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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.

24

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…

26

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