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Connecting Mathematics and Music in Preschool Education
UDK: 373.2 : [51 + 78]Review paper (Pregledni rad)
* Vesna Svalina, Ph.D., Assistant Professor,Josipa Vukelić, univ. bacc. paed.,
Osijek, [email protected]
AbstractIn this paper, we wanted to explore the essential components of mathematics and music and determine the possibility of integrating them into prescho-ol education. Basic mathematical concepts, on which the development of each child’s intellectual ability depends, are formed in the preschool age. By combining components of elements from mathematics and music, we can see their connection in terms of symmetry, values and measurements, and pattern recognition. Through various musical activities, children can acquire certain skills that precede the learning of mathematical operations. Thus, we can practice our comparison making mathematical skill with children by comparing the long and short tones, the treble and the deep tones, the loud and quiet sounds, and the mathematical skill of counting by performing suitable music games, rhymes, and songs in which numbers are mentioned.Counting in rhymes and songs helps a child in learning both the notion of numbers and mathematical operations such as addition and subtraction. Games that combine music and mathematics usually use music as a driving force for a productive and dynamic educational environment. Rhythm and melody help in the process of mathematical thinking as children receive infor-
* CorrespondingAuthor:VesnaSvalina,[email protected],Personalwebsite:www.vesna-svalina.net
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mation directly and as a whole. That is why it is important to connect music and mathematics in preschool education as often as possible.Keywords: early childhood education, games, mathematics, music, prescho-ol children, songs
IntroductionTheconnectionbetweenmusicandmathematicshasbeendebatedsinceancient
times.InancientGreece(around500BC),Pythagorasandhisdiscipleswereattheforefrontofthisandtheybelievedthatanumberisatthebasisofeverythingandthateverythingcanbeunderstoodbynumbersandtheirproportions.Pythagorasintro-ducedthenotionofamusicalintervalinthemusictheory,whichrepresentssimplemathematicalproportionsthatareobtainedbyvibratingstringswhoselengthsarepositionedincertainmathematicalproportions(Plavša,1981).
WhilemusicwasexclusivelyconsideredtobeaformofscienceinancientGree-ce,intheMiddleAgesitwasregardedasbothaformofscienceandanartform.Thestudyofmusicisassociatedwiththestudyofarithmetic,geometryandastronomy.Inmedievaltimes,theprevailingviewwasthatifgeometrygavebirthtoastronomy,thenarithmeticisthemotherofmusic(Chailley,2006).
Inthe16thcentury,musictheoristscontinuedtoargueforPythagoras’sonicnum-bersandtheclaimthatperfectchordswereactuallyintervalsonthemusicalscaleexpressedbytheratiosofnumbers1,2,3,and4.However,thereisstillalongwaytogo.ItalianmusictheoristGioseffoZarlinodeterminedtherelationshipbetweentheindividualstagesofthescaleonthebasisofacousticmeasurementsandcalculations,proportionsofsmallandlargethirds,purequartersandquintilesindifferentpartsofanoctave,hedistinguishedtwobasictypesofchords–amajorandminorthird(partitionintothemajorandminorscale)andproposedthattheoctavebedividedintotwelveequalparts(Partch,1974).
Nowadaysmusicisnolongerviewedasabranchofsciencebutasabranchofart,butthedebateabouttheconnectionbetweenmusicandmathematicscontinues.Mu-sictheoristsandscientistshavefoundthatthereisacorrelationintermsofrhythmandmeasure,intermsofscaleformationandmusicalinstrumenttuning.Toneco-lourisassociatedwiththedistributionofaliquottonestrengths(Andreis,1967),andthedissonanceisaconsequenceoftheshocksgeneratedbyclosealiquotfrequencies(Plomp,andLevelt,1965).
Throughouthistory,composerswerefascinatedbymathematicsandnumbers,and therehavebeenmathematicianswho lovemusicandareactively involved in
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music.JohannSebastianBachwasacomposerwhosemusicalworksshowunusualnumbersandsymmetriesthatcanbemathematicallyexpressed.Thesymbolismofnumbersisexpressedeveninhislastname–BACHwhichisexpressedasnumber14:B=2+A=1+C=3+H=8.Thesumofthesenumbersis14.Number14isalsoincorporatedintohisotherworksthroughthenumberofnotesinthebeatorthenumberofbeatsinthecomposition.Thus,wehave14canonsofGoldberg Va-riationsand14contrapunctusinDie Kunst der Fuge(Currie1974;Rumsey1997;Tatlow,2015).
Ontheotherhand,AlbertEinstein,aprominentGermantheoreticalphysicist,saidthatheoftenthinksaboutmusicandviewshislifeinmusiccategories.Einsteinemphasized that ifhewerenotaphysicist,hewouldprobablybeamusician.Hespentmuchofhisfreetimeplayingmusicandoccasionallyperformedasaviolinistatlocalconcerts.MusicwasatypeofpleasureforEinsteinbutalsoaninspirationforhisscientificwork(Foster,2005;White,2005).
Many authors have written about the connection between mathematics andmusicandthepossibilitiesofcombiningthemineducation(Cheek,1999;Vaughn,2000;Church,2001;Shilling,2002;Fauvel,Flood, andWilson,2006;Geist, andGeist,2008;McDonel,2015).Thebasicsofmathematicalthinkinghavebeenapartofthepreschoolcurriculumintheworldformanyyears,moreprecisely,apartoftheindispensableeducationalparadigmforthepurposeofdevelopingandstrengtheningthelogical-mathematicalintelligence.Music,ontheotherhand,reachesyoungmin-dsmoredirectly–phenomenologically.Rhymes,songsthattellastoryandsoundsofdifferentinstrumentsdevelopthemusicalintelligenceofchildren,buttheyalsoprovidediversityinteaching.
Mathematics in Preschool EducationInordertobeabletobringmusicandmathematicsclosertopreschoolchildren,
weneedtoidentifywhatmainareasofmathematicsarerelevantfortheireducation.Tothisend,wewillusetheclassificationrecommendedbytheNational Council of Teachers of Mathematics1.Thisclassificationappliestopreschoolchildrenuntilthesecondgradeofprimaryschool.Numbersandoperations,algebra,geometry,andmeasurementanddataanalysisaretheareasofcompetencerelatedtoproblemsol-ving,communication,reasoning,connectingandrepresentation.
1 The National Council of Teachers of Mathematics(NCTM)isthelargestmathemati-caleducationorganizationintheworldandoperatesintheUSandCanada.
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Numbers and Operations
Gettingacquaintedwithnumbersandtheirpossibleconnectionsisthebasisandthebeginningoftheformalmathematicaleducation.Therefore,thefieldofnumbersandoperationsisthemainbasisforlearningothermathematicalfields,aswellasforlearningothereducationalfields,especiallymusicandlanguage.
Oneofthenaturalabilitiespresentinhumansistheabilitytoperceiverelativequantities,i.e.torecognizethedifferenceinquantitywithinthedomainofvisualperception.Children,however,mustlearnconventionalpairingandcountingmet-hodstodetectthesedifferencesmorereliably.Inthethirdyearoftheirlife,childrendeterminetheequalityoftwogroupsofobjectsbasedontheirimmediateproximity,andintheirfourthyeartheycanidentifypairsofthesegroupstoconfirmthattheyareequal(Clements,Sarama,andDiBiase,2004).Bycounting,thechildrencom-binethewordandtheveryconceptofnumbertocomeupwithmorecomplexcon-cepts;orderandsize.
Additionandsubtractionintheirbasicformdependonthelearnedcountingskill(Vlahović-Štetić,andVizekVidović,1998).Childrenwholearntocountaccuratelycanclearlydistinguishbetweenthesequenceofsizesandthepossibilityofincrea-singanddecreasingthem,butstillhaveadifficultywithcountingback,indicatingthatthesubtractionprocessismoredifficulttounderstand.Subtractionisbestun-derstoodbyvisualizingtaskswithobjectsonthetableorwithchildrenintheroom.
Thenextdevelopmentstepisassemblyanddisassembly.Preschoolersunderstandtheterms“whole”and“partwhole”.Thefivelinedobjectsclearlyformawhole,andwith theirgrouping thechildrennotice thecreationofseparateunits fromwithintheinitialunit.Therefore,theywillquicklyunderstandthatthenumber5containsthenumbers3and2withinitself.Assemblinganddisassemblingisunderstoodasacombinationofthevisualizationofnumbersandthecountingprocess,andassuchenableschildrentodisassembleandassembleunitsintoallpossiblevariantsandisaprecursorformorecomplexanalyticalthinking.Whenitcomestonumbers,childrenshowanextremefascinationwithbignumbers.Asthelearningprocessofcountingbeginsinthesphereofcountingto10,theareaofhighernumbersandthesystemoftensandhundredsmakesmorerulesforthepotentialassemblyanddisassembly.
Groupingistheprocessofcombiningobjectsintosetsthathaveanequalnumberofcomponents.Asetof20itemscanbedividedinto5setsof4items.Thisprocessleadstoanunderstandingofmultiplication,segmentcountingandmeasuring.Be-causelargenumberscanbehardertounderstandandcanbeconsideredasacombi-nationoftwoseparatenumbers,groupingnumbersintotenshelpswiththistask.Inadditiontocountingbynumbers,inthiswaychildrenlearntosegmentnumbersthat
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areapartofalargerseriesand,asaresultofagroupexercise,managetodealwithdouble-digitadditionoperations.
Geometry
Geometryplaysanimportantroleinthedevelopmentofachild’sspatialorien-tation.Thinkingaboutandunderstandingspaceandmovementasasetorawholehasaninfluenceonthemovementofthechildandtheperceptionofspaceastheirareaofactivity.Theaspectofgeometrythatisthemostevidentinachild’sdailylifeisashape.Thebasictwo-dimensionalformshavebeenlearnedbytheageoffouryears,and,beforestartingthefirstgrade,childrenarealsofamiliarwiththebasicthree-dimensionalforms.Inadditiontothebasicshapes,childrenclearlynoticethespecificsofpolygonsormultifacetedthree-dimensionalshapes.Theynotice,aswithnumberoperations,thatmorecomplexshapesarecomposedofcomplexshapessuchastrianglesorsquares.Asearlyasthefourthyearof life,childrencomeupwithwaystocombineshapestocreatenewones.Attheageoffiveandsix,theyclearlyrecognize similar shapesof different sizes anddifferentlyplaced shapes and canreproducethemincustomcomputerprograms.
Averyimportantpartofspatialorientationisthevisualizationofforms,i.e.thementaldelineationofformsforthepurposeofanalyzingthem.Throughvisualiza-tion,childrenmapthespacearoundthem,determinethedirectionoftheirownandothers’movement,andcreatereferencepointsinthespace.Thesementalimagesoftwoand three-dimensional shapes familiarize thechildrenwith theworldaroundthem,whethertheyarespacesorobjects.Theynoticeregularitiesandirregularitiesinrelationtoprototypeexamplesofforms(Clements,2001;Ho,2003).
Measurement
Thefirst typeofmeasurementoccursby learning tocount, it isupgradedbyadoptingconceptslikelength,weight,orheight.Bythethirdyearoflife,childrenwilldevelop,tosomeextent,anunderstandingoftheseconcepts.Theyhavediffi-cultyinnoticingthedifferencesintheweightsofobjectsmadeofthesamematerialbyvisualobservation,andinsomecasestheyareunabletocomparethelengthofthetwoobjectsbyplacingthemsidebysidefromthestartingpoint(Clements,Sarama,andDiBiase,2004).Attheageoffourandfive,theabilityofchildrentocompareobjectsbyperceptualmeasurementisgreatlyimproved.
Theabilityofthechildtopredicteffectsofhisactionsoncertainobjectsinspacedependsonhisunderstandingofthedifferenceofscalarsizes.Althoughthisunder-standingisseeminglysimpleandintuitive,itmakesitpossibletomakefurthercom-parisonsbyusingreferenceobjectsandtools.Measurementasamethodofanalysis
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andinferencerequiresspecificreferentsorunitsofmeasurement.Thesereferentscanbeofanyarbitrarysize,suchasthesizeofachild’spalmorastandardizedunitofmeasure.Childrenwilllearnthelatterbyusingcentimetrecubesorrulers,toolsthathaveonlyrecentlybecomeusedintheteachingofearlyandlatepreschoolers.Ofcourse,usingsuchtoolsfurtherdevelopsfinemotorskillsandcounting,andtheoutcomeistheabilitytomeasureaccurately(Clements,Sarama,andDiBiase,2004).
Algebra and Patterns
Unlikecounting,geometry,andmeasurement,algebraisnotasemphasizedasanactivefieldofteachingforpreschoolers,andplaysamorecomplementaryrole.Preschoolers’algebraicknowledgesumsuptothepatternrecognitionskillthatisoftheutmostimportancefortheirfurtherintellectualdevelopment.Theabilitytono-ticepatternsdevelopsfromanearlyage,andisobservedinthechild’senvironmentbycomparing,sortingandanalyzingobjectsandbehavioursofchildrenandadults(Mason,Graham,andJohnston-Wilder,2007;Kieran,2018).
Data Analysis
Likealgebra,dataanalysisisnotattheforefrontofdevelopingachild’smathe-matical ability.Data analysis involves classifying, organizing, demonstrating andusinginformationforthepurposeofaskingquestionsandprovidinganswers.Chil-dren sort agroupof items, suchasbuttonsor toys, intogroups according to thepatternstheyhaveobservedinordertodistinguishthemfromothergroups.Theyclearlystatethereasonsfortheirsortingandaskquestionsaboutthepurposeofcer-taindifferences.Childrensorttheirconclusionsbysimplecategoriessuchascolour,size, shape,volume,and thuscombine theability tonoticepatternsandestablishspecificrelationshipsofthings(SelimovićandKarić,2011).
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Music in Preschool EducationMusicalskillsofapreschooleraredevelopedbysinging,performingrhymes,
(active)listeningtomusic,performingrhythmsandmelodiesonrhythmicandmelo-dicpercussion,andcreativemusicalexpression.Atthisage,childrenareintroducedtomusicalconceptsthatrelatetotheexpressiveelementsofapieceofmusic(rhythm,tempo,dynamics,tone,melody,andharmony)throughanapproachofcoursethatisappropriateforpreschoolers.
Tone
Toneisabasicelement,amaterialusedbythemusicarts.Unlike thesoundsthatsurroundusinnature,whicharisefromtheimpropervibrationofelasticbodies,toneisbornfromthepropervibrationofsoundsources(Andreis,1967).Scottpointsoutthat“thedevelopmentofunderstandingofpitchandpitchrelationsisthekeytoapproachingmusicinourculture”(Scott1979,p.87).
Asaphysicallyexplainedphenomenon,toneispoorlyunderstoodbychildrenandismostlyidentifiedwiththetermsound.Onlywhenthetoneisplayedindividu-ally,andinsequencewithothertones,dochildrenslowlyperceiveitassomethingseparateandseparable,buttheypracticeitthroughthewhole,i.e.songsandrhymes.
Whenattemptingtoproducehighpitchedsounds,childrenwilloftentrytoreachthoseheightsbyraisingthechestwithvisiblestrain.Sam(1998)statesthatachildlearnsthisheightbylisteningandplaying.First,thetoneisunderstoodbylistening.Thesametonewillonlybesungwhenthechildisabletodevelophisvocalcords.Intonationreceivesgreaterattentionattheageoffiveandbeyond.Propersingingislearnedusinghearing,andtonesarenolongerabstract,andtheirdurationisaclearlyobservedpattern(Vidulin,2016).
Volume is also avery important componentof a tone that childrengenerallycategorizeintermsofcomfortanddiscomfort.Strongorweaktonescauseexcessiveorinsufficientstimulusinthechild,soamoderatepitchisbestsuitedforpreschoolchildren(Sam,1998).Inadditiontothevolume,thecolourofthetoneisanenhancerof theaestheticexperienceofmusic.Childrennoticeveryclearly thesamesongsandnumbersplayedondifferent instrumentsormedia,butalsonotice thecolourdifferencestheyusuallydescribeinrelationtotheirownexperiencesrelatedtothisnewsource.
Rhythm and Meter
Childrenhaveaspecialinteractiveencounterwithmusicbyusingthebodyasaninstrumentorapercussioninstrument(OrffInstrumentarium).Bydoingthis,the
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childrenareintroducedtotheconceptofrhythmandmeterforthefirsttime,butalsotothecreationofmusic,whichgivesusaninsightintothemotorskillsofthechildcombinedwithhearingandmusicalexperience.Inadditiontoinstruments,rhymesareusedforlearningrhythmandmeterthroughasong.Rhymesteachchildrentofollowacertainmeterandtofollowthedifficultandeasiermeasuringunitatacer-tainspeed(Sam1998).Rhymelearningbeginsintheyoungestkindergartengroupsandplaceslessdemandsonthechildwhenitcomestohearingdevelopment.Evenbeforecoming tokindergarten, children sway, swing, jump, shake, andgenerallymovewiththerhythmofmusic,andtheinfantrespondstotherhythmwithbodyandlimbmovements(Zentner,andEerola,2010).Rhythmicityinchildreniscau-sedbyamovementthatallowsquicklearningofrhymesandthepotentialfornewwordsandrhythmizing,i.e.newrhymesthatcanbecomposedwithchildren.Thephenomenon,whichisgenerallyapplicablebutalsoexpressedinchildhood,oftwoormorepeoplerepeatingthesamerhythmtoharmonizetheirbreathing,heartrate,brainwaves,attentionandmovements,isalsointeresting.Suchanobservationonceagainconfirmsthephenomenologicalpowerofmusicandtheinfluenceithasontheemotionalandpsychophysicaldevelopmentofhumans,sincetheharmonizationofrhythminducesasenseofcommunity,empathyandsocialcohesion.
Tempo and Dynamics
Slow,moderate,fastandveryfastarethebasisformarkingthetempoofeachcomposition,andthereareseverallevelsofsubtledifferencesbetweenthem.Tem-platetagsthatareunderstandableinpreschoolareusuallyslowandfast.Thisdicho-tomyisquicklylearned,andattheageoffiveorsixitisreasonabletoexpectthatchildrenwillalsobeabletodevelopasenseofamoderateperformancespeed(Sam,1998).Childrennoticeveryquicklythatthespeedoftheperformancechangestheverycharacterofthesong,soafastersongwillevokeanoptimisticandjoyfulemo-tionalstateinthem,whileslowersongswillhavearelaxingormemorableimpact(Vidulin,2016).
Aswas already noted in the chapter on tone perception, the ideal volume isbetweenaweakandastrongtone.Liketempo,thedynamicshavebasiclabelsthatdividethemintoveryquiet,quiet,medium-quiet,medium-strong,strongandverystrong,and the ideal range forpreschoolerswouldbebetweenmedium-quietandmedium-strong.Theaestheticexperienceofdynamicsalsoinfluencestheemotionalchangesinchildren.
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Melody
Amelodyisaseriesoftoneswithadifferentpitchanddurationthatareperfor-medconsecutivelyandformameaningfulwholeintermsofmusiccontent.Fromtheexampleofmelodyformation,wecanseethatmusiclivesintheconnectionandparticipationofdifferentelementsofrhythm,harmony,formalframe,dynamicsandcolour(Andreis,1967).
Playingsimpletunesinpreschoolrequiresacertaindegreeofrhythmandinto-nation.Bylisteningtothemelody,thechildreceivesalltheinformationwithoutbe-ingawareofeachindividualelement.However,playingatuneinpreschoolrequiressomeadjustmenttothecomponentsofthetuneforeaseoflearningandteaching,andasufficientlydevelopedvoiceapparatus.
Itisveryimportanttodistinguishbetweenthequalityofaestheticstandardsandtheability tomonitormelody structureswhen listening toa child’sperformance.Onlyafterwehaveestablishedthatthechildhaslearnedthebasicsforfollowingatune,wecanrecognizethedevelopmentofmusicalabilitiesofanindividualchild(Drexler,1938).
Harmony
Harmonyrepresents thechordalorverticalstructureofapiece.Theelementsofharmonyarechordsthatarecontrarytothenotionofmelodicflow.Chordsarecreatedbythesimultaneoussoundofaseriesoftonesthatareplacedoneabovetheother.Theyareproducedoneaftertheother,andeachofthemisgivenameaninginrelationtothechordsthatprecedethem,i.e.theonesthatfollow(Andreis,1967).
Makinganexampleforthismusicalelementforpreschoolersisnotaneasytasksince theconceptofharmony isquiteabstractand the termsof intervals like thethirdandthefiftharetoospecific,butstillnotimpossible.Theeasiestwaytoshowharmonyisthroughadisharmonicrenditionofalreadyknownsongs.Childrenwilleasilynoticethatthecompositionisnotperformedinthe“right”wayorthatitsoun-ds“uncomfortable”.Ifwechangethetonality,childrenwillrecognizethesimilarity,and,inmostcases,willnotconsiderthetransposedsongamistake.However,thisdoesnotmeanthatchildrenwillnotconsiderthesongamistakeifweplaythesongathirdorafourthbelowtheoriginal,sincethesongwillbequitedifferentfromtheoriginal.
Harmonyhasbeenthefocusofmanyresearches,andresultsrelatedtochildrenhavegenerallyproducedduplicate results.Costa-Giomi (1994) states thatharmo-nicskillsonlyemergeattheageofeightornine,andthecurrentpracticeofmostteachersandthemethodologicalliteratureareconsistentwiththeseconclusions.In
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contrast,somestudies(Moog,1976;Zimmerman,1993;Welch,2002)havefoundthatpreschoolerscandistinguishbetweentoneandchord,hearchangesinchords,anddistinguishharmonyfromdisharmony(Berke,2000).
Mutual Elements of Mathematics and Music Afterdeterminingthespecificsofbothareas(mathematicsandmusic)wecan
showacross-sectionofsimilarelementsandtheiruseinpreschooleducation.Themathematical elements of counting, geometry, algebraic patterning, data analysisandmeasurementarecontained,tosomeextent,inaspectsofmusicalelementsliketone,rhythm,melody,tempoanddynamics,andharmony.
Thefoundationforallthemusicalandmathematicalelementslistedsofar,aswellasallhumanfunctions,isinthecerebralactivity.Therefore,thefirstlinkbetweenmusicandmathematicsishiddeninthefunctionsofthebrainareaformathematicalandmusicalthinkingandcreativity.Nowadays,thereisageneralknowledgethattherightcerebralhemisphereisasetofareasinchargeofcreative,artistic,spatialandholisticcharacteristicsandactions,andtheleftdominatesinstructure,organizati-on,analyticalapproachandlogic.So,whatwasdiscoveredbyusingbrainmappingtechnology?
The“musicbrain”consistsofcomplexandwidespreadneuralsystems,aswellaslocallyspecializedareasinthebrain,andtheresultsofinitialstudieswereredu-cedtoactivitiesintherighthemisphereonly.Themostrecentdiscoveriesindicatehemisphere connecting as one of the characteristics specific tomusic processingandaction(Hodges2000).Musicalpotencydoesnotstopatthispoint,sincesomestudieshave shown that engaging inmusiccausesvariouschanges in thehumanbrain.DonaldA.Hodges(2000)summarizesimportantfindingsfromthestudyofmusicandcerebralactivity,andstatesthefollowingpremises:1)thehumanbrainrespondstoandparticipatesinmusic;2)the“musicbrain”startsperformingatbirthand lingers throughoutone’s life:3) continuousmusicpractice frombirthaffectstheorganizationof the “musicbrain”;4) the local specializedareasof themusicbrainare:cognitivecomponents,affectivecomponentsandmotorcomponents;5)the“musicbrain”isextremelyresilient.Thelastpremiseisgenuinelyinterestingasitrelatestothepersistenceofhighmusicalfunctionsinspiteofmental,emotionalordegenerativedifficulties.
The“mathematicsbrain”,likethe“musicbrain”,isnotlimitedforoperationinonlyonehemisphere.However,mathematicalfunctionsaregroupedintheparietallobeandseeminglythe“mobility”ofthesefunctionsislowerthanforthemusicalonesthatoccurinalmostallpartsofthebrain(Cranmore,andTunks,2015).
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Itisalsointerestingthattheactivityofmathematicalareasisstrongerinchil-drenthanadultswhencalculating.Thereasonforthisistheunderstandingofthesemathematical actions that are understoodmore generally andmore abstractly bychildrenthanbyadults,whohaveadoptedtheseprocessesandcansolveproblemswitheaseandlessactivity(Rochaetal.,2005).
Sincemosttasks,whetherrelatedtomathematicsormusic,areintertwinedwithlanguage,movementandemotion,itisimpossibletoignoretheinterconnectionsofallthesesystems.Onepossiblelinkbetweenthemathematicalandthemusicalbrainis thecorrelationofgeometric thinkingand intensemusical exercise. Ithasbeenobservedthatpeoplewhoareactivelypracticingmusichaveabetterunderstandingofbasicgeometricsystems(Spelke,2008).MRItestshaveshownthatmusiciansareabletomanipulatecomputationwithfractionsbetterthanothersbecauseofanincre-aseintheirmemoryandanimprovementintheabstractionofnumericalquantity(Schmithorst,andHolland,2004).Theareawheretheintersectionofmathematicalandmusicalprocessingoccursistheprefrontalcortex,whichisassociatedwithexe-cutiveactionsandmemory,aswellasemotionalreactions.
Perhapsthemostcommonandwidespreadlinkbetweenmusicandmathema-ticsinpreschoolisnoticingpatterns.Musicisahuman’sfirstcontactwithpatterns,whether it isourmother’sheartbeat, listeningtomusicasanewborn,or learningsongsinkindergarten,ourbrainsnoticemusicalpatterns.Thus,musichelpschildrenperformmathematical tasksevenwhenchildrendonotexperience these tasksassuch.Musicisasocial,naturalanddevelopmentallyappropriatewaytoacceleratetheprocessoflearningmathematics(Geist,Geist,andKuznik,2012).Melodyandrhythm,asitscomponents,areclearexercisesthatenhancetheabilitytorecognizepatterns.Itisenoughjusttoexposechildrentomusic,andwecanalreadynoticeacumulationofprogressintermsofpatterns.Wecanseethisatthebeginningoftheharmonynoticingprocess since children in preschool canonly absorb it throughpassivelistening.
Aswithsymmetry,becauseoftheimpossibilityofintroducingtheterminologyofmusictheory,wecannotclearlyvisualizethevaluesoftones,tempo,ordynamicsto children, butwe can relate them to themagnitudes they learned inmathema-ticsandstrivetobringtheseconceptsauditorily.Thechildrenwillclearlyclassifyheightsof two tones,especially if theyareplayed inanascendingordescendingsequenceofatone.Scalesaregreatteachingmaterialsforlearningthevaluesandproportionswithinmusicaschildrenrecognizethesequencestheylearnedbycoun-ting,andconnectdifferentsegmentsofrhythmandmelodywithaddition.
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Songs and Games with Elements of Mathematics and Music
Beforeachildcanlearnandapplymathematics,heorshemustacquirecertainskillsthatprecedethelearningofmathematicaloperations.Makingcomparisonsisanimportantmathematicalskillthatwecanpracticethroughmusicactivities.Whatkindsofcomparisonsdoesmusicinclude?Childrencancomparefastandslowbeats,longandshorttones,highanddeeptones,andloudandquietsounds.
To compare loud and quiet sounds,Voglar (1980) suggests playing theLittle Drummer. In thisgame theeducator is thepuppetLuta,whohas receivedanewdrumasagiftandhitsitalldayandnight.Heplaysloudlyduringtheday,andplaysquietlyatnight,soasnottodisturbotherdollswhiletheysleep.Children(whoalsoplaydolls)shouldwatchthedrummingoftheLutadollandguesswhenitisdaytimeandwhenitisnight.Ifthedrummerisplayingloudly,thepuppetswakeup,accom-paniedbythemovementsofthedrum.Themovementsaresolid,strongandener-getic.Whenthedrummerisplayingquietlythedollsarestillmovinginaccordancewith the rhythm,but themovementsareperformedmore softly,moregentlyandpreparingforsleep.Inthisgame,theeducatorcanperformarhythmthatinvolvestonesofdifferentdurations,butthedynamicsneedstobealteredtobeeitherloudorquiet.Thegameisrepeatedseveraltimes.Ifolderchildrenareinvolvedinthegame,oneofthemmayassumetheroleofthedrummer’sdoll.Figure1showsexamplesofrhythmsthatcanbeperformedonadrum.
Figure 1 Examples of rhythm that can be played on a drum during the game Little Drummer
Throughvariousmusicgameschildrencanexperienceandcomparehighandlowtones.TheHigh and Low Tonesgameisperformedwithchildrenstandingby
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theirchairs.Whentheyhearthelow-octave(a)toneplayedbytheireducatorontheinstrument,everyoneshouldsquat.Whentheyhearthetoneinthefirstoctave(a1)everyoneshouldstandup.Thegamecanalsobeplayedusingblackandwhitepaper.Whenchildrenhearalowtone,theyraisetheblackpaperandwhentheyhearahightone,theyraisethewhitepaper(Domonji,1986).
Wecanalsoperformacertainrhythmonadrumorbyclappingslowly.Wethenaskthechildrentomovealongthatrhythmandtoaligntheirstepswiththepace,atwhichtherhythmisperformed.Afterthatwewilldotheopposite–wewillperformthesamerhythmatafastpace,andthechildrenwilladjusttheirmovementstotherhythmofthenewpace.Thisactivitycanalsoberelatedtoastory.
TheBear and the Beesgamecanbeusedtocomparelongandshorttones.Thegamebeginsbytellingthestoryofabearandabee.Thebearwokeupfromawintersleepand,beingveryhungry,decidedtolookforfoodintheforest.Theeducatorshows that thebear iswalkingslowlyandasks thechildren tomove in thesameway.Eachstepgoestothefirstperiodinthemusicaltact(halfnotes)(Figure2).Thebearsuddenlysawabeehive.Hetookahoneycombfromonebeehiveandstartedeating it.At thatmoment thebees escaped from thebeehive andbegan to circlearoundthebearwhilestinginghim.Whendescribingthemtheeducatormovesfastlikethebees,andoneachperiodperformstwosteps(eightnotevalue)(Figure2).Theeducatorasksthechildrentomoveinthesameway,withtheirarmsheldonthesides,withelbowsbent,asiftheyhadwings.Thebearroared,ranawayandwasnolongerapproachingthebees.Thegamecancontinuewiththeeducatoralternatingtherhythminlongerandshorterdurations,withtherhythmbeingthebearforthelonger duration and then asking the children tomove slowly.When the educatorstartsperformingarhythminshorterdurations,representingbees,childrenshouldmovequickly.Thiscanbechangedseveraltimes(Stefanović,1958).
Figure 2 Rhythms representing the bear and bees in the game The Bear and the Bees
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Childrencanbeinvolvedincomparisongamesbylettingthemlistenandcreateadifferentsoundorrhythmthanwhatweperformedforthem.Forexample,wecanperformalongandhighpitchandaskthechildrentorepeatit.Thenweaskthemtomaketheoppositesound,therefore,lowandloud.Wecanalsoaskthechildrentoperformthesoundonpercussioninstruments.Theywilldothisbyproducingaquietsound,thenaslowsound,thenfastandfinallyslowrhythm.Throughouttheseexercises,thechildrenimprovisetherhythm.
WiththesongSeven Steps2(Figure3),childrendeveloptheabilitytoverballycountfromagivennumberbackandforth.Thepoemcountsfromonetoseven,onetothree,seventooneandseventofive.Bysingingthissong,thechildrenwillfirstlearnthewordsthatrepresentthenumbers,andlatertheywillbegintounderstandthattheyareinterconnected.Numbersonethroughsevencanbecutfromacollageofpaperandhungagainstawallsothatchildrencanindicatethemwhiletheysing.Performanceofthesongcanflowinthewaythatonechildshowsnumbersfromonetosevenandanotherchildshowsseventoone.
Childrencanalsopracticesubtractionthroughvariousmusicgamesandbysin-gingdifferentsongs.OneofthesegamesiscalledLittle Train.Childrenaredivided
2 SuperSimpleSongs–KidsSongshttps://www.youtube.com/watch?v=pTLtcno5_cY
Figure 3 Seven Steps sheet music
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into four groups of five children,which then form a queue of “wagons” in eachgroup,withthefirstchildinthequeuebeingthelocomotive.ChildrenthensingthesongLittle Train (Figure4).
Thentheeducator’scommandfollows:“Thelastwagonsshouldgototheirpla-ce”.Thelastchildinthequeuefromeachgroupgoestotheirplace(Figure5).Theeducatorthenasksthechildrenhowmanywagonsareleft.Childrenrespondaloud–three,two,one,zero.Thegamegoesonuntilthewagonsaregone.Bygraduallyreducingthenumberofchildreninthegame,wewillgetasubtractionexercise.Inaddition to the song, this gamealso featuresmovement and spacehandling.Thecombinationoftheseelementsenableshigh-qualitypatternrecognitionwiththede-velopmentofmotorskills(Čupić,Sarajčev,andPodrug,2017).
WithamusicalgamecalledTen Green Bottles(andasongofthesamename),childrencanpracticesubtractionuptothenumberten(Figure6).Thepoemconsistsofonlyoneverse.Theonlythingthatchangesisthenumberofbottles.Inthefirstversetherearetenbottles,inthesecondnine,inthethirdeight,etc.Thesequencecontinuesuntilthenumberofbottlesiszero.Thefinalverseendswith“Therewillbenogreenbottleshangingonthewall.”Forthismusicgame,itisnecessarytopre-parenumbersfromonetotenonseparatecardsthatchildrenwillputaroundtheirnecksbeforethegamestarts.Childrenwillbeincircleformation,holdinghands,dancingandsingingTen Green Bottles3.
3 BBC–SchoolRadio–CountingSongshttps://www.bbc.co.uk/programmes/p038bdqh
Figure 4 Little Train sheet music
Figure 5 Arrangement of children in space during the performance of the game Little Train
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Eachtimethechildrensaytheverse,“Ifagreenbottleshouldaccidentallyfall”,thechildwhohasthesaidnumberwillfalltothefloor.Therefore,achildwhohasthenumber10fallsonthefloor,childwhohasnumber9willfallafterhim,etc.Whenthechildrenarebetteracquaintedwiththenumbers,theycanplaythegamewithoutcards,sothateachchildrememberswhattheirnumberis.Everythingelseisdonethesamewayaswhenthegameisplayedwithcards.
Conclusion In thispaper,wewanted to explore the essential componentsofmathematics
andmusicanddeterminethepossibilityofintegratingtheminpreschooleducation.Althoughmathematicsandmusicdifferintheirformalteachingmethodsinpres-chool,thereareundeniablelinksthatmaketheminterestingpartnersfordevelopingcognitiveanalyticalskills.
Thephenomenologicalpotencyofmusicandtherationalharmonyofmathema-ticsareevidentinnumerousgames,rhymesandsongs.Throughmusicalactivities,childrencanacquirecertainskillsthatprecedethelearningofmathematicalopera-tions.Wecanpracticetheskillofmakingcomparisonswithchildrenbycomparingthelongandshorttones,thetrebleandthedeeptones,theloudandquietsounds,andtheskillofcounting,addingandsubtractingthembyperformingappropriatemusicgamesorrhymesandsongsthatmentionnumbers.Rhymesandsongshelpconnectthebeatwithactionsandnumbers.
Musicenhancestheoverallbrainactivityandisappropriatefordifferentareas.However, it should be borne inmind that themusical components are explainedpreciselythroughmathematicaltermsandconcepts.Thatiswhyitisimportanttointegratemathematicsandmusicasoftenaspossibleinpreschooleducation.
Figure 6 Ten Green Bottles sheet music
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ReferencesAndreis,J.(1967).Vječni Orfej [The eternal Orpheus].Zagreb:Školskaknjiga.Berke,M.K. (2000).The ability of preschool children to recognize chord changes and
audiate implied harmony. Unpublished doctoral dissertation, University of Arizona,Tucson.
Chailley,J.(2006).Povijest glazbe srednjega vijeka [History of middle ages music].Zagreb:Hrvatskomuzikološkodruštvo.
Cheek,J.M.,andSmith,L.R.(1999).Musictrainingandmathematicsachievement.Adoles-cence,34,136,759-761.[GoogleScholar]
Church,E.B.(2001).Themathinmusicandmovement.Early Childhood Today, 15,4,38-43.[GoogleScholar]
Clements,D.(2001).Mathematicsinthepreschool.Teaching Children Mathematics,7,270-275.Doi:https://doi.org/10.5951/TCM.7.5.0270
Costa-Giomi,E.(1994).Recognitionofchordchangesby4and5yearoldAmericanandArgentinechildren.Journal of Research in Music Education, 42, 1,68-85.Doi:https://doi.org/10.2307/3345338
Clements,D.H.,Sarama,J.,andDiBiase,A.(2004).Engaging young children in mathema-tics: standards for early childhood mathematics education.Mahwah:Erlbaum.[GoogleScholar]
Cranmore,J.,andTunks,J.(2015).Brainresearchonthestudyofmusicandmathematics:AMeta-Synthesis.Journal of Mathematics Education, 8,2,139-157.[GoogleScholar]
Currie,R.N.(1974).ANeglectedguidetoBach’suseofnumbersymbolism-partI.Bach,5,1,23-32.[GoogleScholar]
Čupić,A.,Sarajčev,E.,andPodrug,S.(2017).Edukativneigreunastavimatematike[Edu-cationalgamesinmathematicsteaching].Poučak: časopis za metodiku i nastavu mate-matike,18,70,66-80.[GoogleScholar]
Domonji,I.(1986).Metodika muzičkog vaspitanja u predškolskim ustanovama [Methods of music education in preschool institutions].Sarajevo:Svjetlost.[GoogleScholar]
Drexler,E.N.(1938).Astudyofthedevelopmentoftheabilitytocarryamelodyatthepres-choollevel.Child Development,9,3,319-332.Doi:10.2307/1125444
Fauvel,J.,Flood,R.,andWilson,R.J.(2006).Music and mathematics: from pythagoras to fractals.NewYork:OxfordUniversityPress.[GoogleScholar]
Foster,B.(2005).Einsteinandhisloveofmusic.Physics World,18,1,1.[GoogleScholar]Geist,K.,andGeist,E.A.(2008).Doremi,1-2-3:That’showeasymathcanbe:usingmusic
tosupportemergentmathematics.Young Children,63,2,20-25.[GoogleScholar]Geist,K.,Geist, E.A., andKuznik,K. (2012).The patterns ofmusic: children learning
mathematicsthroughbeat,rhythm,andmelody.Young Children, 67,1, 74-79.[GoogleScholar]
Hodges,D.A.(2000).Implicationsofmusicandbrainresearch.Music Educators Journal, 87, 2,17-22.Doi:https://doi.org/10.2307/3399643
Ho,S.Y.(2003).Youngchildren’sconceptofshape: vanHieleVisualizationLevelofgeo-metrixthinking.The Mathematics Educator,7,2,71-85.
https://repository.nie.edu.sg/bitstream/10497/65/1/TME-7-2-71.pdf
V. Svalina and J. Vukelić: Mathematics and Music... napredak161 (3-4) 411-430 (2020)
428
Kieran,C.(Ed.)(2018). Teaching and learning algebraic thinking with 5 to 12 year olds: the global evolution of an emerging field of research and practice.NewYork,NY:Springer.[GoogleScholar]
Mason, J.,Graham,A., and Johnston-Wilder, S. (2007).Developing thinking in algebra.London:Sage.[GoogleScholar]
McDonel,J.S.(2015).Exploringlearningconnectionsbetweenmusicandmathematicsinearlychildhood.Bulletin of the Council for Research in Music Education, 203,45-62.Doi:https://www.jstor.org/stable/10.5406/bulcouresmusedu.203.0045
Moog,H.(1976).The Musical Experience of the Pre-school Child. (trans.C.Clarke)Lon-don:Schott.[ISBN-10:0901938068];[ISBN-13:978-0901938060]
Partch,H.(1974).Genesis of a Music.NewYork:DaCapoPress.[ISBN-13:9780306801068]Plavša,D.(1981).Muzika: prošlost, sadašnjost, ličnosti, oblici [Music: past, present, perso-
nalities, forms].Knjaževac:NotaKnjaževac.[GoogleScholar]Plomp,R.,andLevelt,W.J.M.(1965).Tonalconsonanceandcriticalbandwidth.Journal
of the Acoustical Society of America,38,548-560.Doi:https://doi.org/10.1121/1.1909741Rocha,F.,Rocha,A.,Massad,E.,andMenezes,R.(2005).Brainmappingsofthearithmetic
processinginchildrenandadults.Cognitive Brain Research,22,359-72.Doi:https://doi.org/10.1016/j.cogbrainres.2004.09.008
Rumsey,D.(1997).Bachandnumerology:‘Drymathematicalstuff’?Literature & Aesthe-tics,7,143-165.[GoogleScholar]
Sam,R.(1998).Glazbeni doživljaj u odgoju djeteta [Music experience in child education]. Rijeka:Glosa.[GoogleScholar]
Schmithorst,V. J., andHolland, S.K. (2004). The effect ofmusical training on the ne-ural correlates of math processing: a functional magnetic resonance imaging studyin humans.Neuroscience Letters, 354, 3, 193-196. Doi: https://doi.org/10.1016/j.neu-let.2003.10.037
Scott,C.R.(1979).Pitchconceptformationinpreschoolchildren.Bulletin of the Council for Research in Music Education,59,87-93.https://www.jstor.org/stable/40317550
Selimović,H.,andKarić,E.(2011).Učenjedjecepredškolskedobi[Thelearningprocessofpreschoolchildren].Metodički obzori,11,6,145-160.[GoogleScholar]
Shilling, W. A. (2002). Mathematics, music, and movement: exploring concepts andconnections. Early Childhood Education Journal, 29, 3, 179-184. Doi: https://doi.org/10.1023/A:1014536625850
Spelke,E.(2008).Effectsofmusicinstructionondevelopingcognitivesystemsatthefoun-dationsofmathematicsandscience.InAsbury,C.,andRich,B.(Eds.),Learning, Arts, and the Brain(pp.17-49).NewYork/Washington,D.C.:TheDanaFoundation.[ISBN:978-1-932594-36-2]
Stefanović,B.D.(1958).Metodika muzičkog vaspitanja predškolske dece [Methods of music education of preschool children].Beograd:Nolit.
Tatlow, R. (2015). Bach’s Numbers: Compositional Proportion and Significance. Cam-bridge:UniversityCambridge Press. [Online ISBN: 9781316105061];Doi: https://doi.org/10.1017/CBO9781316105061
Vaughn,K.(2000).Musicandmathematics:modestsupportfortheoft-claimedrelationship.Journal of Aesthetic Education,34, 3-4,149-166.DOI:10.2307/3333641
429
V. Svalina and J. Vukelić: Mathematics and Music... napredak161 (3-4) 411-430 (2020)
Vidulin,S.(2016).Glazbeniodgojdjeceupredškolskimustanovama:mogućnostiiograni-čenja[Musiceducationofchildreninpreschoolinstitutions:opportunitiesandlimitati-ons].Život i škola [Life and School], 62, 1,221-233.[GoogleScholar]
VlahovićŠtetić,V.,andVizekVidović,V.(1998).Kladim se da možeš – psihološki aspekti početnog poučavanja matematike [I bet you can - psychological aspects of initial te-aching mathematics].Zagreb:Korakpokorak.
Voglar,M.(1980).Kako muziku približiti deci [How to bring music closer to children].Beo-grad:ZavodzaudžbenikeinastavnasredstvaBeograd.[ISBN:86-17-05425-5]
Welch,G.F. (2002).Earlychildhoodmusicaldevelopment. InBresler,L.,andThompson,C.(Eds.),The Arts in Children’s Lives: Context, Culture and Curriculum(pp.113-128).Dordrecht,NL:Kluwer.[ISBN1-4020-0471-0]
White, P. (2005). Albert Einstein: the violinist.The Physics Teacher, 43, 286-288. Doi:10.1119/1.1903814
Zentner,M.,andEerola,T.(2010).Rhythmicengagementwithmusicininfancy.Proceedings of the National Academy of Science, 107,13,5768-5773. Doi:10.1073/pnas.1000121107
Zimmerman,M.P.(1993).Anoverviewofdevelopmentalresearchinmusic.BulletinoftheCouncilforResearchinMusicEducation,116,1-21.
https://www.jstor.org/stable/40318552
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Povezivanje matematike i glazbe u predškolskom odgoju i obrazovanju
SažetakU ovom smo radu željeli istražiti bitne sastavnice matematike i glazbe te utvr-diti mogućnost njihove integracije u predškolski odgoj i obrazovanje. Osnovni matematički pojmovi, o kojima ovisi razvoj intelektualnih sposobnosti svakog djeteta, formiraju se u predškolskoj dobi. Kombinacijom komponenata ele-menata iz matematike i glazbe možemo vidjeti njihovu povezanost u smislu simetrije, vrijednosti i mjerenja te prepoznavanja uzoraka. Raznim glazbenim aktivnostima, djeca mogu steći određene vještine koje prethode učenju matematičkih operacija. Tako možemo vježbati matematičku vještinu uspoređivanja, uspoređujući s djecom duge i kratke tonove, visoke i duboke tonove, glasne i tihe zvukove i matematičku vještinu brojanja izvodeći prikladne glazbene igre, rime i pjesme u kojima se spominju brojevi. Brojanje u rimama i pjesmama pomaže djetetu u učenju pojma broja i ma-tematičkih operacija kao što su zbrajanje i oduzimanje. Igre koje kombiniraju glazbu i matematiku obično koriste glazbu kao pokretačku snagu produktiv-nog i dinamičnog obrazovnog okruženja. Ritam i melodija pomažu u procesu matematičkog razmišljanja jer djeca primaju informacije izravno i u cjelini. Zato je važno što češće povezivati glazbu i matematiku u predškolskom od-goju i obrazovanju. Ključne riječi: glazba, igre, matematika, pjesme, predškolska djeca, rani od-goj i obrazovanje