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IsNNBiggerthanMenette?

ProfessorGinsbooExplainsEverythingYouNeedtoKnowaboutMeasurement

byProfessorGinsboo

Himself

SettingtheStage

Oneday,I,ProfessorGinsboo,wasdeepinthought,strollingalongablankcomputerscreen,

whenalittlegirlemergedtofaceme.Itwasmydearstudent,Menette.

“ProfessorGinsboo,”sheexclaimed,“Ineedyourhelp.MybrothersaysthatheistallerthanIam.ButIknowthatheisdefinitelyshorter,andwhenItrytoexplain,hedoesnotunderstand.Hekeepsinsistingthatheistaller.”

IknewrightawaythatIcouldeasilyhelpMenette.Afterall,whatcouldbesimplerthanmeasuringabrother?

“Menette,Icaninstructyouandyourbrotherintheproperuseofarulersothatyoucanthenmeasureeachotherandthussettlethematter.”

“Ialreadyknowhowtousearulerandsodoeshe.Hesaid,andIquote,‘Usingarulerisboring.Iwanttolearnaboutthemathematicsofmeasurement.Afteryouclarifythe

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underlyingideas,thenwecandiscusswhyIamtallerthanyou.’Thatisabitprecocious,Iknow,butthat’swhathesaid.”

“Doesyourbrotherhaveaname?”

“No,”saidMenette.

“Well,Iguessthatdoesn’tmatter.Infivelessons,Iwillteachyoueverythingyouandyourbrotherneedtoknowabouttheideasunderlyingmeasurement.

Lesson1

Measurementdoesnothavetobeboring

Iproceededasfollows,directingmyremarkstotheabsentbrotherNo-Name(NNforshort)asMenettetooknotes.“Thefirstthingyouneedtolearn,NN,isthattakingmeasurementsdoesnothavetobeboring.Abandonthiscanard.Here’swhy.Imaginethatyou,NN,andMs.Motorcyclehaveagreedtosharethebananathatsheischasing.Onceshecatchesit,yourjobistomeasureitsothatbothofyoureceiveequalshares.Inthiscase,measurementisthekeytosharing,peace,harmony,andbeneficence.”

Menettedidnotseemconvinced.“Unfortunately,mybrotherNNisselfish;heisnotinterestedinsharing,peace,harmony,andbeneficence.”

Ireplied,“Thatisnoproblem.Measurementisalsousefulforcheating!YourbrotherNNcouldbreakthebananaintotwounequalpieces,measurethem,andthenusesubterfuge(Iwillnottellyouhow)togivethesmallerpiecetoMs.Motorcycle.NNwillbeveryhappy.Whatdoyouthink?”

“IthinkthatmakessenseandwillaskmybrotherNNrightaway.Excusemeforaminute.”

MenettewalkedoffsothatoncemoreIwasdeepinthought,strollingalongablankcomputerscreen,thinkingandthinking,andcontemplatingtheuseofmeasurementtocheat.ItgoeswithoutsayingthatI,ProfessorGinsboo,donotcondonecheating.Atthesametime,cheatersshouldbehelpedtorealizetheirfullpotentialsothattheycancheatinamathematicallysoundmanner.

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Menettethenreappearedandsaid,“MybrotherNNwasgratefulforyourlessononcheatingbutdeclaredthathedoesnothimselfcheat.Hepointedoutthatmeasurementcanbeusedforavarietyofotherpurposes,likeweighingadonutordeterminingtheheightofaskyscraper.Measurement,hesaid,canbeconsidered,andIquotehim,‘…ausefultoolfortheimplementationoftheengineeringimpulse,theneedtobuildandcreatenewstructures,gadgets,buildings,andotherappurtenances.’Healsosaidthatpreciseandobjectivemeasurementis‘appliedmathematics,stemmingfromtheancientGreeks,andassuchisbasictoscientificendeavor.’Andbecausebothengineeringandscienceareinherentlyfascinating,measurementcanbe,andIquote,‘alotoffun,like,duh,dependingonthecontext.’Isn’tmybrothersmart?”

Ireplied,“Ifheissosmart,howcomehedoesn’thaveaname?”

Lesson2

HowlongisElly’strunk?

HavinghelpedNNtoappreciatetheexcitementofmeasurement,Iwasnowreadytopresentmyinsightsregardingitsconceptsandmethods.

“Menette,”Isaid,“Iwillbeginbyexplainingthemathematicsofmeasurement.Thegoalofmeasurementistodeterminetheamountofsomecontinuousattributelikethelengthofanelephant’strunk.Todothis,weemployaunitofmeasurement,anexercisewhichdatesbacktoprehistorictimes,whenCavette,thefirstcave-persontoencounteranelephantsaid,‘Wow.That’sabigelephant.Whatalongtrunk!Iwanttotellmycave-mateshowlongitis.’Toaccomplishthisgoal,Cavettefirstcommandedtheelephant,whosenamewasElly,tolaydownandextenditstrunk,whereuponshemeasureditbythenumberofstridesittookhertogofromoneendofthetrunktotheother.

“Themathematicsofmeasurementisstraightforward.ImagineastraightlinecorrespondingtoElly’strunk.ImaginethatwedividethelineintosegmentscorrespondingtothelengthofCavette’sstrides(allofwhichareexactlythesame

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length).ThetotallengthofElly’strunkisthenumberofCavette’sstridesintowhichthelineisdivided.TheCavette-Strideisanon-conventionalunitofmeasurement.

“Laterthecave-peoplethankedCavetteforherhardworkanddedicationtomeasurement,butdecidedtodevelopasharedunitthatdidnotrequiresteppingonelephantsorotherdangerousanimals.Theysettledontheinchandthefoot.Again,imaginethestraightlinecorrespondingtoElly’strunk.Theconventionalmeasuredividesitupintofeetandinches(ormetersandcentimeters.Moreaboutthatlater!)Thelengthoftheline(andElly’strunk)doesnotchange.ButmeasuringlengthbyCavetteStridesgivesadifferentresult(fourCavetteStrides)thanmeasuringlengthbyfeetandinches(6feet4inches)becausecave-persons,needingtofleeallkindsofanimals,haveextremelylengthystrides.

“Thefoot/inchunitofmeasurementisstandardbecauseallofthecave-peopleagreedtouseit.Indeed,generationsofcave-peoplecontinuedtousefeetandinchestomeasurethings.Andbecausethisgroupofcave-peoplelivedinwhatlaterwascalledLondon,themethodissometimescalledtheEnglishmeasurementsystem.

“AtthesametimethattheEnglishcave-peoplecreatedtheirstandardmeasure,theFrenchdevelopedtheirs,whichinvolvescentimeters,meters,andsoon.Later,Iwilldiscussthedifferencesbetweenthetwostandardmeasures(Englishandmetric).”

I,ProfessorGinsboo,pausedsothatMenettecouldassimilatethefullmeaningofmylength-of-Elly’s-trunklesson,andthenaskedhertogiveittoherbrother,NN.

Whenshereturned,MenettereportedthatNNthoughtthatthedetourintohistoricaleventswasnottooboringandthathewantedtolearnmoreaboutstandardunitsofmeasurement.Idecidedtograntthecheekyfellow’srequest,asisshowninthenextlesson.

Lesson3

Conventionalunitsfostercommunicationandunderstanding

“Menette,”Isaid,“listencarefullyandtakenotessothatyoucaneducateyourbrotherNN.Mylessononconventionalunitswillbeintheformofahaikuthatcapturestheessenceofthezenofmeasurement,asfollows.

Conventionalu-nitsarebetterthannon-con-

ventionalunits.

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“Didyounoticehowthefirstlinehasfivesyllables,thesecondseven,andthethirdfive?Thesyllableitselfisaconventionalunit!

“ButIhaveevenmoretoreveal.Conventional,standardmeasurementallowseveryonetosharetheresults.Theunitsarefamiliartoall.WhetheryouareinIndiaorPortugal,youcantalkthesamemeasurementlanguage.”

AlthoughIhadfinishedmylesson,Menettestaredatmeinasomewhatpuzzledmanner.After43seconds(thesecondisastandardmeasure!),shesaid,“Isthatyourfulllesson?Itseemsobvious.”

“Itis.PleaseinformNN.”

Within17secondsshereturned(maybeshedidn’treallygotoseeNN),sayingthatuponhearingthefulllesson(haikuandtherest)NNsmiledandasked“Butwhataboutdifferentkindsofconventionalmeasures,inparticulartheEnglishandmetricsystemsthatyoudescribedinlessontwoandpromisedtoexplain?PleasetellProfessorGinsboothataddressingthistopicisessential.”

CheekylittleNN!Nevertheless,Iexplainedasfollows.

“Forgetaboutthecavepeopleandfocusonthecurrentsituation.Inacountrycalled‘theUSA,’somewhereoverthescreensaverrainbow,citizensusecustomaryortraditionalmeasuressuchasinchesandpounds.USAerssay,forexample,thatthispencilissixincheslongandweighstwoounces.ThesecustomaryunitsderivefromtheirolderEnglishcounterparts.

“Nowadays,mostcountries,andalmostallscientistsregardlessofcountry,usethemetricsystem,inwhichlength,forexample,ismeasuredincentimeters,onehundredofwhichcompriseameter,onethousandofwhichcompriseakilometer,andsoon.Allofthemeasuresarerelatedtooneanotherbymeansofabase-tensystem.Thisisnottrueofthecustomaryunitsofmeasurement.

“Themetricsystemisclearlysuperiortothecustomarysystem,whichisused(atleastattheendof2017itstillwas)byonlythreecountries:Liberia,MyanmarandtheUSA.”(ProfessorGinsbooacknowledgesthathehadtogooglethislastresult.Unscholarlybutconvenient[andperhapseventrue].)

Menettenoddedherheadindeepappreciationfor14seconds,leftandreturnedin19seconds,andsaid,“NNwasgratefulforyourlesson,butnowhasanotherquestion,namely,andIquote,‘Doyouhaveanythingmoreinterestingtoteachme?’

ThricecheekylittleNN!Waituntilheseeswhat’snext.

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Lesson4

Closeenough:ThecaseofthefickleslouchingProfessorGinsboodoll

I,ProfessorGinsboo,thenaddressedMenette:“Iwillnowteachyouanimportantbutnormallyoverlookedfeatureofmeasurement.Afteryouhavemasteredmylesson,youmayconveyittoNN.

“Wearegoingtoplayagame.Inthisgame,youareattemptingtomeasuretheheightofyourfavoriteProfessor-Ginsboo-as-a-young-childdoll(maybeavailablesoonatasitelikehttps://www.ProfGinsbooDoll.com).Likemostdolls(andrealchildren),hewantstokeeptrackofhisheight,soheobtainsameasureofitateachbirthday.Tohelp,youstandhimupagainstthewall,placeyourrulernexttohim,andfindthatthetopofhisheadisacertainnumberofinchesandfractionalpartsofaninchfromthefloor.

“Atthispoint,youmaycongratulateyourselfonajobwelldone.Butsupposethatyoucelebratebyeatingamagicicecreamconethatunbeknownsttoyouhastheeffectofmakingyouforgetyourmeasurement.Youthereforemeasureagainandeatanothercelebratorycone,whichhasthesameforgettingeffectasthefirst.Thecyclestopsonlyafteryouhaveobtained25heightmeasurements,atwhichpointyouhaveexhaustedthesupplyofcones.”

Menetteagreedthatthiswasasplendidgameandbegantoplay.

Whenfinished,Menette,lookingalittlequeasy,reportedback.“LookatthefollowinggraphIconstructed.IfinditdistressingbecausetheresultsshowthatIrecordedseveraldifferentheightmeasurementsfortheverysamedoll.ThismeansthatIgotatleastsomewronganswers.Areallmyanswerswrong?DidIgetanycorrectanswers?”

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Icomfortedher.“Don’tworry,Menette.Thisisentirelytypical.Let’sreviewthesituation.Almostallofyourmeasurementsinvolvedthreevalues,namely714/16,715/16and8inches,andtherestareinfrequent.Soyouweren’tjustguessingormakingrandomselections.Butthinkaboutwhyyoumighthaveobtaineddifferentvalues.

• Whenyoustoodthedollupagainstthewalleachtime,hemayhavemovedorslouchedalittleonsomeoccasionsbutnotothers.Yourmeasurementsmaysimplyhavereflectedtherealitythattheobjectofthemeasurementmaychangeunexpectedly.

• Whenyougottiredoftheactivity,orwereperhapsbloatedbyicecream,youmayhavelostfocus,withtheresultthatyourmeasurementacuitywasdiminished.

• Youmayhavefounditdifficulttoreadthehatchmarksontheruler.Lookatit.Determiningthenumberofsixteenth-inchmarkingsissimplynoteasy.

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“Sochangesinthephysicalrealityofthethingbeingmeasured(theficklestanding-slouchingdoll),fluctuationsinyourconcentration,andtheobjectivedifficultyofthetaskmayhaveresultedininconsistentmeasurements.Andremember,Menette,thatyouarenottheonlymeasureraffectedbythesefactors.”

Naturally,Menettewasoverjoyedwithmylesson.

Butshehadacommentandaquestion.“Iamgladthatmyfluctuatingmeasurementswerenotabnormal,atypical,orotherwiseunusual.YetIdon’tunderstandthepracticalimplicationsofthisinformation.Pleaseinformme.”

“Theimplicationsareasfollows,Menette.“Wemustbetolerantofvariability.Inmanycases,itisappropriatetoapplyamarginoferrortoourmeasurements.Forexample,inthecaseofthefickleslouchingdoll,youmightconsiderthatjudgmentsintherangeof714/16to8inchesarecloseenough.Afterall,ifthegoalistocomparegrowthfromonebirthdaytothenext,1/16ofaninchmatterslittle.IfyoupurchaseanupgradetotheProfessor-Ginsboo-as-a-young-childdollonhisnextbirthday,youwillseethathehasgrownseveralinches,andthatyouneednotbeconcernedwiththosetinyhatchmarks(thesixteenthsofaninch)ontheruler.Ontheotherhand,ifyouwanttomeasurethedailyincreaseinheight,thenfinermeasurementsinvolvingthesixteenthswouldbenecessaryandthereshouldbealmostnomarginoferror.”

SuddenlyMz.Zoler,Menette’steacher,jumpedintothecomputerscreenandexclaimed:

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“Yourideasabouttoleranceforvariabilityareveryimportantforeducation.Teachersneedtounderstandthatonsomeoccasionsandforsomepurposesthereisnosinglerightanswerandthatstudentsmustbeflexibleenoughtoembraceandunderstandtheideasofgoodenoughandmarginoferror.Goodbye.”

Lesson5

Fairmeasurement

Afterabsorbingthedeepideasinlessonfour,Menetteturnedtomeandsaid,“ProfessorGinsboo,Ihavelearnedagreatdeal,butyouhaven’tyetsaidhowIcanconvincemybrotherthatheisindeedshorterthanI.”

IexplainedtoMenettethatsheneedstolearnaboutfairmeasurement,thatis,howtoensurethattheresultsarenotbiased,defective,orincorrect.

“First,Menette,knowwhatyouaremeasuring.Iftwomeasurersgetdifferentresults,youneedtodetermineexactlywhattheyaremeasuring.Forexample,EliyasawBearAsittingonaparkbenchontheleft,andMelinasawBearBsittingonaparkbenchontheright.EliyaandMelinaeachmeasuredtheirpersonalbear.EliyasaidthatBearAisbiggerthanBearB.ButMelinadisagreed,sayingthatBearBisbiggerthanBearA.Whowasright?Both,becauseEliyameasuredheightandMelinameasuredwidth.BearAwastallandskinny,whereasBearBwasshortandfat.”

“Secondistheissueoftheorigin.Ioncegavethefollowingproblemtoayoungchild:

‘Pleaselookatthesetwosticks,”Isaid.“Whichislonger?’

A_______________

B__________

“HepointedtoB.

“Askedtoexplainwhy,hesaidthatBwasobviouslylongerbecauseitstuckouttotherightbeyondstickA.

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“Somakesurethattwoobjectsyouarecomparingbeginatthesameorigin.Thisofcoursewillresultinareversalofthepreviousjudgment:nowAisclearlylonger.

A_______________

B__________

“Third,usethemeasurementtoolproperly.OnechildIknow,MichèleMMM,measuredtheheightofher10pounddumbbellbyplacingtherulernexttothedumbbellasshownbelow.Sheputthe12thinchonthebottomandthezeroonthetop.Readingthisruler,sheconcludedthatthedumbbellwastwoincheshigh.Shedidputboththeweightandtherulerinthesameplaceonthefloorandstoodbothupnexttoeachother.Butshedidnotappreciatethatthenumbersontherulerneedtostartwithzeroonthefloor.”

Menettelookedveryimpressedwithmylesson.Shesaid,“ProfessorGinsboo,Inowunderstandfairmeasurement:IknowIwanttomeasureheight;Iknowthattheoriginneedstobeatzero;andIknowhowusetherulerproperly.

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“Butyouhavenotansweredthequestionwithwhichwebeganoureducativeconversation,namely:howcanIhelpmybrotherunderstandthatheisnottallerthanIam?”

IrepliedthatnowisthetimetobringbrotherNNintotheconversation.Isaid,“I,ProfessorGinsboo,willnowconvinceNN,facetoface,thatheisshorterthanyou.”

Thesurprisingending

NNimmediatelyappearedonthecomputerscreen.I,ProfessorGinsboo,thoughtitwashighlyirregularforMenette,whoisacartoonfigure,tohaveanactualhuman“boy”forabrother.Butbecausewewelcomediversity,Ichosetoignoretheincongruityandproceededwithmylesson.Aftercareful(andlengthy)inspection,IconcludedthatNNwasstandingonachair.

Iasked,“DearNN,Menette’sbrother.WhydoyousaythatyouaretallerthanMenette?”

“IsaythatIamtallerbecauseIamuphereandsheisdownthere.”

“ButNN,youhavetoconsidertheruleaboutbeginningmeasurementattheorigin.StepdownsothatwecanmeasureyouandMenettebothstandingonthefloor.”

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NNdidasIrequested.Iwasdumbfoundedattheresult!NNwasclearlyandconsiderablytallerthanMenette,evenwhenhewasstandingonthefloor!Nomeasurementwasrequiredtoestablishthisfact.

Menetteexclaimed,“Whathappened?”

NNcalmlyreplied,“I’mtaller.Itoldyouso,Menette,myteenysister.”

AndIconcludedthatmyfinallessononmeasurementis:Nevertakeanythingforgranted(especiallyasneakysiblingorastoryending).