sabbatical report by amy gaudia, © june 2019 › ... › sabbaticalreportgaudia2019.pdf ·...
TRANSCRIPT
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SabbaticalReportbyAmyGaudia,©June2019
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MathandMusicConnections:AnexplorationofhowbasicmathematicalskillsmaybeimprovedbyincorporatingmusicandmusicalconceptsintotheAdultBasicSkillsmathclassroom
Part1:FrettingAboutFrets
“Allthingswhichcanbeknownhavenumber;foritisnotpossiblethatwithoutnumberanythingcaneitherbeconceivedorknown.”PhilolausinthefourthcenturyBC
IhavebeenteachingGeneralEducationDevelopment(GED)classesandcoordinating
tutorsintheAdultBasicandSecondaryEducation(ABSE)departmentforover23
years,andhaveexperiencedboththeexcitementandthechallengeofteachingand
interactingwithstudentsatallskilllevelsinmathandlanguagearts.Myassignmentfor
thelastfouryearshasbeenMath1andIamconstantlysurprisedtofindthatmanyof
thesestudentsarelackingthemostbasicarithmeticskills.Likemostinstructors,Iam
alwaysworkingtoimprovemyinstructionandtofindnewandbetterwaystohelpmy
studentsaccesstheirlearningpotentialandgaintheskillstheyneedtomoveforward
towardearningtheirGEDsand/orplacingintocreditclasses.
Onepositiveandinspiringaspectofworkingwiththispopulationisthatmanyofthem
areveryopentotryingnewthingsandgivingthemselvesachancetounderstandthe
interestingmathematicalconnectionsthatIhavepointedouttothem.Forexample,
whenshowingthemaspiralmadefromtheFibonacciseries,theybecamemore
attentiveandcuriousaboutthelessonthatfollowed.Ihavealsobrieflyand
spontaneouslytalkedaboutmusicalrhythmsastheyrelatetosomebasicfractions,and
foundthesametobetrue–greaterinvolvementintheclasslessonandotheractivities.
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AndthusIstartedexploringthepossibilityofincorporatingmusicalconceptsintothis
basicmathclass.
Ifoundthiswonderfulimagetopostonmyofficedoor:
EverydayIwouldarriveatmyofficeintheearlymorning,andstudytheimagewhile
unlockingthedoor.IknewwhatIwaslookingat.Afterall,Ihavebeenplayingthe
guitarfor45yearsincludinganinformalstudyofmusictheory.Idefinitelyhada
thoroughunderstandingthatthehalfwaypointalongaguitarfretboardproduceda
soundanoctavehigherthantheopenstringaswellasthecorrespondingpitch
frequencies.Butalltheothernumbersonthisimagewerepuzzling.Ihadageneral
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ideaofwhatitwasallaboutbutIcouldnotmakesenseofallthoseratios.Insteadof
researchingit,IwascertainthatifIjuststaredattheimageandpondereditlong
enough,Icouldfigureitallout.Couldthisbewhatsomeofmystudentsaredoingin
class?
Andsothefirstweeksofmysabbaticalprojectweredevotedtoplacingmyselfinthe
roleofstudentwithaswirlingcloudofnumbersbeforeme,andadeterminationto
solvethemystery.IreadseveralaccountsofthetuningsystemdevelopedbytheGreek
mathematicianPythagorasandalthoughthisissometimescreditedtosomeofhis
contemporaries,itisusuallyreferredtoasPythagoreantuning.Hebelievedthatmusic
hadadivinerelationshipwithnumbers.Thestoryistoldthatwhenheheardthe
differentsoundsproducedbythehammersoftheblacksmithsworkingnearby,hewas
curiousaboutthefrequenciesofthepitchesandhowsomeofthepitchessoundedvery
nicetogether.Helearnedthatthemassesofthehammerswereinsimpleintegerratios,
whichhethenattemptedtoduplicatewithstringsattachedtoablockofwood.He
theorizedthatallconsonantsounds,notesthatsoundpleasantoragreeabletothe
humanear,havefrequenciesinsimplesmallintegerratioswithoneanother.Hefirst
dividedthestringinhalfandfoundthatitmadeabeautifulandconsonantsounding
pitch.Itisknowntodayasanotethatisanoctavehigher,andcanbecomparedto
voiceswhenayoungchildandamancansingtogetheronthesamenote.Whenhethen
dividedastringat2/3thelengthoftheoriginal,itsoundedbeautifulwhenplayed
simultaneouslywiththeoriginalfull-lengthstring.Infact,itwassolovelythattodayit
iscalleda“perfectfifth.”InWesternmusic,thisisthefifthnoteofan8note,ordiatonic,
scale.Pythagorascontinuedbuildinghisscalebystackingthe2/3and1/2ratiosina
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processthatdoeseventuallyrevealtheotherratiosthatwereconfusingtome.For
example,hewantedanotethatwouldfitwithinthescaleofthelowandhighoctaveand
havethesame2/3ratiowiththehighoctave.Sohetookthe½lengthstringand
multipliedbytheinverseof2/3.(Heusedtheinversebecausehewascreatingastring
thatwouldbelongerthatthe½lengthstring.)1 /2 x 3/2 = 3 /4
Andsoastringwithalength¾oftheoriginalstringbecamewhatisknowntodayasthe
fourthnoteinthediatonicscale.
Theguitarimagewasstillpuzzlingmeforanotherreason.Withsomebasicknowledge
ofpitchfrequencies,andharmonics,IwonderedhowthosePythagoreanratiosareused
tomeasurethedistanceofallthefretsonmodernguitars.Myunderstandingwasthat
theratiosofPythagorasresultedinasystemofnotesandscalebuildingthatwould
eventuallynotworkoutfortransposingtootherkeysandotheroctaves.Forexample,
theintervalcreatedbythefirstandthirdnotesofhisscale(nowknownasamajor
third)wereslightlyoffanddissonant.Andso,equaltemperamenttuningwas
developedtoremedythoseproblemsbyshorteningandlengtheningthePythagorean
intervalsasneeded.AccordingtotheEncyclopediaBritannicaandOxfordMusicOnline,
thetheoryofequaltemperamentwasfirstpublishedin1584byChuTsai-Yuofthe
Mingdynasty.Itgrewinpopularityduringthefollowingtwocenturies,andbecame
quitefavoredbysomecomposerssuchasJ.S.Bach,wellknownforhisworkTheWell-
TemperedClavier,acollectionofpreludesandfugueswritteninall24keys.Thereare
manytuningsystemsfoundinotherpartsoftheworldthatbuildscalesofsmaller
intervals,orspacesbetweenthenotes.InIndiantheory,theoctaveisdividedinto22
tones,orsrutis.Thiswillbeagreatsubjectforfurtherinvestigation.
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InMathematics,Music,andtheGuitarbyDavidHornbeck,theauthorexplainsingreat
detail,howthemethodusedbyPythagorasresultedinadjacentnotesofthescalenot
beingequallyspaced.Thismadeitdifficulttotransposemusic.Transposingis
sometimesneededformusictobeplayedorsunginahigherorlowerkey.Hegoeson
toillustratethedevelopmentoftheformulausedin12toneequaltemperament(12
TET)tuning.
AlthoughthisequationisfarabovetheleveltaughtinmyMath1class,theideaofguitar
buildingaspartofamathcurriculumisquiteappropriate.TheNationalScience
FoundationSTEMGuitarProjecthasbeendoingthisforover8years.
“DuringtheNSFgrantcycles,theSTEMGuitarProjecthasexceededinitialestimatesof
facultyimpactedbyrecruitingover450STEMeducators,withanadditional500faculty
exposedvianationaleducationconferences.Thusfar,thiseffortisimpacting
over20,000studentsnationallyoverthe8yearsbecauseoffacultymembersadopting
oradaptingthecurriculumdevelopedthroughtheproject.”
MyfirstintroductiontotheSTEMGuitarprojectiswhatinspiredmetotrymyown
didgeridoobuildingproject,whichinturnresultedinmyinterestinfurtherexploring
thepossibilitiesofinvolvingmusicinthebasicskillsmathclass.Thedidgeridooisan
indigenousAustralianhollowtubewindinstrument.Intheproject,themusicconcept
thatweexperiencedwaspitchanditsrelationshiptothelengthofthetubes.Inthe
futureIwouldliketoexpandtheprojectintobuildingasimplemarimba,apercussion
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instrumentwithasetofwoodenbarslikeakeyboard.Ipredictthiswillbeavery
engagingwayforstudentstostudyPythagoreantuningandtheconceptofratios.
Inmypersonalmusicallife,Inowhaveagreaterunderstandingofthedifficultywe
musiciansoftenfaceintuningourinstrumentsandwhymusiccansometimessound
betterinonekeycomparedtoanotherkey.Ialsohaveadeeperunderstandingofhow
thesechallengesareexperienceddifferentlyonmyfrettedinstrument,theguitar,
comparedtothe(unfretted)cello.
Thisistheformulapresentlyusedforfretspacingonguitars:
Dn=[(L–Dn-1)÷17.817]+Dn-1
https://archive.siam.org/careers/pdf/guitar.pdf
PartII:TheQuadrivium
“Musicissonaturallyunitedwithusthatwecannotbefreefromitevenifwesodesired.”--Boethius
MyinterestintheQuadrivium,thefourclassicalliberalartsofnumber,geometry,music
andcosmologyislargelyduetoitsfocusonthesoulorinmyterms,the“wholebeing.”
DuringtheMiddleAgesandrightonthroughtheRenaissance,peopleattributed
everythingtothedivineandfoundmeaningfulconnectionsbetweennumbers,beauty,
planetarymotions,sounds,andallofthenaturalworld.Dr.ChristopherPerrinof
ClassicalAcademicPressdescribesclassicaleducationas“thecultivationofaffections
orloves”and“thecultivationofthesoulontruth,goodnessandbeauty.”
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Inmyworkteachingbasicskillstoadults,Ihavefoundmanystudentstobeverytuned
intophilosophicalideas,spirituality,mysticism,andpersonalgrowthtopics.My
experiencewiththemhasledmetobelievethatsomething“opensup”andtheybecome
morepresentandinvolvedwhenmakingmathematicalconnectionstoart,music,
beauty,andmuchmore.Healthandnutrition,socialactivism,andmanyotheraspectsof
lifebecomemoreinteresting.Imaginethatyouare28yearsold,sittinginawindowless
classroomwitholdwornoutwalls,tryingtomemorizebasicmultiplicationfacts!
Somethingisgreatlyneeded.
ItisBoethius(SaintAniciusManliusSeverinusBoëthius)c475CE,whoiscreditedwith
thefirstliteratureonteachingthefourmathematicalsciences:arithmetic,music,
geometry,andastronomy.Hewasdeeplyintriguedbythephilosophicalaspectofthe
studyofarithmeticandmusic.Boethiuscontributedabodyofwritingonmathematics
thatgreatlyinfluencedtheintellectualevolutionoftheWestduringthehighMiddle
Ages,about1000CE–1250CE.Hewasnotthatinterestedincreatingnewconcepts.
Onthecontrary,hewasfascinatedbytheideaoffindingeternallawsintheuniverse.
DeInstitutioneMusicaisanessaywrittenbyBoethiusandsometimesreferredtoashis
musicaltreatise.Itwasverypopularamongphilosophersandmusictheoristsofthe
BaroqueandRenaissanceperiodsduetoitsfocusontheinterconnectednessofmusic,
planetarymotions,ethics,mind,soulandaninclusivephilosophyofharmony.He
classifiedmusicinthreeparts:
Musicamundana–musicoftheworld;tobeunderstoodwithoutbeingheard
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Musicahumana–body/mind/spiritharmonyMusicainstrumentalis–instrumentalmusic
Theintertwiningofarithmeticandmusic,tohim,providedthebasisforunderstanding
theuniverse.Unfortunatelyhewasimprisonedandexecutedfortreasonasaresultof
hiseffortstoconfrontthecorruptionfoundintheRomancourts,andaccordingtothe
StanfordEncyclopediaofPhilosophy,hewasalsoaccusedofengaginginmagic.
However,duringhisfinalyearsBoethiuswrotetheConsolationofPhilosophy,forwhich
heismostwellknown.
Inthefinaldaysofmysabbatical,IhappeneduponTheCambridgeHistoryofWestern
MusicTheory,CambridgeUniversityPress2008.Thisbookwassomehowmade
availableonlineasafree1002pagePDF.Notonlyisthereachapterthatspecifically
goesintogreatdetailonmusictheoryandmathematics,aquickreviewofitsentire
contentsrevealsaverythoroughanddeeplyresearchedaccountofallthetopicsIhave
beenexploringinthisproject.Insection5,Thetransmissionofancientmusictheoryinto
theMiddleAgesbycontributingauthorCalvinM.Bower,Ifoundthisreferencetothe
workofMarcusTulliusCiceroandtheplatonictraditions;“Theratiosthatgovernedthe
highestorderofthephysicaluniverseandthemetaphysicalworlditselfwerethosethat
determinedmusicalconcord,andthedegreetowhichsensualmusicwasshapedby
theseratios,wasthedegreetowhichthesoulwasledawayfromranksensualityto
contemplateeternaltruths.”Ibelievethatdevelopingagreaterunderstandingofthe
historyofmusictheoryfromthisperspectivewillprovidearichandmeaningful
foundationfortheintegrationofmath&musicincoursecontent.
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Allofthesciencesasweknowthemtodayhavebeenrelatedtonumber,andinsome
sense,tomusic.Theage-oldconnectionbetweenmusicandsciencedatesbacktothe
ancientGreeks.ItwasPhilolausinthefourthcenturyBC,whostated,“Allthingswhich
canbeknownhavenumber;foritisnotpossiblethatwithoutnumberanythingcan
eitherbeconceivedorknown.”
Throughouttime,thinkersandthosewhowerephilosophicallyinclinedhave
recognizedsomeconnectionbetweenthemechanismsofthephysicaluniverse,
especiallythemotionsofcelestialbodies,andtheestheticsofmusicalharmonyand
rhythm.BeginningwithPythagoraswhothoughtthattheSun,theMoonandthevisible
planets"hummed"astheypursuedtheirpathsinthesolarsystemandcontinuingon
throughJohannesKepler'sLatinworkHarmonicesMundi,(TheHarmonyoftheWorld)
in1619,tothemasterpiecebyGuyMurchiein1961,MusicoftheSpheres,musicand
sciencehavebeenseenaspartofoneorganicwhole.
ByKepler'stime,astronomyhadalreadybeenseparatedfromastrology,largely
becauseoftheinventionofthetelescopetowhichKeplercontributedgreatly.Withthis
newtoolandhisbetterunderstandingandhisdiscoveryoftheso-called"thirdlawof
planetarymotion,"Keplersawmusicalharmonyingeometricalformsandphysical
phenomena.Hehadakeeninterestinmusic,whichincludedfamiliaritywithgreat
composersofhistime,andhehadanalmostmysticalbeliefinwhathecalled"the
mournfulsong"oftheEarth'smotionaroundthesun.
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Kepler’snotation(copyofacopyfromtheoriginaltextofHarmonicesMundi)
AmodernreproductionofKepler’simage
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Kepler’sideasinfluencedIsaacNewton,whohadstudiedmusicaltextandonce
commented,“Pythagoras’sMusickoftheSphereswasgravity.”In1961,GuyMurchie
wroteMusicoftheSpheres,whichfollowedhisSongoftheSkyin1954.Boththese
bookswerecombinationsofscientificclevernessandmusicalunderstandingand
appreciation.Bothbuiltuponandspeculatedabouttheage-oldlinkbetweenthehard
dataofthephysicalworldandtheetherealbeautyofmusic.
Apollo,theMuses,theplanetaryspheresandmusicalmodes
https://commons.wikimedia.org/wiki/File:The_music_of_the_spheres.jpg-
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https://joedubs.com/the-platonic-and-pythagorean-solids/
PartIII:ThePowerofMusic
Backtool’Pythagoras!Thepowerofmusicwassoacceptedbytheancientsthatithas
beennotedthattheyusedcertainmelodiestopromoterestfulsleep.Differentmusical
modeswereuseddependingonwhatwasneededandsouponrisingafteradeepsleep,
inordertoleavebehindthestupor,theywouldlistentomusicinamodethatwould
stimulatealertness.Pythagorastaughtthatyoucouldhealbyusingsoundand
harmonicfrequenciesandheisbelievedtobethefirstpersontoprescribemusicas
medicine.
Inthebook,ThisisyourBrainonMusic,DanielJ.Levitintakesusonajourneyintothe
modernresearchandanalysisofwhatmayhavealreadybeenbelievedinancient
Greece.Hestates,“Fromaone-dimensionalcontinuumofmoleculesvibratingat
differentspeeds,ourbrainsconstructarich,multidimensionalpitchspacewiththree,
four,orevenfivedimensions(accordingtosomemodels).Ifourbrainisaddingthis
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manydimensionstowhatisoutthereintheworld,thiscanhelpexplainthedeep
reactionswehavetosoundsthatareproperlyconstructedandskillfullycombined.”
Levitinprovidesasummaryofmusichistory,ofhowweareaffectedfroma
neuroscienceperspective,andhowweevolvedtohavemusicalpreferences.Hemakes
referencetoresearchonfetusesandhow,accordingtoAlexandraLamont,co-authorof
ThePsychologyofMusicalDevelopment,atoneyearofageababywillrecognizeasong
theyheardinutero.Levitinsupportsthisbysayingthatourbrainsareimprintedby
everysongwehear.Mostinterestingtomeiswhenheelaboratesquiteabitonthe
subconsciouspredictionsthatwearemakingwhenlisteningtomusicandhowthis
determineshowmuchorhowlittleweenjoythesound.IcansaythatIdon’tentirely
agreewiththat.Frommypersonalexperience,inparticularwhenlisteningtothemusic
ofJ.S.Bach,Ihaveexperiencedsomethingquitedifferent.Forexample,thefirsttimeI
listenedtoBach’sPreludefromPartitaBWV997,therewasnothingatallpredictable
andinfactitalmostsoundedliketheopeningnoteswerefromoneofthemodesthatI
amnotaccustomedtolisteningto.Itsoundedpeculiarandasiftheperformermight
havebeenmakingmistakes.Yet,Iwasextremelydrawnintothemusic.Itwasquite
pleasingtomeandunlikeanythingIhadeverheardandIwascompelledtolistenall
thewaythrough.Thiswillneedtobeexploredfurther.
Levitinremindsusthatwhilesomescientistsarguethatthesolepurposeofmusicis
simplyforpleasureseeking,mostofthescientificcommunitywillagreethatmusic
playedasignificantroleinthedevelopmentofhumanspeechandlanguage.
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Theauthorexplains“multipletracetheory”andhowwecanlookatbrainscansthat
showareasofthebrainlightingupwhilelisteningtomusic.Thescansshowhowwe
processtheabstractinformationsuchasmelodyandtimbre,comparedtothemore
specificinformationlikethevocabularyusedinthelyrics,whenlisteningtomusic.
Levitintellsusthatthesebrainscanstudiescanshowushowanoldchildhoodmemory
canbetriggeredbyhearingasongfromthattimeperiod.
TheMozarteffect,ornot?
AlthoughIwouldlovetobelievethatlisteningtoclassicalmusicmakesaperson
smarter,thetruthisthattheresearchclearlyshowsthatthereisnotenoughevidence,
andtheclaimisconsideredcontroversial.Ifitweretrue,IsuspectthatmyownIQ
wouldbenearly250becauseofalltheBach,Mozart,Beethoven,Chopin,andScarlattiI
havelistenedto!ThemostlegitimatesourceIfoundonTheMozartEffect,isfromthe
NCBIdatabase.TheNationalCenterforBiotechnologyInformationadvancesscience
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andhealthbyprovidingaccesstobiomedicalandgenomicinformation.TheMozart
effect,J.S.JenkinsJRSM,2001,isapublicationfromtheUKbasedJournaloftheRoyal
SocietyofMedicineandistheresultofscientificstudy.
Thestudyexplainsthebrainscienceandgivessomesimpleexamplesthatareeasyto
understandsuchaswhichpartsofthebrainareresponsibleforspecificaspectsof
musiclikepitch,timbre,andrhythm.Itprovidesdetailsoftheexperimentsanddoes
pointoutthatforsomepatientswithepilepsytherewasafavorableresponsetothe
soundofMozart’spianosonataK448.Somehadasignificantimprovement.Brain
“spikes”associatedwithseizureactivityhaddecreasedsignificantly.
ThestudyevenincludedexperimentswithratsthatwereexposedtoMozart’sK448
Sonata,whitenoise,silence,andthemusicofcontemporaryminimalistcomposerPhilip
Glass.Theyusedfoursamplesthatwereaestheticallyverydifferentsounding.Thisis
becausesomecriticsofthetheoryclaimedthattheimprovementseeninprevious
experiments(withhumansubjects)listeningtoMozart,wasduetothe“enjoyment
arousal”factor.
AlthoughsomeresultsoftheJenkinsstudyshowedanimprovementinspatial-temporal
reasoninginafewofthesubjects,theconclusionisthatthereisnotenoughevidenceto
provethatlisteningtoclassicalmusiccanimproveintelligenceorscholasticaptitude.
However,itseemstomethatcreatinganenvironmentsuchasaclassroom,witha
componentthatislikelytobeenjoyablewouldcertainlyincreasethechancesofstudent
engagementandpossiblyamorerelaxedandopenmind.Ihaveinfactexperimented
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with5minutesofmusicappreciationatthebeginningofmathclass,andithadobvious
positiveeffects.Itwasn’tMozartorBach,butinthenearfutureIamplanningona
morefocusedandintentionaluseofthemusicappreciationactivity.
PartIV:MusicintheMathClassroom
“Philosophyiswritteninthisgrandbook,theuniverse,whichstandscontinuallyopentoourgaze.Butthebookcannotbeunderstoodunlessonefirstlearnstocomprehendthelanguageandreadthelettersinwhichitiscomposed.Itiswritteninthelanguageofmathematics,anditscharactersaretriangles,circles,andothergeometricfigureswithoutwhichitishumanlyimpossibletounderstandasinglewordofit;withoutthese,onewandersaboutinadarklabyrinth.”Galileo
Rhythm
Introductorymusictheory
Thescienceofsound
Tuningandtemperaments
12tonemusic
Symmetryinmusic
Mathematicalmodernmusic
ThislistofmusictopicsisthebeginningstageofwhatIwouldliketodevelopintoa
supplementalcurriculumformy11-weekbasicmathskillsclassforadults.Inthelast
sectionofmysabbaticalprojectIlookedatwhatotherinstructorsandexpertshave
beendoingtoimprovemathinstructionspecificallywithmusicalconnections.First,I
contactedlocalelementaryschools,andABSEinstructorsfromnearbycommunity
collegesanddidnotfindanyonewhoiscurrentlydoingthisoranythingevensimilar.I
feelweneedtodevotesometimeandenergyintorevivingthewaysoftheQuadrivium.
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Weneedtoaccelerateourcreativeenergiesandmakeanefforttoexpandthemuch
neededbasicmathcurriculumbuiltuponstateapprovedlearningstandardsby
integratingmusicaswellasart,nature,health,andphilosophy.
GarethE.Roberts,MathematicsprofessorattheCollegeoftheHolyCrossinWorcester,
MA,publishedandimplementedhismathematicsandmusiccoursewithsuccessand
positiveoutcomes.Heexplainsthatinhisexperience,moststudentswhodopoorlyin
thebeginninglevelmathclassesinhighschoolarethennotabletoaccessothermore
interestingmathtopics.Robertssaysthatinhismathematicsandmusiccourse,
“studentswhodidnotthinkofthemselvesasmathematicallyinclinedhaveunearthed
hiddentalentsandinterestsinthearea.”Thetitleofhis320-pagebookFromMusicto
Mathematics:ExploringtheConnectionsfromthepublishersofJohnsHopkinsUniversity
Pressisnearlythesameasthetitleofmyprojectandwillsurelybeonmysummer
readinglist.Althoughcurriculumdevelopmentisnotthepurposeofthissabbatical,I
havecertainlyfoundsomeinterestingplacestostartandsamplesoflessonideasthat
arequiteabitmoreinterestingthanwhatIhadexperimentedwithinthedidgeridoo
classproject.
InMusic:AMathematicalOfferingbyDaveBenson,theauthorgoesintodepthonthe
scienceofsoundandthewaythehumanearworks.Heteachesusingreatdetailabout
sinewaves,periodicity,theharmonicsofvibratingstringsandmuchmore.Thisis
essentiallyamanualonthephysicalscienceofsoundandmusic,completewiththe
complexmathematicalequationsthatexplainthebehaviorsofvariousmusical
instrumentsinanorchestra.Heprovidesexamplesofwavemotions,andwhatismeant
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byacousticpressureanddisplacementinwindinstruments.Benson’sbookis
fascinatingwithitsvastnessofscientificknowledge,manyexamples,illustrations,and
multiculturalperspectives.Followingeachsection,thereisanexerciseforthereader.
Forexample,afterdescribingthevibrationalpatternsondrummembranes,complete
withdiagrams,equations,andratios,heasks,“Whatdoesasquaredrumsoundlike?”
AlthoughmostofthemathandscienceinBenson’sbookisatalevelthatisdifficultfor
myunderstanding,Ihavebeenabletoextractsomeideasthatcanbeusefulinmy
futuremathandmusiccoursecontent.Thefollowingareexamplesofmusicalconcepts
thatcouldbebuiltintoamathlessononsymmetry.
Reflectionalsymmetryappearsinmusicintheformofinversionofafigureorphrase.
Thelowerlineisobtainedbyinvertingtheupperline.
Thisisthemusicalequivalentofthepalindrome.
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Eachbaroftheupperlineofthelefthandisinvertedtoformthenextbar.
HelpfulResearch
TolearnmoreabouthowIcandevelopmyideas,Iamveryhappytohaveafoundan
extensivepeerreviewedresearchstudythatwaspublishedin2013.Thestudy,
ElementaryTeachersIntegrateMusicActivitiesintoRegularMathematicsLessons:Effects
onStudents’MathematicalAbilities,investigatedtheintegrationofmusicinto
mathematicslessonsandwasexaminedwithpreandposttesting.Theeffectsofthe
integratedcurriculumclearlyshowedpositiveresultsinavarietyofmathematical
abilities.
Theauthorsofthisresearchstudybegantheirinvestigationasaresultofthebodyof
researchthatpresentsaconvincingcaseagainsttraditionalmathcurriculumand
teachingmethods.TheywenttotheworkofGarner’smultipleintelligencesandseveral
othermotivationaltheoristsfortheframeworkandthendesignedandadministeredan
interventioninvolvingtwoelementaryschoolteacherswithsimilardemographic
backgrounds,fromtwoseparateschools.Thestudentsineachoftheirclasseswere
alsodemographicallysimilar.Overafive-weekperiod,thestudentsparticipatedin
activitiesthatintegratedvariousmusicconceptssuchasmusic composition, withthe
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usualmathlessons.Thecontentoftheselessonsincludednumbersense,mathematical
reasoning,measurement,statistics,andothertopicsthatarealsotaughtinABSEMath
1.Theresearchdesigntools,procedures,anddatacollectionandanalysisseemsowell
donethatitinspiresandencouragesmetomoveforwardinmyplanofbringingmusic
tothemathclassIamcurrentlyteaching.However,itisimportanttonotethatIwill
needtoworkonadaptationsandmodificationsthattakeintoconsiderationthe
differencesinthelearningneedsbetweenchildrenandadults.Iwillberevisitingthis
studywhenIbeginmyMath&Musiccurriculumwork,astheresearchreportincludes
thelessonsandactivitiescarriedoutbytheteachers.
AdultLearning
Commonsenseandconventionalwisdomhavelongheldthatchildrenarebetter
learnersthanadults.Afterall,formorethanthreecenturiesJean-JacquesRousseau's
“tabularasa”concept,the"emptyslate,"dominatededucationaltheoryandeventoday
stillseemshardtorefute.Children's"slates"arerelativelyemptywhileadultshave
plentywrittenontheirs,andmoreover,alotofitisjustplainwrong.Iflearningis
conceivedofaswritingonanemptychalkboard,thenthereislittleroomforadultsto
learnbecausetheir"chalkboards"arerelativelyfullwhilethechild'semptyslateis
practicallybeggingtobewrittenupon.Butrecentresearchhaschallengedthetabula
rasaview,andinfact,adultsmaylearnaswellas,ifnotbetterthanchildren.
AdultLearningTheory:EvolutionandFutureDirectionsbyShananB.Merriam,isanin
depthanalysisofadultlearningandincludesathoroughreviewofthethreemajor
theoriesthathaveevolved,andthemorerecentcontextsthatarethefoundationsof
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understandingthelearningneedsofadultsintoday’sadulteducationclassrooms.
Merriam’sworkismostrelevanttomeinmyexplorationbecauseshewritesaboutthe
holisticelementsofteachingandlearningwiththeemphasisoncreativity,thearts,
healthandwellbeing.Sheisanadvocateoftransformativelearningandreportsthat
thisareaofadultlearningtheoryisbackedwithanamplesupplyofresearchedbased
literatureincludingajournalofinternationalconferences.Inherclaimthataholistic
approachshouldincludethespiritualdimensionsofthehumanbeing,shemakesaclear
distinctionbetweenreligionandspiritualityandofferssomeopinionastowhytherole
ofspiritualityinlearninghasnotbeenwidelyaccepted.Merriamalsocommentsonthe
multi-culturalperspectivesoflearningthatareincontrastwiththeformalschoolingof
theWestanditsrationalcognitiveframework.Shestates,“Ourbody,ouremotions,and
ourspirit(whatisoftenreferredtoasholisticlearning),arealsoimportantavenuesfor
learningorknowledgeconstruction.”Inconcluding,theauthorprovidesacompelling
argumentforthisholisticviewasshealsowritesaboutmeaning-makingandhow
adultsenteraclassroomwitha“meaning-makingagenda”.Indefiningthis,she
specificallypointstotheroleofimagesandsymbolsintheconstructionofknowledge
andregardingtheimagesandsymbolsshestates,“whichoftenemanatefromthe
deepestcoreofourbeingandcanbeaccessedandmanifestedthroughart,music,or
othercreativework.”
InclosingIwilladdthatbasedonmyrenewedunderstanding,andonsomeofthe
literaturethatIamplanningtostudy,myhopesforcombiningmusicandmathwill
involvemorethanjustthelearningprocessasstudentsworktowardtheirgoalsin
college.Integratingtheartsingeneral,andteachingfromaperspectiveofthe
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Quadriviumandtheelementsofclassicaleducationwill,inmyopinion,providea
strongerfoundationforinstillingthevalueoflife-longlearninginourstudents.More
specifically,thecombinationofmusicandmathinABSEbasicskillsisunique,andasa
resultofthissabbaticalwork,myplansformovingforwardindevelopingmyideasand
futurecoursecontentarewellsupported.
SchoolObservations
Afterseveralattemptstovisitlocalschoolswhereteachersmaybeintegratingmusic
andmathinstruction,Iwasunabletofindanyonewhoispresentlydoingsomething
thatwouldberelatedtothisproject.Icontacted3elementaryschoolsinthe4Jdistrict
aswellastenABSEinstructorsfromcommunitycollegesaroundthestatesofOregon
andWashington.MostoftheABSEinstructors,inonewayoranother,repliedbyletting
meknowthatalthoughtheywerenotcurrentlyintegratingmusicandmaththeyfound
myprojecttobeveryinteresting.TheadministratorofoneoftheEugeneelementary
schoolsrespondedpositively,informingmethattheinquirywouldbepassedonto
instructors.Afterasecondattempttoconnectwiththatschool,Istillreceivedno
replies.Iamassumingthattheyaresimplytoobusy!
Musicianship
Mycompetenceasamusicianandmyunderstandingofitstheoryandaestheticsis
crucialtomycontinuingplanofintegratingmusicandmath.Tocontinuemygrowth
andfoundationfortheproject,Ihavecommittedtodailymusicpracticeandforthis
sabbaticalthiswasproposedas2-3hourseachday.
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NotesonmyweeklymusicpracticeWeek1:Setgoals 1.Buildmyclassicalrepertoire Learnonenewpieceevery2-3weeks Improvetechniqueasweaknessesarediscovered 2.Improvepitchandtoneonthecello 3.Study: Baroquefugue Africanrhythms ClassicalIndianmusicWeek2: Learnednewguitarpiece:RondeauinAbyDavidKellner,1670–1748
• Enjoyabletoplay• NeedstobeperformedatafastertempothanIamaccustomedto
Workedonpitchwiththecello• Needtoeliminatethesheetmusicandjustfocusonsound
Week3: Improvedcellobowspeed Workedontempowithguitar Inprogress:Understandingthefugue ListenedtoandanalyzedJ.S.Bach-FugueBWV1001arrangedfor classicalguitarWeek4: Learnednewguitarpiece:AdelitabyFranciscoTàrrega,1852-1909
• Difficultduetomordentsthatrequiregreataccuracy• Soundsbestataslowertempo
Understandingthefugue• Watchedaninstructionalvideoonhowtocomposeafugue• Questioningmathconceptsthatcanbeapplied
Week5: Workedonpitchwiththecello
• Playscalesalongwithpre-recordedcellodrones Studythetechniqueofproducingrighthandharmonicsontheguitar
• Watchedseveralinstructionalvideos ListenedtoandanalyzedJ.S.Bach-ToccataandFugueinDminorBWV565Week6: Improvedcellobowspeedandcontrol
• Learnedto“divein”deeperintothestrings ListenedtoNorthAfricanrhythms
• DRUM&AfricanRhythm,Part1|MokhtarSamba
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Watchedvideo-MusicisMath!TeachingRhythmsasFractionstoMiddle ElementaryWeek7: BeganavolunteerpositionatRiverBendHospitalinthe SoundsforHealingprogram
• Weeklyperformanceonguitarinthehospitallobby-- “Tocreateahealingandrelaxingenvironmentforpatientsand families…throughthegiftofmusic.” Inprogress:ReadingTheClassicalIndianJustIntonationTuningSystemWeek8: Extensiverehearsalofmyguitarrepertoireinpreparationfor RiverBendHospital
• SelectallpiecesthataresuitablefortheSoundsforHealingprogram• Addmyownarrangementsofsomepopmusic
Continuedperfectingtherighthandharmonics ListenedtoAnoushkaShankar-IndianClassicalRagaWeek9: Learnednewguitarpiece:GymnopèdieNo.1byErikSatie,1866–1925
• Soundsbestwhenslightlyflatteningthebandgstrings• Producesameditativestate• Addedrighthandharmonicstotheending
Week10: Watchedavideo,J.S.Bach-TheArtOfFugueBWV1080 Firstattemptatcomposingasimplefugue ContinuedstudyofclassicalIndianmusic
• Exploredmelodicpatternstouseinoriginalcomposition Watchedvideo-MathematicsofAfricanDanceRhythmsWeek11: Beganlearningnewguitarpiece:MoonRiverbyHenryMancini,1924–1994
• Easyandenjoyable• EspeciallyappropriatefortheSoundsforHealingprogram
Composingafugue--tobecontinued…
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