» » Acquaintance with geometry as one of the main goals of teaching mathematics to preschool children


Acquaintance with geometry as one of the main goals of teaching mathematics to preschool children

>MINISTRYOFEDUCATIONOFTHEREPUBLICOFBELARUS

>EDUCATIONALESTABLISHMENT BELARUSIANSTATE

>PEDAGOGICALUNIVERSITYNAMEDAFTERMAXIMTANK

>FOREIGNLANGUAGESDEPARTMENT

>ACQUAINTANCEWITHGEOMETRYASONEOFTHEMAINGOALS >OFTEACHINGMATHEMATICSTO

>PRESCHOOLCHILDREN

>Executedby:

>student ofmagistracydepartment

>YuliandreevnaDunai

(>tel.: 8-029-3468595)

ScientificSupervisor:

>Professor

>Doctor ofpedagogical science,

>I.V.Zhitko

EnglishSupervisor:

>Doctor of Psychology

>AssociateProfessor

N. G.Olovnikova

>Minsk, 2009


>CONTENTS

>INTRODUCTION

I.HISTORICALPATTERNSANDPERSPECTIVESOFTEACHINGMATHEMATICS INPRIMARYSCHOOL

II.THEPURPOSESANDCONTENTOFMODERNMATHEMATICALEDUCATION INPRIMARYSCHOOL

III.THEMETHODSOFCHILDSACQUAINTANCEWITHGEOMETRICSHAPES

>CONCLUSION

>CONTENTS

>REFERENCES


>INTRODUCTION

 

Youngchildren ">do"mathspontaneously intheirlives and intheirplay.Mathematicallearningforyoungchildrenismuchmorethan thetraditionalcounting andarithmeticskills.Itincludes avariety ofmathematicalsections ofamongwhich theimportantplacebelongs togeometry.We've allseenpreschoolersexploringshapes andpatterns,drawing andcreatinggeometricdesigns,takingjoy inrecognizing andnamingspecificshapestheysee.Thisisgeometry anarea ofmathematics thatisone of themostnatural andfunforyoungchildren.

>Geometryis thestudy ofshapes,bothflat andthreedimensional, andtheirrelationships inspace.

>Preschool andkindergarten >childrencanlearnmuchfromplayingwithblocks,manipulatives (>Jensen and'Neil),differentbutordinaryobjects (JulieSarama, Douglas H.Clements),boxes,snacks andmeal (>EllenBoothChurch).Also cardgames,computergames,boardgames, andothers allhelpchildrenlearngeometry.

>This problemisrelevantbecause thegeometricalconceptsshouldbeformedsinceearlychildhood.Geometricalconceptshelpchildren toperceive the world.Also itwillprovidefuture success inacademicachievement :as therudiments ,childrenlearn inprimaryschool,from thebasisforfurtherlearning ofgeometry. Gamemethodshelpchildren tounderstandsomecomplexphenomena ingeometry.Theyalsoarenecessaryfor thedevelopment ofemotionally-positiveattitudes andinterest to themathematics andgeometry.


I.HISTORICALPATTERNSANDPERSPECTIVESOFTEACHINGMATHEMATICS INPRIMARYSCHOOL

>Throughouthistory,mathematicalconcepts andsystemshavebeen developed inresponse toreal-lifeproblems.Forexample, thezero,whichwasinventedby theBabyloniansaround 700.,by theMayansabout 400a.d., andby theHindusabout 800a.d.,wasfirstused tofill acolumn ofnumbers inwhichtherewerenonedesired.Forexample,an 8 and a 3 next toeachotheris 83;butifyouwant thenumber toread 803 andyouputsomethingbetween the 8 and 3 (>otherthanemptyspace), itismorelikely tobereadaccurately (>Baroody, 1987).When itcomes tocounting,tallying, orthinkingaboutnumericalquantity ingeneral, thehumanphysiologicalfact oftenfingers andtentoeshasled in allmathematicalcultures tosomesort ofdecimalsystem.

>History'searlyfocus onappliedmathematicsis aviewpointwewoulddowell toremember today. Afewhundredyearsago auniversitystudentwasconsiderededucatedifhecoulduse hisfingers todosimpleoperations ofarithmetic (>Baroody, 1987);nowweexpect thesame ofanelementaryschoolchild. Theamount ofmathematicalknowledgeexpected ofchildren todayhasbecomesoextensive andcomplex that itiseasy toforget thatsolvingreal-lifeproblemsis theultimategoal ofmathematicallearning. Thefirstgraders inSuzanneColvin'sclassesdemonstrated theeffectiveness oflying instruction tomeaningfulsituations.

>Itspossible torecall thatmorethan 300yearsago,Comeniuspointed out thatyoungchildrenmightbetaught tocountbut that ittakeslongerforthem tounderstandwhat thenumbersmean. Today,classroom researchsuchasSuzanneColvin'sdemonstrates thatyoungchildrenneed tobegivenmeaningfulsituationsfirst andthennumbers thatrepresentvariouscomponents andrelationshipswithin thesituations.

Theinfluences of JohnLocke andJeanJacquesRousseauarefelt todayaswell.Lockeshared apopularview of thetime that the worldwas afixed,mechanicalsystemwith abody ofknowledgefor all tolearn.Whenheappliedthisview toeducation,Lockedescribed theteaching andlearning processaswritingthis world ofknowledge on theblank-slatemind of thechild.Inthiscentury,Locke'sviewcontinues tobe apopularone.Itisespeciallypopular inmathematics,where itcanbemoreeasilyargued that,atleastat theearlylevels,thereis abody ofknowledgeforchildren tolearn.

B. F.Skinner, whoappliedthisview to aphilosophy ofbehaviorism,referred tomathematicsas ">one of thedrillsubjects."WhileLockerecommendedentertaininggames toteacharithmeticfacts,Skinnerdevelopedteachingmachines andaccompanyingdrills,precursors totoday'scomputerizedmathdrills.Onecritic ofthisapproach tomathematicslearninghassaid that,while itmaybeusefulformemorizingnumberssuchasthose in atelephonelisting, ithasfailed toprovide apowerfulexplanation ofmorecomplexform: oflearning andthinking,suchasmemorizingmeaningful information or problemsolving.Thisapproachhas, inparticular,beenunable toprovide a sounddescription of thecomplexitiesinvolved inschoollearning,like themeaningfullearning of thebasiccombinations orsolvingwordproblems (>Baroody, 1987).

>Rousseau'sviews of howchildrenlearnwerequitedifferent,reflecting hispreferencefornaturallearning in asupportiveenvironment.During thelateeighteenthcenturyas today,thisviewarguesforreal-life,informalmathematicslearning.Whilethisapproachismorecloselyaligned tocurrentthinkingabout thewaychildrenlearnthanis theLocke/Skinnerapproach, itcanhave theundesiredeffect ofgivingchildrensolittleguidance thattheylearnalmostnothingat all.

Theview thatseemsmostsuitableforyoungchildrenis thatinspiredbycognitivetheorists,primaryamongthemJeanPiaget. Threetypes ofknowledgewereidentifiedbyPiaget (>Kamii andJoseph, 1989), all ofwhichareneededforunderstandingmathematics. Thefirstisphysical, orempirical,knowledge,whichmeansbeingable torelate to thephysical world.Forexample,before achildcancountmarblesbydroppingtheminto ajar,sheneeds to know how to hold amarble and how itwillfalldownwardwhendropped.

The secondtype ofknowledgeislogico-mathematical, andconcernsrelationshipsascreatedby thechild.Perhaps ayoungchildholds alargeredmarble inonehand and asmallbluemarble in theother.Ifshesimplyfeelstheirweight andseestheircolors,herknowledgeisphysical (orempirical).Butifshenotes thedifferences andsimilaritiesbetween the two,shehasmentallycreatedrelationships.

Thethirdtype ofknowledgeissocialknowledge,whichisarbitrary anddesignedbypeople.Forexample,namingnumbersone, two, andthreeissocialknowledgebecause, inanothersociety, thenumbersmightbeichi,ni,san oruno,dos,tres. (>Keep inmind,however, that the realunderstanding ofwhatthesenumbersmeanbelongs tologico-mathematicalknowledge.)

>ConstanceKamii (>Kamii andDeClark, 1985), aPiagetianresearcher,hasspentmanyyearsstudying themathematicallearning ofyoungchildren.Afteranalyzingteachingtechniques, theviews ofmatheducators, andAmericanmathtextbooks,shehasconcluded thatoureducationalsystemoftenconfusesthesethreekinds ofknowledge.Educatorstend toprovidechildrenwithplenty ofmanipulatives,assuming thattheywillinternalizemathematicalunderstandingsimplyfromthisphysicalexperience.Oreducatorsignore themanipulatives andfocusinstead onpencil-and-paperactivitiesaimedatteaching thenames ofnumbers andvariousmathematicalterms,assuming thatthissocialknowledgewillbeinternalizedas realmathlearning.Somethingismissingfrombothapproaches,saysKamii.

>Traditionally,mathematicseducatorshavenotmade thedistinctionamong thethreekinds ofknowledge andbelieve thatarithmeticmustbeinternalizedfromobjects (>asif itwerephysicalknowledge) andpeople (>asif itweresocialknowledge).Theyoverlook themostimportantpart ofarithmetic,whichislogico-mathematicalknowledge.

>In thePiagetiantradition,Kamiiargues that ">childrenshouldreinventarithmetic." Onlybyconstructingtheirownknowledgecanchildrenreallyunder standmathematicalconcepts.Whentheypermitchildren tolearn inthisfashion,adultsmayfind thattheyareintroducingsomeconceptstooearlywhileputtingothers offtoolong.Kamii's researchhasledher toconclude,asSuzanneColvindid, thatfirstgraders Endsubtractiontoodifficult.Kamiiarguesforsaving ituntillater,when itcanbelearnedquickly andeasily.Shealsopoints tostudies inwhichplacevalueismasteredbyabout 50percent offourthgraders and 23percent of a group of secondgraders.Yetplacevalue andregroupingareregularlyexpected of secondgraders!

>Asanexample ofwhatchildrencandoearlierthanexpected,Kamii (1985)points totheirdiscovery (orreinvention) ofnegativenumbers, aconcept thatdoesn'tevenappear inelementarymathtextbooks.Based onherexperienceswithyoungchildren,Kamiiargues that itisimportant toletchildrenthinkforthemselves andinventtheirownmathematicalsystems.WithPiaget,shebelieves thatchildrenwillunderstandmuchmore,developing abettercognitivefoundationaswellasself-confidence:children whoareconfidentwilllearnmore in thelongrunthanthose whohavebeentaught inways thatmakethemdistrusttheirownthinking. . . .Children whoareexcitedaboutexplainingtheirownideaswillgomuchfarther in thelongrunthanthose whocanonlyfollowsomebodyelse'srules andrespond tounfamiliarproblemsbysaying, "Idon't know how todo itbecause Ihaven'tlearned it inschoolyet."

>Inrecentyears, theNationalCouncil ofTeachers ofMathematics (>NCTM)hasgivenmuchconsideration to theinternationalfailure ofAmericanchildren inmathematics, andhasdevised aset ofstandards thatecho, inmanyways, thePiagetianperspective ofKamii. The Curriculum andEvaluationStandardsforSchoolMathematics (1989)preparedby theNCTMaddresses theeducation ofchildrenfromkindergarten up.Some of themoreimportantstandardsare:

■  >Childrenwillbeactivelyinvolved indoingmathematics. >NCTMseesyoungchildrenconstructingtheirownlearningbyinteractingwithmaterials,otherchildren, andtheirteachers.Discussion andwritinghelpmake newideasclear.Languageisatfirstinformal, thechildren'sown, andgraduallytakes on thevocabulary ofmoreformalmathematics.

■  Thecurriculumwillemphasize abroadrange ofcontent. >Children'slearningshouldnotbeconfined toarithmetic,butshouldincludeotherfields ofmathematicssuchasgeometry,measurement,statistics,probability, andalgebra.Study in allthesefieldspresents amorerealisticview of the world inwhichtheylive andprovides afoundationformoreadvancedstudy ineacharea.Allthesecontentareasshouldappearfrequently andthroughout theentirecurriculum.

■  Thecurriculumwillemphasizemathematicsconcepts. >Emphasis onconceptsratherthan onskillsleads todeeperunderstanding.Learningactivitiesshouldbuild on theintuitive,informalknowledge thatchildrenbring to theclassroom.

■  >Problemsolving andproblem-solving,approaches toinstructionwillpermeate thecur>riculum. >Whenchildrenhaveplenty ofproblem-solvingexperiences,particularlyconcerningsituationsfromtheirownworlds,mathematicsbecomesmoremeaningful tothem.Theyshouldbegivenopportunities tosolveproblems indifferentways,createproblemsrelated todatatheyhavecollected, andmakegeneralizationsfrombasic information.Problem-solvingexperiencesshouldlead tomoreself-confidenceforchildren.

■  Thecurriculumwillemphasize abroadapproach tocomputation. >Childrenwillbepermitted tousetheirownstrategieswhencomputing,notjustthose offeredbyadults.Theyshouldhaveopportunities tomakeinformaljudgmentsabouttheiranswers,leading totheirownconstructedunderstanding ofwhatisreasonable.Calculatorsshouldbepermittedastools ofexploration.Itmaybe thatchildrenwillcomputebyusingthinkingstrategies,estimation, andcalculatorsbeforetheyarepresentedwithpencils andpaper (>AdaptedfromTrafton andBloom, 1990).

TheNational Associationfor theEducation of YoungChildren, initspositionstatementregarding >Developmental /AppropriatePractices (>Bredecamp, 1987),arrivesatviews ofteachingmathematics toyoungchildren thatreflectthose ofConstanceKamii and theNCTM.Theirpositionregardinginfants,toddlers, andpreschoolersis thatmathematicsshouldbepart of theday'snaturalactivities:countingchildren in theclass orcrackersforsnacks,forexample.For theprimarygradestheyaremorespecific,identifyingwhatisappropriate andinappropriatepractice.Table 1summarizestheirguidelines.

>Table 1. >APPROPRIATEMATHEMATICS INTHEPRIMARYGRADES (>THENAEYCPOSITION)

>APPROPRIATEPRACTICE >INAPPROPRIATEPRACTICE

>Learningisthroughexploration,

>discovery, andsolvingmeaningfulproblems

>Noncompetitive,impromptuoral

">mathstumper" andnumbergamesareplayedforpractice.

>Mathactivitiesareintegratedwithothersubjectssuchas science andsocialstudies >Learningisbytextbook,workbooks,practicesheets, andboardwork
>Mathskillsareacquiredthroughplay,projects, anddailyliving >Mathistaughtas aseparatesubjectat ascheduledtimeeachday

Theteacher'sedition of thetextisusedas aguide tostructure

>learningsituations andstimulate

>ideasforprojects

>Timedtests onnumberfactsaregiven andgradeddaily

>Manymanipulativesareused

>includingboard, card, and

>paper-and-pencilgames

>Teachersmovesequentiallythrough thelessonsasoutlined in theteacher'sedition of thetext
Onlychildren whofinishtheirmathseatworkarepermitted touse thefewavailablemanipulatives andgames

>Competitionbetweenchildrenis

>used tomotivatechildren tolearn

>mathfacts.

TheNCTMStandards, theNAEYCpositionstatement, andstudieswithyoungchildrencarried outbysuchresearchersasConstanceKamii andSuzanneColvinbringus totoday's bestanalysis of howchildrenlearnmathematics. Theconclusiontheseresearchers andtheoristshavereachedarebasednotonly ontheirworkwithchildren,but ontheirunderstanding ofchild development [6,pp. 426 - 436].


II. >THEPURPOSESANDTHECONTENTOFMODERNMATHEMATICALEDUCATION INPRIMARYSCHOOL

>Oftenchildrenquestion theimportance oflearningmathematics.Now thathandheldcalculators andhomecomputersarecommonlyavailable,questionsabout therelevance oflearningmathhavebecomelouder.Nevertheless,educatorscontinue tomakemath the secondmosttime-consumingsubject inelementaryschool (>afterreading). Thereasonsforteachingmatharemany, and thegoals ofgeneraleducationrequire thatmathbe amajorpart of thecurriculum.

Thegoals ofmatheducation changeslowlyfromgrade tograde.Mostchildrenrequire all thetimefrompreschoolthrough the end ofgrade 6just tolearn themeaning ofwholenumbers,fractions, anddecimals and how toperformoperationswiththem (>Ofcourse, anumber ofothermathematicalideasarealsotaughtalong theway).Althoughactualcomputationscan oftenbedonewith acalculator,answersare ofnousewithoutanunderstanding ofbasicmathprocesses.

>Businesspeople whoareinvolved insettingpricesfind thatelementaryalgebraishelpful.Geometryismorethanuseful inplanningmanysewingprojects.Scientists of allkinds,includingbiologists andsocialscientists,needcalculus tosolveproblems anddo research.

>As aresult,high-schoolmathcoursesarelargelydesigned toprovide thebasics thatareneeded insuchsituations and topreparestudentsforcollege.Somecollegesrequire allstudents totakemathematics,butmanyhavemathrequirementsonlyforstudents ofscience,engineering, andadvancedplanningfor business.

>Nearlyeveryonestartslearningmathematicsbeforegoing toschool.Whentelevisionfirstbecamepopular in the1950's,somepeoplejoked thatchildrenwerecoming tokindergartenalreadyable tocountatleastas highas thenumbers on thechannelselector.But thejoketurnedseriouswhenpeoplerealized thatveryyoungchildrenreallywerelearning tocountfrom TV,especiallyiftheywatchededucationalshowssuchastoday's ">SesameStreet." Thetotsalsolearnedcolors,shapes, anddirectionssubjects thatusuallyform alargepart of thekindergartenmathematicsprogram [7,p. 13].

>Mathematicslearningissequentialoneideabuilds onanother.Consequentlymathematicsistaught innearly thesamesequence inalmosteveryschool in the United States [7,p. 29].

>Inpreschool ( 2till 5years) thechildrengaininformalpracticewithcounting andshapes.So,one of thefirstgoals of akindergartenmathematicsprogramis topresentnumbers andcounting inways thatshow howwords,meanings, and thesymbols thatrepresentthemarerelated. Thesymbols,suchas thenumerals 1through 10areespeciallyimportantbecausemanychildrencancountcorrectlybeforetheyareable togetanymeaningfrom thesymbols [7,p. 14].

>Theyalsolearn themeaning ofwordssuchastop, in, andleft.Preschoolsputmuchemphasis ongames andactivities thatusesimplecounting. Reading andwritingnumeralsarealmostnevertaught.

>Not allchildrengo topreschool.All of thetopicscoveredat thatlevelaretaughtagain inkindergarten andgrade I. Theschoolscannotassume that allchildrenwillhavehad thesameearlymathexperiences.

Today,nearly allchildren in the United Statesgo tokindergarten ( 'years). Thebeginningpart ofkindergartenfocuses oninformalexperiencessimilar tothose inpreschool.Later in theyear,moreformalexperiencesstart.Sometimesbooks orkitsareused toorganizemathematicslearning,butmanykindergartenteachersbelieve that itistooearly toaskchildren toworkwithbooks orevenwithspecificmathematicsmaterials.Someclassroomsmayhave acomputerwithmath-related software tohelpteachearlymathconcepts.

>Childrenlearn twoways tocomparenumbers.Thus,evenbeforetheylearn theorder of thenumbers,childrencanunderstand thatsomenumbersarelargerthanothers and thatsomenumbersaresmaller [7,p. 30].

>Anotherimportantearlyskilliswritingnumerals.Thisskillisessentialbecause itenableschildren tocommunicate onpaperwiththeirteachers andwithothers inlater life.

>Althoughunderstanding themeaning ofnumbersis themaingoal ofbeginningmathematicseducation, itisnot theonlygoal.Therearenumeroussubgoals.Inkindergarten orgrade 1, thefirstsubgoalmaybe toteachsuchbasicconceptsastop,left, andbefore.Theseideashavemanyimportantusesbothinside andoutside theclassroom.Forexample, ateacherwilllosemuchtimeexplainingifchildrendonotunderstand asimpledirectionsuchas ">Lookat thepictureat thetop of thepage."

>Anotherearlygoalis toteachchildren tounderstandpatterns.Becausemathematicsis thestudy ofpatterns, theteaching ofpatternrecognitionhasbecome apart of thestandardcurriculum inkindergartenthroughgrade 3.Atfirst,childrenarepresentedwithsimplepatterns tocomplete.Thentheymove on tomorecomplexpatterns of thesamekind.

>Theseexercisesaresometimescalledattributestudiesbecausesuchattributes (orcharacteristics)asbeingcolored ornot andbeingsquare ornotare thefocus ofattention inaddition to theattribute ofpattern.Studentsmayuseblocks orpictures todotheseexercises.

>At thesametime,studentsmaybeginsomesimpleworkwithgeometry.Onegoal ofbeginninggeometryis toteachchildren torecognize themostsimpleshapesthesquare, thecircle, thetriangle, and therectangle.Teachingsuchbasictermssimplifiesclass roomexplanations andlays thefoundationforfutureworkwithgeometry.Also,someshapesareusedwhenfractionsareintroduced.

>Childrenrespondbetter tothree-dimensionalshapesthantheydo totwo-dimensionalpictures inbooks.Thisprobablyoccursbecause,asidefromprintedmaterials andtelevision, theshapesaroundthemactuallyarethree-dimensional.Therefore,mosteducatorsbelieve thatearlyexperienceswithgeometryshouldincludesuchsolidshapesas thecube,sphere,cone,cylinder, andpyramid.Infact, thesolidfiguresareoftentaughtfirst, and thetwo-dimensionalshapesareexplained interms of thesolids. Asquare,forexample,isoneface of acube.This ">analytic"approach togeometryisusuallynottried inkindergarten,but itmaybestartedasearlyasgrade 1.

>Anothergeometricconcept thatisnearlyalwaystaughtearlyissymmetry,specificallywhatmathematicianscall linesymmetry. Thereasonforincludingsymmetryatthislevelis that itisuseful indesign.Also, itisnotdifficultforyoungchildren torecognize thedifferencebetweensymmetricalobjects,suchas thecapitalletters A and B, andanasymmetricalobject,suchas theletter F.

>Itisclear thatfor A and B, thedotted lineseparates twoparts thatareidentical.Onepartis thereflection of theother. Nosuch linecanbedrawnfor F.

>Anothergoal ofearlyelementaryeducationis toteachchildrenaboutmeasurement.Thisinvolvesmanyskills andconcepts.Often thefirstnotiontaughtis that ameasurementis anumber ofstandardunits.Itiseasiest toexplainthisideabymeasuringstraightlines [7,pp. 15-17].

>So, formalkindergartenmathprogramusuallystartsatleastsixweeksinto theyear andoftenaslateas thebeginning of the secondterm.Itgenerallyincludescounting;ordering andcomparingnumbers;preparationforaddition andsubtraction;comparingsize:preparationfortellingtime; theconcept of apenny;recognizingsquares,triangles, andcircles;suchconceptsastop,bottom,front,back. in. on,between,left, andright (>whichareneeded toexplainlessons);classifyingobjects; andrecognizingpatterns.Mosttimeisspent oncounting andlearning tounderstandnumbersthroughone-to-onematching andcomparing andorderingnumbers.

>Everygradehas amajorgoalformathematicsachievement,although itisnotalwaysformally defined.Thus,forkindergarten, themajorgoaliscounting [7,p. 31].


III.THEMETHODSOF >CHILDSACQUAINTANCEWITHGEOMETRICSHAPES

>Preschoolers'intuitiveknowledge ofgeometryfrequentlyexceedstheirnumericalskills.Bybuilding onstrengths andinterests thatarealreadypresent,youcanfosterenthusiasmformath andprovide alogicalcontext todevelopnumberideas.

>Byagesix,childrenoftenhavestableyetlimitingideasaboutshapes.Four-year-oldTinatellshermother, ">That'snot asquare.It'stoobig. Asquarelookslikethis."HerfriendCharlieadds, ">Triangleshave tobethisway.That'snot atriangle.It'stooupsidedown."Broadenchild'sunderstandingbypointing out avariety ofexamples squares thataremanysizes andtriangles thatare ">long," ">skinny," ">fat," andturned inmanydirections. Youcanalsoencouragedeeperthinkingaboutshapesnotjustthroughhands-onactivities anddiscussions,butthroughpicturebookssuchas TheGreedyTrianglebyMarilynBurns [1,pp. 5-6].

>Geometryis thestudy ofshapes,bothflat andthreedimensional, andtheirrelationships inspace.Geometryentersinfants'livesfrombirthastheyattempt tomakesense of theshapes intheirenvironment:cribbars,stuffedanimals,mother'sbreast andface, thedoor to thebedroom.Geometricshapesbecomesome of thefirstintentionalscribblesmade inyoungchildren'sdrawings, andtheydelight intheirawareness of theshapesaroundthem.

>Suchnaturalinterestdeservesencouragement andinformalteaching intervention.In the1950s, twoDutcheducatorsdeveloped atheory ofstagedevelopment ingeometricunderstanding. ThevanHieletheoryhasgainedacceptance in the United States inrecentyears, andapplies tochildrenfrom theearlyyearsthrough highschool.Animportanttenet of thetheoryis thatchildrendonotgrowthrough thestagesautomaticallybut,withteacherassistance,willdosocompetently (>Teppo, 1991).Whatchildrenareexposed to in theearlyyearssets thestageforlearning ingeometrythroughouttheir entireschoolexperience.Through theprimarygrades,childrenareat theearliest,visual,stage inwhichtheyexploretheirenvironment tolearn toidentify theshapeswithin it.Activitiessuchasdescribing,modeling,drawing, andclassifyinghelpthemdevelop aspatialsense [6,p. 457 ].

Douglas H.Clementsofferssuchreceptionforformation ofknowledge ofchildrenaboutshapes.In apreschoolclassroomyoumightsee ascenelikethis: AteacherchallengesMichelle andDebbie tousetheirbodies tomake ashapetogether. The girlssitdownfacingeachother andstretchtheirlegsapart.Withfeettouching,theycreate adiamond.Anotherchildtakes a look,sees thediamondshape, andsays: ">Ifweputsomeoneinside,wecanmake twotriangles."Immediately,theyaskRay toscrunch in andlieacross themiddle.

>Itworks. Adiamondcanbedividedinto twotriangles.Michellenotes thatthere's ashape thathassixsides, andshewants totrymakingone ofthose.Anotherchildmayeven know that theshapeiscalled ahexagon.After abriefdiscussion,Michellegetsfiveotherchildrentogether.Then,underherdirection,they allliedown on thefloor andcreate asix-sidedshape [1,p. 5].

>EllenBoothChurch ( aformerprofessor ofearlychildhood ) >suggestsparents toapplyfollowingreceptions in thecourse ofacquaintance ofchildrenwithshapes:

Sort the house:Collect avariety ofhouseholdobjects,likebottlecaps andenvelopes, andinviteyourchild tosortthemintodifferentpiles oneforcircles,rectangles, andso on.Inviteher togo on atreasurehuntaround the house tofind ">onemorething"foreachpile.

Cutshapesandwiches:Usecookiecutters tomakedaintyteasandwichesfor aShapeTeaParty!Cut thebreadwith thecutters,thenspread theshapeswithcream cheese.Tryfilling thebreadsliceswithfoods ofdifferentshapes,like aroundtomatoslice onsquarebread or atriangle of cheese onroundbread.

Makepuzzles:Uselarge,coloredfilecards,folders, orposterboard tomakeshapepuzzlestogether.Cut outbigbasicshapes,likecircles ortriangles, andthencuteachinto two orthreepieces.Yourchildcandecorate theshapepuzzleswithcrayons andthenput thepuzzlesbacktogether.

Exploreletters:Huntforshapeswithin theletters of thealphabet.Write out thealphabet incapitalletters, andhaveyourchildfindwhichshapesmake upeach. Youcanalsoshowher howshapesformasyouwrite;inviteher toexperimentbydrawinglargeshapes ofherown andturningthemintoletters.

Paste apicture:Provideyourchildwithpaste and avariety ofcut-outpapershapes indifferentsizes andcolorsforcreatingpictures.Encouragehim tocombineshapes tocreatedesigns orfamiliarimages.Forexample,hemightmake ashapepersonwith acirclehead, asquarebody, andrectanglelegs.

Searchthroughstories:Manychildren'spicturebookauthor-illustratorsboldlyusebasicshapes intheirillustrations (>booksbyTanaHoban,EricCarle, and LeoLionniareparticularlygoodplaces tostart).Asyoureadwithyourchild,askher topoint out theshapesshefinds ineachpicture.

Take awalk:Moveshapeplayoutsidebytaking awalkwithcardboardcut-outs of atriangle,circle,square, andrectangle.Yourchildcanmatchthem toobjectslikesigns,plants,doors, andcartires.Takealongyourdigitalcamera tosnapphotos of theshapes.Thenprintthem andmake a familyshapebook [2,p. 12].

>EllenBoothChurchconsiders thatboxesare therawmaterials ofcreativethinking.Exercises: >Big andLittleSorting, >Boxes andLidsMatching, >SerialBoxes, Fill '>Em Up, >Make aShapeFeely-Box, >TreasureBoxes, >Shadow BoxCollages, >PlayStore, >You-in-the-Box andothers [3,pp. 9-10].

>AlsoEllenBoothChurchoffers tousesnacks andmealtime toteachbigideaswithtaste andease. Thekitchenisfilledwithmanywonderfulfoods andcookingtools in avariety ofcolors,sizes, andshapes.Itis theperfectlaboratoryforexploringsome of thefirsttopicschildrenlearn inschool:color,shape, andsize.Understandingtheseconceptsisimportantbecauseyourchildusesthem inobserving,comparing, anddiscussing allshesees andencounters. Theability tonotice,use, andvoicesimilarities anddifferencesareat theheart ofbeginningmath, science, andreadingskills.Forexample,receptionforacquaintancewith theshape.Eat asquaremeal.Wehave allheard of theimportance ofeating asquaremeal ofhealthyfoods,butwhynothave areally ">square"meal?Servewaffles (>big andlittlesquares)with asidedish ofpineapplechunksforbreakfast.Have asnack ofsquare cheeseslices onsquarecrackersplaced on asquarenapkin.Asyouarepreparing andenjoyingyourmeals,askyourchild tonotice thesimilarities anddifferencesbetween thedifferentsquares. Helphernotice that all thesquareshavefoursides,butcanbevarioussizes.For afunchallenge,giveyourchild aslice ofpre-wrapped American cheese.Assheunwraps it,askher howshecanfoldher cheesesquareinto atriangle (>point topoint).Oneway tofocus on aparticularcoloris tohave ameal all in thesamecolor.Thiswillhelpyourchild tonotonlyfocus onlearning thename of aparticularcolor,butalso itwillhelphersee themanydifferentshades of aparticularcolor.Forexample,not allorangesare theexactsameshade andso on [4,p. 3].

>JulieSarama, PhD, and Douglas H.Clements, PhDofferaresomeactivitiesparentscantryathome tosupportmathlearning:

1.Playwithmanydifferentbutordinaryobjects.Childrenstretchtheirimaginationswhentheyplaywithordinaryobjects.Many ofthesehaveinterestinggeometricproperties.Forexample,somecylindricalobjects,suchaspapertowelrolls and toiletpaperrolls,canbelookedthrough,rolled, andused torepresentobjectssuchastowers in acastle.Alltheseactivitiesdevelop thefoundation ofunderstandingthree-dimensionalshapes.

2.Playwith thesameobjects indifferentways.Creativity andthinkingareenhancedifchildrenplaywith thesameobject indifferentways.When theboxis acontainer,then a house,thenstairs,thencutapartinto a trackfor acar,childrensee therelationshipsbetweenshapes,real-worldobjects, and thefunctionstheyserve.

3.Playwith thesametoysagain andagain.Somematerialsaresobeneficial that allchildrenshouldplaywiththemagain andagainthroughouttheirearlyyears.Blocksfor allages,Duplos, andLegos,at therightage,canencouragechildren tobuildstructures,learnaboutshapes andcombinethem,comparesizes, andcount.Theyalsolearn tobuildmentalimages,plan,reason, andconnectideas.Sand andwaterplayareinvaluableforlearning thefoundations ofmeasurementconcepts.Creatingpatterns anddesignswithstringingbeads,blocks, andconstructionpaperdevelopgeometric andpatterningideas.Puzzlesdevelopspatialthinking andshapecomposition.

4.Everydayobjectscanbefun,asisplayingwithconstructivetoysagain andagain.However,buyingtoomanydifferenttypes ofcommercialtoyscandecreasechildren'smathematicalthinking andcreativity.Lesscanbemore!Rotatingtoyskeepschildreninterested.

5.Countyourplayfulactions.Manygames andplayfulactivitiesjustcall outforcounting.Howmanytimescanyoubounce aballoon in theairbefore ittouches theground?Howmanytimescanyouskiprope?

6.Playgames. Cardgames,computergames,boardgames, andothers allhelpchildrenlearnmathematics.Theycountdots oncards andspaces tomove.Countinghelpsthemconnectonerepresentation ofnumbers toanother.Theylearn toinstantlyrecognizepatterns ofnumbers,suchas thedotpatterns ondice ordominoes.Somegamesinvolveusing atimer.Concentration andBingoinvolvematching.Checkers andCandy Landinvolvespaces andlocations.

7.Playactivegames.Beanbagtossing,hopscotch,bowling, andsimilargamesinvolvemoving anddistances.Most of thegamesinvolvenumbers andcountingforscorekeeping,too.Gamessuchas ">MotherMay I?"involvecategories ofmovement. ">Follow theLeader"canbeplayedusingmathconcepts,suchasannouncingyouwilltakefivelargestepsbackward,then twosmallsteps to theside.

8.Discussmathplayfully.Mathwillemergewhenyouhelpyourchildrensee themath intheirplay.Talkaboutnumbers,shapes,symmetry,distances,sorting, andsoforth.Doso in aplayfulway,commenting onwhatyousee in thechildren'sconstructiveplay.

9.Provideampletime,materials, andteachersupportforchildren toengage inplay, acontext inwhichtheyexplore andmanipulatemathematicalideaswithkeeninterest [5,pp. 10-11].

>Preschool and >kindergarten >childrencanlearnmuchfromplayingwithblocks (>Jensen and'Neil, 1982).Theycan

compare andseriateshapes.Startwithsingleshapes andhavechildrencompare twopiecesaccording tosize.Later,morepiecescanbeaddeduntilseveralitems ofoneshapecanbeseriated;

classify andnameshapes.Provide twoshapesatfirstthen,aftermuchpractice,add athird andfourth.Makelargeloops ofyarn on thefloor orprovideboxes toplace theblocks in. Agoodgame toplayfornamingisPass theBlock.Childrensit in acircle and,as musicplays,theypassblocks inonedirection.When the musicstops,eachchildnames theshapehe orsheholds;

trace andfeelshapes.Childrentracearound ablockwithpencil,thencolor it inifdesired.With alittlepractice,theycansuperimposedifferentshapes ononepaper,coloring insome of thesections.Blockscanalsobeusedasitems in a ">mysterybag."Two orthreefamiliarshapesareplaced in abag andchildrentaketurnsreaching in,feeling ablock, andidentifyingitsshape.

>Planefigurescanbeexploredthroughactiveinteraction.Coloredtapeislaid on thefloor ingeometricshapeslargeenoughforchildren towalk on.Ask thechildren tojump,walk,crawl, andso onacrossspecificshapes.Theymightcount thenumber ofchildren whocan fit inonetriangle or thenumber ofsteps ittakes towalk theperimeter of asquare.

>Kindergarten and >primary >childrencontinue tolearn bestfromworkingwithmanipulatives andmayfind theillustrations inmathtextbooksconfusing.Somematerials thatareappropriatearetiles,patternblocks,attributeblocks,geoblocks,geometricsolids,coloredcubes, andtangrams. Computergames inwhichgeometricshapesappearfromdifferentangleshelpchildrenovercometheirmisunderstandings ofbookillustrations,whichmayshow ashapefromonlyone or twoviewpoints.Someappropriateactivitiesarebuildingstructureswithvarioustypes ofblocks toenhancespatialvisualization.Folding andcuttingactivitiessuchasorigami orsnowflakemaking,exploring theindoor andoutdoorenvironments toidentifyshapes andanglesmadebypeople andnature,readingmaps,makinggraphs,playingTic-Tac-Toe,Battleship, andothergames thatusegridsystems [6,pp. 458- 459].

>Alldaylongthereareopportunitiesforchildren toincreasetheirawareness ofmathematics in the worldaroundthem.Mathematicalphenomenamaynotalwaysbeasobviousasthose oflanguage,however.Thus, theteachermusttakeextracare toincludemathematicallearningwhenevernaturallearningsituationsarise.Oftenthismeansrecognizingopportunities toincorporatemathematicsintootherareas of thecurriculum.


>CONCLUSION

 

>Mathematicallearningforyoungchildrenismuchmorethan thetraditionalcounting andarithmeticskills; itincludes avariety ofmathematicalconcepts (>classification,ordering,counting,addition andsubtraction,measuring,geometry).

>Childrenmaybeginsomesimpleworkwithgeometry inprimaryschool.Maingoal ofbeginninggeometryis toteachchildren torecognize themostsimpleshapesthesquare, thecircle, thetriangle, and therectangle.Teachingsuchbasictermssimplifiesclassroomexplanations andlays thefoundationforfutureworkwithgeometry.

>Byagesix,childrenoftenhavestableyetlimitingideasaboutshapes.Itspossible tobroadenchild'sunderstandingbypointing out avariety ofexamples squares thataremanysizes andtriangles thatare ">long," ">skinny," ">fat," andturned inmanydirections.

>Thusteachersmustkeep inmind thatchildrenlearngeometrymosteffectivelythroughactiveengagementwithtoys,blocks,puzzles,manipulatives,drawings,computers andteachers!

>Itspossible todevelopdeeperthinkingaboutshapesnotjustthroughhands-onactivities anddiscussions,picturebooksbutthroughplaying.Inprimaryschoolplayingisusedas themainmethod ofteaching.

>Hugeexperience ofusinggames andplayingexercisesduring thechildrensstudying ofmathematics (andgeometry)isaccumulated inpractice ofworking ofpreschoolorganizations.


>REFERENCES

 

1. DouglasClements. >ReadyforGeometry!Fromanearlyage,childrenmakesense of theshapestheysee in the worldaroundthem // International Journal ofMathematicalEducation. Science and Technology. 2006. - 2,pp. 5-6.

2.EllenBoothChurch.Exploringsimpleshapessets thestageforcreativethinking // International Journal ofMathematicalEducation. Science and Technology. 2007. - 11,pp. 12-13.

3.EllenBoothChurch.Boxesare therawmaterials ofcreativethinking! // International Journal ofMathematicalEducation. Science and Technology. 2006. - 10,pp. 9-10

4.EllenBoothChurch.Color,Shape, andSize.Usesnacks andmealtime toteachbigideaswithtaste andease // International Journal ofMathematicalEducation. Science and Technology. 2007. - 8,pp. 2-5.

5.JulieSarama, Douglas H.Clements.Someactivitiesteacherscantry tosupportmathlearning// International Journal ofMathematicalEducation. Science and Technology. 2005. - 1,pp. 10-11.

6.SuzanneLowellKrogh.Educating YoungChildren.Infancy toGrade Three. New York.:McGraw-Hill, Inc., 1994. 605p.

7. The WorldBook ofMath Power.Volume 1.LearningMath. Chicago.: WorldBook, Inc., 1995. 420p.


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