The Child's Discovery of Space: From Hopscotch to Mazes: An Introduction to Intuitive Topology PDF
Preview The Child's Discovery of Space: From Hopscotch to Mazes: An Introduction to Intuitive Topology
The childs | discovery ofspace Jean and Simonne Sauvy From hopscotch to mazes: an introductionto intuitive topology Translated from the French by Pam Wells with a new introduction by Bill Brookes mere og) OSL [2 Ib The Child’s Discoveryof Space JeanandSimonneSauvy The Child’s Discovery of Space From hopscotch to mazes: an introduction to intuitivetopology Jean and Simonne Sauvy TranslatedbyPamWellswithan introductionbyBillBrookes Penguin Education PenguinEducation ADivisionofPenguinBooksLtd, Harmondsworth,Middlesex,England PenguinBooksInc,7110AmbassadorRoad, Baltimore,Md21207,USA PenguinBooksAustraliaLtd, Ringwood,Victoria,Australia PenguinBooksCanadaLtd, 41SteelcaseRoadWest, Markham,Ontario,Canada PenguinBooks(N.Z.)Ltd, 182~190WairauRoad,Auckland10,NewZealand FirstpublishedinFrancebyEditionsCasterman,1972 FirstpublishedinGreatBritainbyPenguinEducation,1974 Copyright©EditionsCasterman,1972 Thistranslationcopyright©PenguinEducation,1974 Introductioncopyright©BillBrookes,1974 MadeandprintedinGreatBritainby C.Nicholls&CompanyLtd SetinMonotypeTimes Thisbookissoldsubjecttotheconditionthat itshallnot,bywayoftradeorotherwise,belent, re-sold,hiredout,orotherwisecirculatedwithout thepublisher’spriorconsentinanyformof bindingorcoverotherthanthatinwhichitis publishedandwithoutasimilarcondition — includingthisconditionbeingimposedonthe subsequentpurchaser Contents Introductiontothistranslation 7 Authors’preface 17 1 Topologyandthedevelopmentofthechild’sintelligence 21 2 Therealmoftopology 27 3 Exploringtheworldoftopology:linesandsurfaces 31 4 Exploringtheworldoftopology:volumes 57 5 Topologicalmapsanddiagrams 62 6 Fromsetsofpointsto‘abstracttopologicalspaces’ 85 Conclusion 91 References 93 Furtherreading 95 Introductiontothistranslation Children learning mathematics in the early years ofschool are introduced to numbersas a principal activity. They learn tech- niques for handling, using and applying numbers which have developed overseveralhundredyears, and thestrength ofthese traditional techniques springs directly from the actual control thattheygiveovertheenvironment. The basic element in the power ofnumber is uniqueness of succession; given some objects arranged for counting, then, no matterhowtheyarecounted,weshallalways arriveatthesame finalnumber.Ifthisuniquecorrespondencedidnotexistthenwe shouldneverhavethatsimplicityoforderingwhichfinallyallows usconservationofnumber. This seems obvious, yet those ofus who have workedclosely withyoungchildrenknowtheimmensevarietyofwaysinwhich theycometotermswiththislevelofhandlingofnumber.Tobe- gin to understand the reasons behind this variation, we could look at those experiences thought to be necessary forcounting todevelop.Forinstance,whatismeantbythephrase‘someob- jectsarrangedforcounting’?Surely, allobjectsarearrangedfor counting in some way or another? In order to decide whether theyareindeedarranged ornot, weshouldbegintolookatob- jects and how they appear: battery hens, pigs in sties, cows in fields,antsinanest....Therelationshipofplacetoobjectseems to be part ofthearrangement. We haveto be aware ofcertain kindsofconnectionsbeforewecanbecertainaboutcounting. Asixyear oldoutinthecountryforapicnicwentofftocount thecars.Hedidthisveryquickly.Hewasthenaskedtocountthe people, but declined, sayingthat ‘they wouldn’t be inthesame place’.Whenaskedwhathemeant,hereplied,‘Supposesomeone wasthereandyoucountedhimandthen hewentawayandsome- oneelsecamethere,youwouldn’tcounthim.” Thesixyearold’s. abilitytodifferentiatecountableobjectsandnoncountableobjects 8 Introductiontothistranslation isconnectedwithhisperceptionoftherelationshipsbetweenthe partsofspacetowhichhisattentionhasbeendrawn. Thepowerandsimplicityofcountingsometimesmakesusover eager to see it as an achievement. It seems so obvious and we thereforetendtojudgeitsachievementbycertaincharacteristics. Whenyoungchildrenrecitethenumbernames1,2,3, 4,..., in theircorrectorderandanindexfingermovesattherightmoment foreachsounduttered, wecanrecognizeakind ofcontrol. Un- fortunately, ifatthesametimethechild’sfingerdoesnotcorres- pond withtherow ofbuttons that wehaveput out forhim, we oftenreactbysaying ‘Hecan’t reallycountyet.’ Withthisstate ofdevelopment, there is no doubt that the child recognizes for himselfsome characteristic ofthe uniqueness ofcounting even thoughhecannot,atleastintheinstancestheteachersees,apply ittoobjectsoutsidehimself. During the period when children are learning to count, they experience the world in a wide variety of ways. Some of these waysarerelevant tocounting, someare not.Whichare import- ant?Towhom?Toachildconsciousofthecomplexityofmaking patterns,countingthearrangementsmaybeirrelevant.Ifcounting onlyappearstohimasoneofthemanywaysofseeingtheworld, thenheis as likely to choose other patterns as he is to choose the appropriate one. Whatwillmakecounting patternsimport- ant? | Usuallyweattempttomaketheactofmeasuring,orcomparing insomeway,amotiveforexperiencingcounting.Whohasmost? Least?Whoistaller,heavier?Whohasmostpocketmoney?We use the possession ofproperty, buying andselling to develop a strongmotivationtocount. This is notsurprisingin oursociety and it shouldnotbesurprisingto discoverthat, where asociety hasnoproperty,thereisnocountingeither!Aclosestudyofsuch societies, for example some aborigine tribes in Australia and certaintribesinAfrica,reveals thisfact. Yetthesesocieties sur- viveandhencemustinsomewayexerciseacontrolovertheirown interrelationshipsandtheirphysicalenvironment. Itisclearthatskillsinhandlingnumbersareimportantforus, butoursocietyisgrowingmorecomplexandtherearesignsthat, Introductiontothistranslation 9 inthedrivetohandlenumbersefficiently,weareexcludingmeth- ods ofcontrolwhichwesorelyneedto dealwiththis increasing complexity. Thebasiccharacteristicofnumberisthatofuniquesuccession. With an increasing numberofsituations where the connections thatwehavetomakeareneitheruniquenorsimplyordered, we mustlookoutforothermeans.Itisnolongerpossibleforaper- sontoavoidbeinginvolvedwithotherpeopleinalargenumber ofdifferent ways. Theseways differin qualityand ourabilityto survive will, in theend, depend on howskilful wearein dealing with relationships at a large number of differing levels. The child’s world is no longer a simple immediate world as, for in- stance, he can now have the experience ofan active TV screen fromaveryearlyage. Justasourabilitytocountcaneventuallyleadtohighlyskilled methodsofcontrol,soitcanbearguedthatincreasinglymorein- sight will be needed into the quality oftheconnections thatcan bemadeintheworldinordertosurvive. Jean and Simonne Sauvy are deeply concerned with these developmentsandwithhowchildrengrowandlearn. Theirwork is based on the Decroly School, near Paris. This experimental schoolwasestablishedaftertheWarandis aplacewhereteach- ers havecooperated in developingmethods ofworkappropriate tochildrentoday.InthisbooktheSauvysofferoneexplanationof elementarytopologicalideas,involvingsimplenotionsofconnec- tion,neighbourhoodandconsequentialdevelopments.Infollow- ingchildrenastheyexplorespace,wecanseetheworldinwhich theylearntocount,toconnectobjectsandtodistinguishdifferent waysofconnection. Butthroughout,theaimofthisbookistoencouragepeopleto think of space in new ways. In everyday life the word ‘space’ seemsto be used fortwo quite different ideas. When peopleare asked what theymean byspace, theyusuallyanswerintermsof vastness, space travel, extending out from the Earth’s surface, wideopenvistas....Thisisanaturalresponseandthereaderwill feelsomesympathyfor the notion ofspace as large and empty. Thereisanotheruseof‘space’whichisrarelymentioned,probably 10 Introductiontothistranslation because it is a much more prosaic use of the word. Its use is illustrated bysuch phrases as ‘Make a space for me’, ‘Is there anyspace?’ ‘Watchthisspace’, ‘Spaceouttheplants!’, ‘Check thatthereisenoughspace’,‘Makesomemorespace’,andsoon. Thisis a verydifferentuse ofthewordand weusuallyassociate someactivityorotherwitheachofthephrases. Itisnotsurprisingthat,whenasked‘Whatdoyouthinkspace is?’, peopleanswerwiththemoreintangiblenotion ofvastness, becausetheyarebeingaskedforadescriptionandofferthemost generaldescriptionofspacethattheycanthinkof.Itisnotimme- diatelyobvioushowtodescribethespaceorspacesthatarerefer- redtointhesecondsense,for,eachtimeitisused,itwillbelinked withtheassociated activity—gardening, sittingaround,carpar- king,advertising,organizingmeetingsandsoon. Thedifferences betweenthetwo meanings havebeen stressed, butwhat, bycontrast, do theyhaveincommon?Bothareasso- ciated with emptiness, but this creates problemsas it is one of those ‘something-for-nothing’ words. The other day someone remarkedontrivialincidentbysaying‘Hethrewanemptybag ofnuts atme’. Atfirst sightit seems that theslightlyridiculous notestruckbythis sentenceshouldbechanged. But, ifonesug- gests ‘He threw a bag at me’, this would not be an adequate description. Whatabout ‘a paper bag’? Even here there would stillbeatemptationtoask‘Wasthereanythinginit?’. Sowere- treattosaying ‘Hethrewanemptypaperbagatme’. Nowit is unclearwhatkind ofpaperbagso ‘Hethrewatmeabagwhich usuallyholdsnuts, butwhichin thiscasewasempty.’Ithinkin theendIprefertheobviousambiguityofan‘emptybagofnuts’. Allthephrasesusedearliertodescribespacecomefromspecific situationsinvolvingquitespecificactions.Theirusepointstoour definingaspaceforeachaction. Theactionitselfwilldefinethe space.Thecarparkisasimplydefinedandlimitedspaceinterms ofcarsbeingparked. Ifitwas usedforagameoffootballwhen notbeingusedasacarpark,thenitwouldbeadifferentspace. Wecancontinuetothinkofactionsdefiningspaceandthiscan beveryhelpful,butourideaofspacehasbecomedistortedaswe havelearnttomeasure. BythisImeanthatwecannowthinkof Introductiontothistranslation 11 lengths, areas and volumes and this gives us a definite idea of spacebeingidentifiedbymeasurements.Thisisusuallydonefora practicalpurpose,butwehavebecomesousedtoknowingabout measurementsthatwethinkthisiswhatspaceisabout. Thisseemstobeathirdwayofthinkingaboutspacewhich,in reality,istheconsequenceofgeneralizingaparticularformofthe second,concernedwithaframeworkinwhichthingscanbedone. The making of this framework in terms ofcommon measures developedfromthetimeswhensuchactivities,whichnowrelyon common measures, demanded their own frameworks. It is diffi- cult, for instance, to ask now whether the idea of length (as number) came before or after the invention of the measuring ruler. Measuringappearsto beawayofdealingpracticallywith many different situations, but there are still far more activities undertakenwhichdonotrelyon formal measurementthanthose thatdo. This book deals with matters that reveal themselves more by investigation and thinking than bylearningsome simple techni- calskillssuchashandlingaruler;butthetechniques,asaconse- quence of what is introduced, are not acquired immediately. Herewearemoreconcernedwiththataspectofspaceconcerned withmovement andpossiblyhavingsomerulesformoving. For example, in a game ofchess, the ‘moves’ ofthe pieces and the pawnsgiveaframeworktothegamedescribingwhatcanhappen andmakingthegamepossible.Theagreementbetweentwopeople toplayisanagreementthatexistsforthedurationofthegamein the space governed by the rules. It is easy to see that all formal gamesare like this and perhaps the idea canbe extendedtoless formally defined activities, which individually show constraints that both defineand allow. Writingis suchanactivitywhere, in one sense, a piece ofpaper provides the space, theliving room, forthewriter,butinamorecomplexwaythewriterisheldbyhis skillandinthis‘agreement’hedemonstratesthespaceoftheskill ofwriting. Also in a very rich sense we both exist in space and makespace.Theactorwhomovesonthestageinsuchawayasto engagetheaudience in imaginativeresponses, making the scene fullerandmorevivid,isaddingtothesimplenotionofthemea-
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