[Barbara Oakley] A Mind for Numbers(Book Fi) PDF

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“Beinggoodatscienceandmathematicsisn’tjustsomethingyouare;it’ssomethingyoubecome.Thisusers’guidetothebrainunmasks themysteryaroundachievingsuccessinmathematicsandscience.Ihaveseenfartoomanystudentsoptoutwhentheyhitaroughpatch Butnowthatlearnershaveahandyguidefor‘knowingbetter,’theywillalsobeableto‘dobetter.’” —SHIRLEYM ALCOM,HEADOFEDUCATIONANDHUMANRESOURCESPROGRAMS,AMERICANA SSOCIATIONFORTH ADVANCEMENT OFSCIENC “AMindforNumbersisanexcellentbookabouthowtoapproachmathematics,science,oranyrealmwhereproblemsolvingplaysa prominentrole.” —J.M ICHAELSHAUGHNESSY, PAST PRESIDENT OFTHEN ATIONALCOUNCILOFTEACHERS OFM ATHEMATIC “Ihavenotbeenthisexcitedaboutabookinalongtime.Givingstudentsdeepknowledgeonhowtolearn willleadtohigherretention andstudentsuccessineveryfield.Itisagiftthatwilllastthemalifetime.” —ROBERT R.GAMACHE,PH.D.,ASSOCIATEV ICEP RESIDENT ,ACADEMICAFFAIRS,STUDENT AFFAIRS,ANDINTERNATIONA RELATIONS,U NIVERSITY OFM ASSACHUSETTS,LOWEL “AMindforNumbershelpsputstudentsinthedriver’sseat—empoweringthemtolearnmoredeeplyandeasily.Thisoutstandingbook alsoausefulresourceforinstructionalleaders.GiventheurgentneedforAmericatoimproveitsscienceandmatheducationsoitcansta competitive,AMindforNumbersisawelcomefind.” —GEOFFREYCANADA , PRESIDENT ,HARLEMCHILDREN’ SZON “Aningeniouslyaccessibleintroductiontothescienceofhumancognition—alongwithpracticaladviceonhowtothinkbetter.” —JAMESTARANTO, THEWALL STREETJOURNA “It’seasytosay‘worksmarter,notharder,’butBarbaraOakleyactuallyshowsyouhowtodojustthat,inafast-pacedandaccessible bookthatcollectstipsbasedonexperienceandsoundscience.Infact,I’mgoingtoincorporatesomeofthesetipsintomyownteaching  —GLENNH ARLANREYNOLDS,BEAUCHAMPBROGANDISTINGUISHEDP ROFESSOROFLAW,THEU NIVERSITYOFTENNESSE “AMindforNumbersisasplendidresourceforhowtoapproachmathematicslearningand,infact,learninginanyarea.Barbara Oakley’sauthoritativeguideisbasedonthelatestresearchinthecognitivesciences,andprovidesaclear,concise,andentertainingroad mapforhowtogetthemostoutoflearning.Thisisamust-readforanyonewhohasstruggledwithmathematicsandanyoneinterestedin enhancingtheirlearningexperience.” —DAVIDC.GEARY ,CURATORS’P ROFESSOROFP SYCHOLOGICALSCIENCESANDINTERDISCIPLINARYN EUROSCIENCE U NIVERSITY OFM ISSOUR “Forstudentsafraidofmathandscienceandforthosewholovethesubjects,thisengagingbookprovidesguidanceinestablishingstudy habitsthattakeadvantageofhowthebrainworks.” —DEBORAHSCHIFTER, PRINCIPALRESEARCHSCIENTIST ,SCIENCEANDMATHEMATICS PROGRAMS,EDUCATIONDEVELOPMEN CENTER,INC

J EREMYP.TARCHER/PENGU IN

PublishedbythePenguinGroup PenguinGroup(USA)LLC 375HudsonStreet NewYork,NewYork10014 USA•Canad a•UK•Ireland •Australia•NewZealand•Ind ia•SouthAfrica•China penguin.com APenguinRandomHouseCompany Copyright©2014byBarbaraOakley Penguinsupportscopyright.Copyrightfuelscreativity, encouragesd iversevoices, promotesfreespeech, andcreatesavibrantculture.Thankyouforbuyinganauth editionofthisbookand forcomplyingwithcopyrightlawsbynotreproducing,scanning, ordistributinganypartofitinanyformwithoutpermission.Youaresupp writersand allowingPenguintocontinuetopublishbooksforeveryreader. MostTarcher/Penguinbooksareavailableatspecialquantitydiscountsforbulkpurchaseforsalespromotions, premiums,fund -raising,and ed ucationalneeds.Sp booksorbookexcerptsalsocanbecreatedtofitspecificneeds.Fordetails,write:[email protected]. LibraryofCongressCataloging-in-PublicationData Oakley,BarbaraA. Amindfornumbers:howtoexcelatmathandscience(evenifyouflunkedalgebra)/BarbaraOakley, Ph.D. p.cm. Includesbibliographicalreferecncesand index. ISBN978-1-101-62161-5 1.Mathanxiety.2.Mathematics—Studyand teaching—Psychologicalaspects.3.Educationalpsychology.I.Title. QA11.2.O3320142014003665 501'.9—dc23 Version_1

AMindforNumbersisdedicatedtoDr.RichardFelder,whosebrillianceandpassionhave launchedextraordinaryimprovementsworldwideintheteachingofscience,mathematics, engineering,andtechnology.Myownsuccesses,likethoseoftensofthousandsofother educators,growoutofhisfertileeducationalapproaches.Ilmigliormaestro.

TheLawofSerendipity:LadyLuckfavorstheonewhotries

contents PRAISEFORAMINDFORNUMBERS TITLEPAGE COPYRIGHT DEDICATION EPIGRAPH FOREWORDbyTerrenceJ.Sejnowski,FrancisCrickProfessor,SalkInstituteforBiologicalStudies PREFACEbyJeffreyD.Karpicke,JamesV.BradleyAssociateProfessorofPsychologicalSciences,PurdueUniversity NOTETOTHEREADER

1OpentheDoor 2EasyDoesIt: WhyTryingTooHardCanSometimesBePartoftheProblem

3LearningIsCreating: LessonsfromThomasEdison’sFryingPan

4ChunkingandAvoidingIllusionsofCompetence: TheKeystoBecomingan“EquationWhisperer”

5PreventingProcrastination: EnlistingYourHabits(“Zombies”)asHelpers

6ZombiesEverywhere: DiggingDeepertoUnderstandtheHabitofProcrastination

7ChunkingversusChoking: HowtoIncreaseYourExpertiseandReduceAnxiety

8Tools,Tips,andTricks 9ProcrastinationZombieWrap-Up 10EnhancingYourMemory 11MoreMemoryTips 12LearningtoAppreciateYourTalent 13SculptingYourBrain 14DevelopingtheMind’sEyethroughEquationPoems 15RenaissanceLearning 16AvoidingOverconfidence: ThePowerofTeamwork

17TestTaking 18UnlockYourPotential AFTERWORDBYDAVIDB.DANIEL,PH.D.,PROFESSOR,PSYCHOLOGYDEPARTMENT,JAMESMADISON UNIVERSITY ACKNOWLEDGMENTS ENDNOTES REFERENCES CREDITS INDEX

foreword

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ourbrainhasamazingabilities,butitdidnotcomewithaninstructionmanual.You’llfindthat manualinAMindforNumbers.Whetheryou’reanoviceoranexpert,youwillfindgreatnew waystoimproveyourskillsandtechniquesforlearning,especiallyrelatedtomathandscience. HenriPoincaréwasanineteenth-centurymathematicianwhooncedescribedhowhecrackeda difficultmathematicalproblemthathehadbeenintensivelyworkingonforweekswithoutsuccess.H tookavacation.AshewasgettingonabusinthesouthofFrance,theanswertotheproblemsudden cametohim,unbidden,fromapartofhisbrainthathadcontinuedtoworkontheproblemwhilehe wasenjoyinghisvacation.Heknewhehadtherightsolutioneventhoughhedidnotwritedownthe detailsuntilhelaterreturnedtoParis. WhatworkedforPoincarécanworkforyoutoo,asBarbaraOakleyexplainsinthisinsightful book.Surprisingly,yourbraincanalsoworkonaproblemevenwhileyouaresleepingandarenot awareofanything.Butitdoesthisonlyifyouconcentrateontryingtosolvetheproblembefore fallingasleep.Inthemorning,asoftenasnot,afreshinsightwillpoptomindthatcanhelpyousolv theproblem.Theintenseeffortbeforeavacationorfallingasleepisimportantforprimingyour brain;otherwiseitwillworkonsomeotherproblem.Thereisnothingspecialaboutmathorscience inthisregard—yourbrainwillworkjustashardatsolvingsocialproblemsasonmathandscience problems,ifthatiswhathasbeenonyourmindrecently. Youwillfindmanymoreinsightsandtechniquesabouthowtolearneffectivelyinthisfascinatin andtimelybook,whichlooksatlearningasanadventureratherthanhardlabor.Youwillseehowyo canfoolyourselfaboutwhetheryouactuallyknowthematerial;youwillfindwaystoholdyour focusandspaceoutyourpractice;andyouwilllearntocondensekeyideassoyoucanholdthem moreeasilyinyourmind.Masterthesimple,practicalapproachesoutlinedhereandyouwillbeabl tolearnmoreeffectivelyandwithlessfrustration.Thiswonderfulguidewillenrichbothyour learningandyourlife. —TerrenceJ.Sejnowski,FrancisCrickProfessor,SalkInstituteforBiologicalStudi

preface

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hisbookcanmakeaprofounddifferenceinhowyoulookatandunderstandlearning.Youwill learnthesimplest,mosteffective,andmostefficient techniquesresearchersknowabouthowto learn.Andyou’llhavefunwhileyou’redoingit. What’ssurprisingisthatalotoflearnersuseineffectiveandinefficientstrategies.Inmy laboratory,forexample,wehavesurveyedcollegestudentsabouttheirlearning.Theymost commonlyusethestrategyofrepeatedreading—simplyreadingthroughbooksornotesoverand over.Weandotherresearchershavefoundthatthispassiveandshallowstrategyoftenproduces minimalornolearning.Wecallthis“laborinvain”—studentsareputtinginlaborbutnotgetting anywhere. Wedon’tengageinpassiverereadingbecausewearedumborlazy.Wedoitbecausewefallpre toacognitiveillusion.Whenwereadmaterialoverandover,thematerialbecomesfamiliarand fluent,meaningitiseasyforourmindstoprocess.Wethenthinkthatthiseasyprocessingisasign thatwehavelearnedsomethingwell,eventhoughwehavenot. Thisbookwillintroduceyoutothisandotherillusionsoflearningandgiveyoutoolsto overcomethem.Anditwillintroducegreatnewtools,suchasretrievalpractice,thatcanhavea powerfuleffectinboostingthe“bangforyourbuck”fromyourtimespentinlearning.It’sadeeply practicalyetinspiringbookthathelpsyouseeclearlywhysomeapproachesaresomuchmore effectivethanothers. We’reontheedgeofanexplosioninknowledgeabouthowtolearnmosteffectively.Inthisnew worldofinsight,you’llfindAMindforNumberstobeanindispensableguide. —JeffreyD.Karpicke,JamesV.BradleyAssociateProfessorofPsychologicalSciences,Purd Universit

notetothereader

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eoplewhoworkprofessionallywithmathandscienceoftenspendyearsdiscoveringeffective learningtechniques.Oncethey’vefiguredthesemethodsout,great!Theyhaveunwittingly passedtheinitiationritesneededtojointhemysterioussocietyofmathandsciencepractitioners. I’vewrittenthisbooktolayoutthesesimpletechniquessothatyoucanimmediatelybeginusing them.Whattakesyearsforpractitionerstodiscoverisnowatyourfingertips. Usingtheseapproaches,nomatterwhatyourskilllevelsinmathandscience,youcanchange yourthinkingandchangeyourlife.Ifyou’realreadyanexpert,thispeekunderthementalhoodwill giveyouideasforturbochargingsuccessfullearning,includingcounterintuitivetest-takingtipsand insightsthatwillhelpyoumakethebestuseofyourtimeonhomeworkandproblemsets.Ifyou’re struggling,you’llseeastructuredtreasuretroveofpracticaltechniquesthatwalkyouthroughwhat youneedtodotogetontrack.Ifyou’veeverwantedtobecomebetteratanything,thisbookwillhel serveasyourguide. ThisbookisforhighschoolstudentswholoveartandEnglishclassesbutloathemath.Itismea aswellasforcollegestudentswhoalreadyexcelinmath,science,engineering,andbusiness,butwh suspecttherearementaltoolstobeaddedtotheirlearningtoolkits.It’sforparentswhosechildrena eitherfallingoffthemathtrackortryingtorockettowardmathandsciencestardom.It’sforthe frazzlednine-to-fiveworkerwhohasn’tbeenabletopassanimportantcertificationtest,andforthe night-shiftconveniencestoreclerkwhohasdreamedofbecominganurse—orevenadoctor.It’sfo thegrowingarmyofhomeschoolers.It’sforteachersandprofessors—notonlyinmath,science, engineering,andtechnology,butalsoinfieldssuchaseducation,psychology,andbusiness.It’sfor theretireewhofinallyhasthetimetoembracenewknowledgeincomputing,forexample,orthe intricaciesofgreatcooking.Andit’sforreadersofallageswholovetolearnalittleabout everything. Inshort,thisbookisforyou.Enjoy! —BarbaraOakley,Ph.D.,P.E.,Fellow,AmericanInstituteforMedical&BiologicalEngineeringan VicePresident,InstituteforElectricalandElectronicsEngineers—EngineeringinMedicinean BiologySociet

{1} openthedoor

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hataretheoddsyou’dopenyourrefrigeratordoorandfindazombieinthere,knitting socks?Theoddsareaboutthesamethatatouchy-feely,language-orientedpersonlikeme wouldendupasaprofessorofengineering. Growingup,Ihatedmathandscience.Iflunkedmywaythroughhighschoolmathandscience courses,andonlystartedstudyingtrigonometry—remedialtrigonometry—whenIwastwenty-six yearsold. Asayoungster,eventhesimpleconceptofreadingaclockfacedidn’tseemtomakesensetom Whyshouldthelittlehandpointtowardthehour?Shouldn’titbethebighand,sincethehourwas moreimportantthantheminute?Didtheclockreadtenten?Oronefifty?Iwasperpetuallyconfused Worsethanmyproblemswithclockswasthetelevision.Inthosedaysbeforetheremotecontrol,I didn’tevenknowwhichbuttonturnedthetelevisionon.Iwatchedashowonlyinthecompanyofmy brotherorsister.TheynotonlycouldturntheTVon,butcouldalsotunethechanneltotheprogram wewantedtowatch.Nice. AllIcouldconclude,lookingatmytechnicalineptitudeandflunkinggradesinmathandscience wasthatIwasn’tverysmart.Atleast,notthatway.Ididn’trealizeitthen,butmyself-portraitasbein technically,scientifically,andmathematicallyincapablewasshapingmylife.Attherootofitallwas myproblemwithmathematics.Ihadcometothinkofnumbersandequationsasakintooneoflife’s deadlydiseases—tobeavoidedatallcosts.Ididn’trealizethenthatthereweresimplementaltricks thatcouldhavebroughtmathintofocusforme,tricksthatarehelpfulnotonlyforpeoplewhoare badatmath,butalsoforthosewhoarealreadygoodatit.Ididn’tunderstandthatmytypeofthinkin istypicalofpeoplewhobelievetheycan’tdomathandscience.Now,Irealizethatmyproblemwas rootedintwodistinctlydifferentmodesforviewingtheworld.Backthen,Ionlyknewhowtotapon modeforlearning—andtheresultwasthatIwasdeaftothemusicofmath. Mathematics,asit’sgenerallytaughtinAmericanschoolsystems,canbeasaintlymotherofa subject.Itclimbslogicallyandmajesticallyfromadditionthroughsubtraction,multiplication,and division.Thenitsweepsuptowardtheheavensofmathematicalbeauty.Butmathcanalsobeawicke stepmother.Sheisutterlyunforgivingifyouhappentomissanystepofthelogicalsequence—and missingastepiseasytodo.Allyouneedisadisruptivefamilylife,aburned-outteacher,oran unluckyextendedboutwithillness—evenaweekortwoatacriticaltimecanthrowyouoffyour game. Or,aswasthecasewithme,simplynointerestorseemingtalentwhatsoever. Inseventhgrade,disasterstruckmyfamily.Myfatherlosthisjobafteraseriousbackinjury.We ended up in a hardscrabble school district where a crotchety math teacher made us sit for hours in th

swelteringheatdoingroteadditionandmultiplication.Itdidn’thelpthatMr.Crotchetyrefusedto provideanyexplanations.Heseemedtoenjoyseeingusflounder.

MeatagetenwithEarlthelamb.Ilovedcritters,reading,anddreaming.Mathandscienceweren’tonmyplaylist.

Bythistime,Inotonlydidn’tseeanyuseformath—Iactivelyloathedit.Andasfarasthe scienceswent—well,theydidn’t.Inmyfirstchemistryexperiment,myteacherchosetogivemylab partnerandmeadifferentsubstancethantherestoftheclass.Heridiculeduswhenwefudgedtheda inanattempttomatcheveryoneelse’sresults.Whenmywell-meaningparentssawmyfailinggrade andurgedmetogethelpduringtheteacher ’sofficehours,IfeltIknewbetter.Mathandsciencewer worthless,anyway.TheGodsofRequiredCourseworkweredeterminedtoshovemathandscience downmythroat.Mywayofwinningwastorefusetounderstandanythingthatwastaught,andto belligerentlyflunkeverytest.Therewasnowaytooutmaneuvermystrategy. Ididhaveotherinterests,though.Ilikedhistory,socialstudies,culture,andespeciallylanguage Luckily,thosesubjectskeptmygradesafloat. Rightoutofhighschool,Ienlistedinthearmybecausetheywouldactuallypaymetolearn anotherlanguage.IdidsowellinstudyingRussian(alanguageI’dselectedonawhim)thatanROT scholarshipcamemyway.IheadedofftotheUniversityofWashingtontogetabachelor ’sdegreein Slaviclanguagesandliterature,whereIgraduatedwithhonors.Russianflowedlikewarmsyrup—m accent was so good that I found myself on occasion mistakenly taken for a native speaker. I spent lot

oftimegainingthisexpertise—thebetterIgot,themoreIenjoyedwhatIwasdoing.AndthemoreI enjoyedwhatIwasdoing,themoretimeIspentonit.Mysuccessreinforcedmydesiretopractice, andthatbuiltmoresuccess. ButinthemostunlikelysituationIcouldhaveeverimagined,Ieventuallyfoundmyself commissionedasasecondlieutenantintheU.S.ArmySignalCorps.Iwassuddenlyexpectedto becomeanexpertinradio,cable,andtelephoneswitchingsystems.Whataturningpoint!Iwentfrom beingontopoftheworld,anexpertlinguist,incontrolofmydestiny,tobeingthrownintoanew technologicalworldwhereIwasasstuntedasastump. Yikes! Iwasmadetoenrollinmathematicallyorientedelectronicstraining(Ifinishedatthebottomof theclass),andthenoffIwenttoWestGermany,whereIbecameapitiablecommunicationsplatoon leader.Isawthattheofficersandenlistedmemberswhoweretechnicallycompetentwereindemand Theywereproblemsolversofthefirstorder,andtheirworkhelpedeveryoneaccomplishthe mission. IreflectedontheprogressofmycareerandrealizedthatI’dfollowedmyinnerpassionswithou alsobeingopentodevelopingnewones.Asaconsequence,I’dinadvertentlypigeonholedmyself.If stayedinthearmy,mypoortechnicalknow-howwouldalwaysleavemeasecond-classcitizen. Ontheotherhand,ifIlefttheservice,whatcouldIdowithadegreeinSlaviclanguagesand literature?Therearen’talotofjobsforRussianlinguists.Basically,I’dbecompetingforentry-leve secretarial-typejobswithmillionsofotherswhoalsohadbachelor ’sofartsdegrees.Apuristmight arguethatI’ddistinguishedmyselfinbothmystudiesandmyserviceandcouldfindmuchbetter work,butthatpuristwouldbeunawareofhowtoughthejobmarketcansometimesbe. Fortunatelytherewasanotherunusualoption.OneofthegreatbenefitsofmyservicewasthatI hadGIBillmoneytooffsetthecostsoffutureschooling.WhatifIusedthatsupporttodothe unthinkableandtrytoretrainmyself?CouldIretoolmybrainfrommathphobetomathlover?From technophobetotechnogeek? I’dneverheardofanyonedoinganythinglikethatbefore,andcertainlynotcomingfromthe phobicdepthsI’dsunkto.Therecouldn’tpossiblybeanythingmoreforeigntomypersonalitythan masteringmathandscience.Butmycolleaguesintheservicehadshownmetheconcretebenefitsof doingso. Itbecameachallenge—anirresistiblechallenge. Idecidedtoretrainmybrain. Itwasn’teasy.Thefirstsemesterswerefilledwithfrighteningfrustration.IfeltlikeIwaswearin ablindfold.Theyoungerstudentsaroundmemostlyseemedtohaveanaturalknackforseeingthe solutions,whileIwasstumblingintowalls. ButIbegantocatchon.Partofmyoriginalproblem,Ifound,wasthatIhadbeenputtingmy effortforthinthewrongway—liketryingtoliftapieceoflumberwhenyou’restandingonit.I begantopickuplittletricksaboutnotonlyhowtostudybutwhentoquit.Ilearnedthatinte...