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WHATYOUNGCHILDRENKNOWANDNEEDTOLEARNABOUTPATTERNANDALGEBRAICTHINKINGHERBERTP.GINSBURG

Whatdowemeanbypattern?Canyoungchildrenengageinalgebraicthinkingtounderstandpatterns?Howshouldwehelpyoungchildrentothinkaboutandunderstandpatterns?

REGULARITY,REGULARITY,EVERYWHERE

Youngchildren,evenbabies,areexposedtomany,manyregularitiesintheirworldsandinthemselves.Theyencounterregularitiesthatarefixedanddonotmove,likestripesonclothing.Otherregularitiesunfoldovertime,likesongs,orcontinueindefinitely,likecountingbytwos.

Babies:Onetypeofregularityinvolvesvisualpatternsintheimmediateenvironmentwhichareavailableforbabiestoseeandexamine.Indeed,researchshowsthatbabiesprefertolookatpatterns,ratherthanatuniformscenes,perhapsbecausepatternsprovideopportunitiestodiscerndifferences.BabyCassiemightdiscernthebeautifulpatternsonherBoppy(apillowonwhichshecanrecline).Theelephantsareevenlyspacedinlinesandrows,theelephant’searsareroughlyinthemiddleofeachbody,andoutofeachofthetrunksspoutsahorn-shapedstreamofwater.WereCassietoexamineherownwardrobeshecouldfindthathertopboastsstripesofthreecolors.Inshort,theenvironmentprovidesawealthofregularitiesfromwhichbabiesmay(ormaynot)extractpatternandotherinformation.BabiesallovertheworldliveinenvironmentsthatareascomplexasCassie’s,althoughdifferentfromit.

Cassiewillalsohearregularitiesinmusic,intherepeatingsequencesofsounds.Herparentswillsingtunessuchas“Hushlittlebaby,don’tsayaword,Papa’sgonnabuyyouamockingbird,”and“Therewasafarmerhadadog,andBingowashisname,oh.B-I-N-G-O…”Maybeshewillevenhearhermother’ssteadyheartbeatduringnursing.

Preschoolers:Bythetimechildrenareinpreschool,theyhaveexperiencedanincrediblylargenumberofregularities.Someinvolvespatialrelationsandshapes,asisthecaseinthispictureofRabbit’shomeinapreschoolclassroom.Achildinthisclassroomcouldseetheregular,orderlyrowsandcolumnsofrectangles,eachwithheightapproximatelydoublethebase;therampdividedintoapproximatesquares,separatedby“lines”(thinrectanglesofwood),andascendingatabouta45°angletothefloor.Thesteps,whichareallthesame,ascendinaregularpattern.Inthebackground,thegreenrectangularbuckets(prisms)arestackedneatly

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ontoponeanother,eachrisingaconstantamountovertheothers,untilthispatterniscrownedinpurple.Ofcourse,childrenwouldnotdescribetheseregularitiesinsuchadultterms.Buttheregularitiespervadethechild’squotidianenvironmentandsetthestageforlearningaboutshapes,patterns,andmuchelse.

Learninglanguagealsoinvolvesmanyregularities:Englishpluralsofnounsmostlyhaveansattheendbutindividualnounsdonot;forexample,birdsandbird,orbeesandbee.Therearefewexceptionstotherule,asinthecaseofchildandchildren.Englishindicatesthepasttensebyaddingedtoverbs:“Ikickedtheball.”Again,thereareexceptionsasin“Ifoundtheball,”insteadof“Ifindedtheball.”Youngchildrenclearlynoticetheregularities,asisevidentfromsomeoftheirmistakes,like,“IgoedtothestorewhereIfindedthemooses.”Indeed,themistakesareevidencethatthechildrens(I’mworkingonmyplurals)notedtheregularitiesandusedthemtoformarulewhichtheyovergeneralized.

Theperceptionofregularitiesisacognitiveuniversal:althoughtheircontentmayvary,childreneverywhere,regardlessofcultureandsocioeconomicstatus,encounterhugenumbersofregularitiesanddevelopanintuitive,informalunderstandingoftheregularitiestheyexperience.

PATTERN

Weusethewordpatterntorefertosome(butnotall)regularities.Patternisaverycomplexandgeneralterm,likeunderstanding.Inearlymatheducation,wetendtofocusontwobroadtypesofpatterns:dynamicandstatic.Wemightsaythatdynamicpatternsaregoingsomewhere,butstaticpatternsarehappywheretheyare.Dynamicpatterns,likeasong,unfoldovertime.Adynamicalternatingpatterncaninprincipleextendindefinitely.Bycontrast,staticpatterns,likestripesonadress,arecompleteinthemselves.Lookingatthedress,youcanseeallthestripes,thefullpattern,andtherearenomorestripes,nothingmoretosee.

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PATTERNINVARIOUSBRANCHESOFMATHEMATICS

Patterniseverywhere:inmusic,onclothing,andsoon.Patterniseverywhereinanothersensetoo.Patternisatthefoundationofmanydifferentmathematicaltopics.Forexample,countingbytwosinvolvespattern:thenumbersgetbiggerbyaconstantamountandextendindefinitely.Thesimplehundredschartshowsthepatternofthebase-tensystemofthewholenumbers.Shapeshavepatternstoo,asshownbyablocktowersymmetricalintwodimensions.Inotherwords,thestudyofpatternshouldbeanessentialpartofteachingandlearningalmostanymathematicaltopicinearlyeducation.

UNDERSTANDINGPATTERN

Thegoalofearlymatheducationshouldbetohelpchildrendeepenandmathematizetheirunderstandingofpattern.Theyneednotonlytoperceivethepatternonthedress,butalsotodescribeitinexplicitterms.Inthemostgeneralsense,childrenneedtofocusonthestructuresandrulesthatunderlieovertsurfacephenomena(justasthelinguistattemptstouncoverthegrammaticalrulesthatunderliespeech).Ifchildrenunderstandpatterns,theywillbeabletodescribethem,reproducethem,extendthem,fillinmissingelements,andcreatenewpatterns.PreschoolerscanlearnmuchmorethanaroterecitationofABABpatterns,whichisoflimitedimportanceinearlymatheducation.

ALGEBRAICTHINKINGANDALGEBRA

Whenchildrensearchforthestructuresandrulesunderlyingpattern,theyarebeginningtoengageinalgebraicthinking.Thechildmightdiscoverthateachtowerinaseriesoftowersincreasesbytwoblocks,sothatthetowerafterthelastoneconstructed,whichis,say,Lblockstall,mustbeL+2blockstall.Andthenextoneafterthatmustbe(L+2)+2,andsoon.Althoughshedoesnotwriteanythingorusewrittensymbolism,thechildisengagedinsimplealgebraicthinkingthatexpressesageneralrule.

Eventually,inearlyelementaryschool,thechildwhohasconstructedatowerbeginningwithtwoblocksandthenincreasingitsheightbytwoblockseachtime,candeterminethatthefifthtowerintheseries,evenifshehasnotyetconstructedit,mustbe5X2blockshigh.Andafterthat,thenextstepistorealizethatthetowerinthenthposition,whateveritis,musthavenX2blocks.

Althoughthewrittenstatementofthisruleusingmathematicalsymbolismmaynotbelearnedforsometime,thethinkingisalgebraic.Inthissense,wecansaythatpatternandalgebraicthinkingprovidethegatewaytotheformalalgebrataughtinlatergrades.

WHATPATTERNSSHOULDYOUNGCHILDRENSTUDY?

Childrenshouldlearnaboutavarietyofpatterns,bothstaticanddynamic.Theseinclude:

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• alternatingpatterns,includingo theshapes:circle-square-circle-square…o thecolors:red-blue-blue-red-blue-blue…o themusicalsounds:do-do-re-re…,

• growingpatterns,forexample,theincreasingheightofblocktowers,• numberpatternssuchasskipcounting,groupingbytens,andsimplenumbertables(like

thehundredschart,whichcancontainagreatdealofrichmathematics),• equivalenceofpatterns,forexampletheequivalencebetweenanalternating-colors

patternandapatternofalternatingshapesorevensounds,• musicalpatternssuchasascendinganddescendingmelodies,• spatialrelations,likethesymmetriesinvolvedinblockconstructions,• andmethodsforrepresentingpatterns:bydrawingsorsymbols(likeAandB)and

eventuallythewrittensymbolismthatisattheheartofalgebra.

Childrenalsoneedtolearn,atleastimplicitly,thatpatternsareabstractideasthatcanapplytoalmostanything,beitnumbers,shapes,orunicorns.Wecanusetheideaofpatterntounderstandaseriesofmusicalnotesorthesettingofthesun.Conversely,wecanusecandiesormonstersoranumberlinetoillustrateapattern.Patternscantakemany,manyforms.Patternsaremuch,muchmorethanthetraditional(andvaluable)ABAB.

HOWSHOULDWETEACHPATTERNS?

Oneapproachistoseizeuponteachablemomentsthatariseintheeverydayenvironment.Forexample,theconstructionsthatchildrencreateintheirartworkandblockplayoftencontainpatterns.Theteachercanaskchildrentodescribetheircreationsandperhapseventalkabouthowtheconstructionsinvolvepatterns.“Tellmeaboutwhatyoudidwiththeseblocks.Whydidyouputtheseoverhereandtheotheronesoverhere?Doyouseeapattern?Whydoyouthinkit’sapattern?”

Anotherapproachisguidedinstruction,inwhichtheteacherusesvariousmanipulativesorothertools(apianoorthevoice,orpaperandcrayon,ordigitaldevices)tosystematicallydevelopvariousideasaboutpattern.“Let’susetheseblockstomakeastaircase.”“Trytomakethispartofthehousethesameasthatpart.”

Regardlessofthespecificcontent,manipulatives,andtoolsused,teachersshouldemphasizethinkingandcommunicatingaboutpatternsandthestructuresunderlyingthem.Theuseofstandardwrittensymbolsisusuallynotappropriateforchildrenatthisagelevel,butdescribing,communicating,andthinkingareessentialfromtheoutset.

Therefore,nomore,orperhapsonlyalightappetizerof:ABABABABABABABABABABABABABABABABABABABABABAB!