connecting mathematics and music in preschool education

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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]

[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

V. Svalina and J. Vukelić: Mathematics and Music... napredak161 (3-4) 411-430 (2020)

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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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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

connecting mathematics and music in preschool education

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