative iinteger, iit iis iimportant ito inote ithat ione
iblack ibead iand ione iwhite ibead icancel ieach iout. iSo i–1 i+ i1 ihas ia ivalue iof i0.
i
Example i2. iTo iadd i4 i+ i(–7), i4 iis ientered ifirst iby imoving i4 iwhite ibeads ito ithe
ileft ias ishown iin iFigure i4. iThen i(–7) iis ientered ias iportrayed iin iFigure i5.
i
i
Figure i4. iRepresentation iof i4
i
12
Figure i5. iRepresentation iof i4 i+ i(–7)
In iFigure i6, ithe ifour ipairs iof iblack iand iwhite ibeads iare iequal ito izero. iAfter
ireturning ithe ifour izero ipairs ito itheir ioriginal iposition, ionly ithree iblack ibeads iare ileft.
iThus, ithe ianswer ito i4 i+ i(–7) iis i(–3) ias idisplayed iin iFigure i6.
i
Figure i6. iAfter ithe i4 izero ipairs iare
iremoved, ithe ianswer ito i4 i+ i(–7) iis i(–3)
i
i
Figure i7. iRepresentation iof i(–4)
In iExample i2, ithe iaddends ihave iunlike isigns ibut ithere iare imore inegative
i(black ibeads) ithan ipositive iintegers i(white ibeads). iWhen ithe ipairs iof iblack iand iwhite
ibeads iare ireturned ito itheir iinitial ilocation, iblack ibeads iwill ibe ileft isince ithere iare
imore iblack ithan iwhite ibeads. iThe iact iof iremoving izero ipairs iof ibeads isuggests
isubtraction iof ithe iabsolute ivalues iof ithe iintegers. iThe isum iwill itake ithe isign iof ithe
iinteger iwith igreater iabsolute ivalue.
i
Example i3. iTo iadd i(–4) i+ i7, i(–4) iis ientered ifirst iby imoving i4 iblack ibeads ito
ithe ileft ias iillustrated iin iFigure i7. iThen i7 iis ientered ias idisplayed iin iFigure i8.
i
In iFigure i8, ithe ifour ipairs iof iblack iand iwhite ibeads iare iequal ito izero. iAfter
ireturning ithe ifour izero ipairs ito itheir ioriginal iposition, ionly ithree iwhite ibeads iare ileft.
iThus, ithe ianswer ito i(–4) i+ i7 iis i3 ias iportrayed iin iFigure i9.
i
In iExample i3, ithe iaddends ihave iunlike isigns ibut ithere iare imore ipositive i(white
ibeads) ithan inegative iintegers i(black ibeads). iWhen ithe ipairs iof iblack iand iwhite ibeads
iare ireturned ito itheir iinitial ilocation, iwhite ibeads iwill ibe ileft isince ithere iare imore iwhite
ithan iblack ibeads. iThe isum iwill itake ithe isign iof ithe ipositive iinteger.
i
13
i
Figure i8. iRepresentation iof i(–4) i+ i7
Figure i9. iAfter ithe i4 izero ipairs iare
iremoved, ithe ianswer ito i(–4) i+ i7 iis i3
i
Effectiveness iof ithe iManipulative iMaterial
i
The ieffectiveness iof ithe iimprovised imanipulative imaterial iwas idetermined
iby imonitoring istudents‟ iperformance iwhen iusing ithe imanipulative: ithe icorrected
istudents‟ ierrors, ilevel iof iimprovement iit icaused, iand iretention iof iskills iin iadding
iand isubtracting iintegers.
i
Corrected students’ errors in addition of integers
In the pretest,I the ipercentage iof ierrors icommitted iby ithe istudents iin ithe ieight
icategories iranged ifrom i37% ito i88%, iwith ian ioverall ipercentage iof ierror iof i68%
which reveals that they experienced much difficulty in addition of integers.
In the posttest, their errors ranged from 7% to 45%, which indicates that their
difficulty was reduced by 33% to 60% in some categories and was eliminated in
adding negative integers and zero, and two negative integers. The overall percentage
of errors of 24% was reduced by 44% in the posttest.
Level of improvement
Figure i10 ishows ithe iperformance iof ithe istudents iin iaddition iof iintegers. iIn ithe
ipretest, ithe idistribution iof itheir iscores iis ipositively iskewed isince imost istudents ihad
ilow iscores. iThis iverifies ithat ithe istudents iare ilow iperforming iin iaddition iof iintegers
i(mean i= i8.12). iOn ithe icontrary, ithe idistribution iof itheir iscores iin ithe iposttest i1 iis
inegatively iskewed isince imost iof ithe iscores iare ihigh i(mean i= i18.96). This high
performance of the students resulted from the use of the manipulative material. This
confirms that the use of the improvised manipulative material greatly improved their
i
14
performance in addition of integers. Moreover, their performance became even better
in the posttest 2 or delayed posttest (mean = 22.04) ias icompared ito ithe iposttest i1.
40
35
30
25
20
Pretest
Posttest 1
Posttest 2
15
10
5
0
Pretest
Posttest 1
Posttest 2
i
0-5
11
0
0
.6 - 10
29
3
0
.11 - 15
10
10
5
16 - 20
0
14
9
21 - 25
0
23
36
Figure i10. iStudents‟ iPerformance iin iAddition iof iIntegers
To ishow istatistically ithat ithe iuse iof ithe imanipulative imaterial isignificantly
iimproved istudents‟ iperformance iin iaddition iof iintegers, it-test ibetween imeans iof
icorrelated isample iwas iconducted. iTable i1 ishows ithe iresults iof ithe it-test.
i
The computed t-value for the difference between the means (10.84) of the
pretest and posttest was 15.7218, which was very significant at i0.00 ilevel ito ireject ithe
inull ihypothesis ithat ithere iis ino difference between their scores before and after being
taught with the use of the improvised manipulative material.
Table i1
iT-Test ibetween ithe iMeans iof iPretest iand iPosttest iin iAddition iof iIntegers
iPretest
i%
iPosttest
i%
iDifference
i%
it
idf
i
i
Mean
i
Std. Dev.
I
i
8.12
3.01
32.48
I
18.96
I
75.84
I
4.62
10.84
I
4.88
I
43.36
I
p
i
15.72 I49 I.00
I
I
The findings of the study that the use of manipulative material helps
students concretize and understand abstract mathematical ideas which result to
improved performance are similar to those of Chester i(1991), Cotter i(1996), Fueyo
and Bushel i(1998), Garrity i(1998), Mercer, Miller, and Witzel i(2003), Nalipay
i(1995), Raymond and Leinenbach i(2000).
15
Retention
Another indicator of the effectiveness of the improvised manipulative
material is the retention of skills in addition of integers the students gained.
Students‟ retention of skills is measured by comparing their posttest and delayed
posttest scores. iThese iscores iwere istatistically itested iusing it-test iof idifference
ibetween imeans iof icorrelated isamples. iThe iresults iare ipresented iin iTable i2.
Table i2
iT-Test iof iDifference ibetween iMeans iof iPosttest iand iDelayed iPosttest
iin iAddition iof iIntegers
iDelayed
iPosttest
i%
i%
iDifference
i%
it
idf
ip
iPosttest
iMean
I18.96
I75.84
I22.04
I88.16
I3.08
I12.32
i7.48
I49
I.00
iStd. Dev.
I4.62
I3.53
I2.91
i
The statistical analysis in Table 3 shows the significance of the difference
between means (3.08) in the posttest and delayed posttest in addition of integers.
The computed t-value is 7.4775. The probability of alpha accounted by t is 1.21 
10–9. iThis iis ivery isignificant ito ireject ithe inull ihypothesis ithat ithere iis ino difference
between their posttest and delayed posttest scores. The performance of the students
in addition of integers still improved significantly even a week after the teaching
session with the use of manipulative. Their scores further improved in the delayed
posttest because the use of the improvised manipulative promotes visualization and
enhances retention of the concepts and processes of integer addition.
The use of manipulative materials allows students to learn by doing.
Manipulative use has indeed been found to yield positive outcomes for learner‟s
understanding in different levels of mathematics learning from elementary to
college levels. The results of the study confirm the benefits of using manipulative
materials. They help college students understand and create mental images of
concepts and processes involved in addition of integers i(Dunlap i& iBrennan,i1979)
since it provides them a concrete basis in building, strengthening, and connecting
representations of mathematical ideas i(Clements,i1999). Students‟ learning is
enhanced since they are actively, physically and mentally involved in the learning
process. The use of the improvised manipulative also supports students‟ transition
from concrete to representational to abstract thinking (visualization) which
16
enhanced their retention about addition of integers i(Mercer, iMiller, i& iWitzel,
i2003).
In addition to promoting conceptual understanding and lasting learning of the
topics covered in the study, ithe iresearcher ialso iobserved ithat the students who used
the manipulative material were more engaged, motivated and participated actively
during the discussion i(Nalipay, i1995; iSteele, i1993). Although attitude toward
manipulative use was not directly evaluated in this study, the students commented
that they enjoyed using the manipulative because it is easy to use, fun, very helpful,
very comfortable, and highly interactive i(Garrity, i1998; iRaymond i& iLeinenbach,
i2000; iRust, i1999), and that their confidence in adding integers improved (Raymond
i& iLeinenbach, i2000).
Implications of the Study to Mathematics Teaching and Learning
The isubjects in the study who are college freshmen and have been taught how
to add integers since their first year high school still have numerous errors and
misconceptions. This means that their high school Math teachers failed to diagnose
and correct their errors.
Low performers have the ability to learn provided that they are properly
guided and appropriate materials and intervention strategies are used to address their
difficulty i(Berches, i2005; iDante, i2002; iGamido, i2001; iTimbol, i2006; iVasquez, i2005).
In this study, the use of the manipulative material is effective in improving students‟
understanding of integer addition. This suggests that mathematics teachers should
explore effective materials and methods of helping students, especially the low
performers comprehend math concepts so that they will be enabled to succeed in their
future math studies. One of the possible ways of helping them is through the use of
concrete materials.
Students often describe Math as a boring and terrifying subject. This is due to
their difficulty to visualize and understand its underlying symbolic concepts and
processes. The use of appropriate manipulative is one way of actively engaging them in
the learning process. They can meaningfully touch and move the concrete materials to
make visual representations of mathematical concepts and to see connections between
the abstract and concrete which promote systematic transition from concrete to pictorial
to abstract levels of thinking. When students understand the mathematical abstractions,
they will become confident and motivated in learning the subject which will make them
17
realize that Math is also interesting and enjoyable. Thus the use of manipulative is a
good device in promoting cognitive, psychomotor and affective learning
Furthermore, the use of manipulative materials supports the learning style of
kinesthetic learners who need to move some parts of their body to learn and of tactile
learners who need something to manipulate to fully understand the lesson.
In iplanning ito iuse imanipulative, ithe iteacher imust ibe isure ithat ithe imanipulative
imust iaccurately iillustrate ithe iactual imathematical iconcepts iand iprocesses ibeing itaught,
imust iinvolve imoving iparts ito iillustrate ia iprocess, iand imust ibe iused iindividually iby
ieach istudent. iMost iimportantly, ithe iteacher imust iknow ihow ito iuse ieffectively ithe
imanipulative isince ithe iuse iof imanipulative ialone iis inot ithe ikey ito isuccessful
iunderstanding. iThe imanner ia iteacher ipresents ithe ilesson iusing imanipulative ihas ia igreat
iinfluence ion ithe iresult i(Hinzman, i1997, icited iby iRoss, i2006; iMoyer, i2001).
i
CONCLUSION iAND iRECOMMENDATIONS
i
iThis istudy iaimed ito idevelop iand ivalidate ian iimprovised imanipulative imaterial
as iremediation idevice iin iteaching iaddition iof iintegers. iThe imanipulative imaterial is
effective in correcting students‟ errors and reducing their difficulties in addition of
integers. The high performance of the students in addition of integers is due to the use
of the manipulative material. This validates that the use of the manipulative material
highly improved their performance in addition of integers. The performance of the
students in addition of integers still improved significantly a week after the teaching
session with the use of manipulative. iThis isupports that the use of the manipulative
promotes visualization and enhances retention of the concepts and processes involved
in addition of integers.
i
Based lon lthe lfindings lof lthe lstudy, lthe lfollowing lrecommendations lare
lforwarded. lMathematics lteachers lshould l diagnose lstudents‟ ldifficulties lin ldifferent
lMath ltopics lin lorder lfor lthem lto lhave lbasis lin ldevising land limplementing
lappropriate
land
leffective
lways
lof
l helping
lthem
lovercome
ltheir
llearning
ldifficulties. lThe lmathematics lteachers lin lhigh lschool land lcollege lcould luse lthe
limprovised lmanipulative lmaterial lwith ltheir llow lperforming lstudents lto lhelp
limprove ltheir lstudents‟ lperformance land lretention lof lconcepts land l skills lin ladding
lintegers. lThis lalso laddresses lthe llearning lstyle lof lkinesthetic land ltactile llearners.
l
18
The lmathematics lteachers lare lencouraged lto ladopt land ldevise lmanipulative lmaterials
lthat lwill lfacilitate lstudents‟ lcomprehension lof labstract lmathematical lconcepts. lThis
lwill lnot lonly lenhance ltheir lcreativity land lresourcefulness lbut lwill lalso lmake lMath
lteaching land llearning leasy, linteresting land lmeaningful. lThe lmathematics lteachers
lshould lattend lseminar-workshops lin ldeveloping land lusing leffectively lof lmanipulative
lmaterials lthat lwill lcater lto lthe lvarious lneeds lof lthe llearners. lA lmodule lon laddition lof
lintegers lwith lthe luse lof lthe lmanipulative lmaterial lmay lbe ldeveloped land lvalidated
lfor lGrade l6 lpupils land lhigh lschool lfreshmen lwho lstudy lthe loperation lfor lthe lfirst
ltime. lSchool ladministrators lshould lsupport lthe lconduct lof lresearches lon lthe
ldevelopment lof lmaterials lfor lclassroom lteaching. lAuthors lof lMath lbooks, lespecially
lin lthe lelementary land lsecondary llevels lshould lintegrate lthe luse lof lappropriate land
lresearch-based lmanipulatives lin lpresenting lconcepts lin lelementary land lintermediate
lalgebra lso lthat llearners lcan lsee lrepresentations lof lthe lconcepts lin lconcrete land
lpictorial llevels. lThis lwill lallow lthem lto lmeaningfully lmanipulate lobjects lthat
lembody labstractions, land lto lsee lconnections lbetween lthe labstract land lconcrete
lwhich lpromote lsystematic ltransition lfrom lconcrete lto lpictorial lto labstract llevels lof
lthinking.
l
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