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[{"key": "dc.contributor.advisor", "value": "Lehrb\u00e4ck, Juha", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Haapala, Laura", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2021-12-13T06:51:32Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2021-12-13T06:51:32Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2021", "language": "", "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/78945", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4m\u00e4n tutkielman tarkoituksena on tutustuttaa lukija reaalilukujen historiaan, Dedekindin leikkauksiin sek\u00e4 siihen, kuinka reaaliluvut m\u00e4\u00e4ritell\u00e4\u00e4n opiskelijoille lukion ensimm\u00e4isen\u00e4 opiskeluvuonna. Lukijalle pyrit\u00e4\u00e4n my\u00f6s avaamaan sit\u00e4, millainen prosessi matemaattisen k\u00e4sitteen ymm\u00e4rt\u00e4misen taustalla oikeastaan on. Reaalilukuja on osattu k\u00e4ytt\u00e4\u00e4 sujuvasti matematiikassa antiikin Kreikan ajoista l\u00e4htien, vaikka niille ei ole ollut olemassa t\u00e4sm\u00e4llist\u00e4 m\u00e4\u00e4ritelm\u00e4\u00e4. Lopulta 1870-luvulla pieni joukko lahjakkaita matemaatikoita kykeni luomaan reaaliluvuille useamman erilaisen t\u00e4sm\u00e4llisen m\u00e4\u00e4ritelm\u00e4n, joista suosituimpia k\u00e4ytet\u00e4\u00e4n edelleen. Saksalainen Richard Dedekind k\u00e4ytti m\u00e4\u00e4ritelm\u00e4ss\u00e4\u00e4n niin kutsuttuja Dedekindin leikkauksia, joihin p\u00e4\u00e4st\u00e4\u00e4n tutustumaan tarkemmin t\u00e4ss\u00e4 tutkielmassa. \nReaalilukujen t\u00e4sm\u00e4llisen m\u00e4\u00e4ritelm\u00e4n ymm\u00e4rt\u00e4minen vaatii jo pitk\u00e4lle kehittynytt\u00e4 matemaattista ajattelukyky\u00e4 ja t\u00e4st\u00e4 johtuen reaaliluvut esitell\u00e4\u00e4n lukiolaisille usein vain hyvin yleisell\u00e4 tasolla ilman, ett\u00e4 menn\u00e4\u00e4n t\u00e4sm\u00e4llisiin yksityiskohtiin. Matemaattisen k\u00e4sitteen ymm\u00e4rt\u00e4miseen kuuluu monta erilaista vaihetta, joiden yhteydess\u00e4 k\u00e4ytet\u00e4\u00e4n erilaisia apukeinoja, kuten ajattelumalleja. N\u00e4m\u00e4 mallit n\u00e4yttelev\u00e4tkin t\u00e4rke\u00e4\u00e4 roolia matematiikan opiskelussa, sill\u00e4 niiden avulla uusia haastavia asioita pystyt\u00e4\u00e4n j\u00e4sent\u00e4m\u00e4\u00e4n selke\u00e4mp\u00e4\u00e4n ja informatiivisempaan muotoon. Monelle lukiolaiselle reaaliluvun k\u00e4site on viel\u00e4 verrattain ep\u00e4selv\u00e4, mik\u00e4 todenn\u00e4k\u00f6isesti johtuu sek\u00e4 opiskelijoiden viel\u00e4 rakenteilla olevasta matemaattisesta ajattelutaidosta ett\u00e4 havainnosta, jonka mukaan irrationaaliluvut ja yleisesti lukualueen laajentaminen ovat opiskelijoille vaikeita aiheita. Lukijalle my\u00f6s kerrotaan mink\u00e4laisin keinoin voitaisiin helpottaa t\u00e4t\u00e4 opiskelijoiden l\u00e4pik\u00e4ym\u00e4\u00e4 ymm\u00e4rt\u00e4misen prosessia. \nTutkielman lopussa on toteutettu pienimuotoinen vertailu siit\u00e4, kuinka lukion pitk\u00e4n matematiikan oppikirjoissa reaaliluvut esitell\u00e4\u00e4n opiskelijoille. Vertailun seurauksena l\u00f6ydet\u00e4\u00e4n huomattavia eroja muun muassa irrationaalilukuihin liittyv\u00e4ss\u00e4 tiedon m\u00e4\u00e4r\u00e4ss\u00e4 sek\u00e4 asioiden esitysj\u00e4rjestyksess\u00e4. Pohdintaa k\u00e4yd\u00e4\u00e4n my\u00f6s siit\u00e4, mit\u00e4 mahdollisia syit\u00e4 n\u00e4ille oppikirjoihin p\u00e4\u00e4tyneille sis\u00e4ll\u00f6llisille ratkaisuille voisi olla.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2021-12-13T06:51:32Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2021-12-13T06:51:32Z (GMT). No. of bitstreams: 0\n Previous issue date: 2021", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "44", "language": "", "element": "format", "qualifier": "extent", "schema": "dc"}, {"key": "dc.format.mimetype", "value": "application/pdf", "language": null, "element": "format", "qualifier": "mimetype", "schema": "dc"}, {"key": "dc.language.iso", "value": "fin", "language": null, "element": "language", "qualifier": "iso", "schema": "dc"}, {"key": "dc.rights", "value": "In Copyright", "language": "en", "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.subject.other", "value": "Dedekindin leikkaukset", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Reaaliluvut : historiaa, teoriaa ja pedagogiikkaa", "language": "", "element": "title", "qualifier": null, "schema": "dc"}, {"key": "dc.type", "value": "master thesis", "language": null, "element": "type", "qualifier": null, "schema": "dc"}, {"key": "dc.identifier.urn", "value": "URN:NBN:fi:jyu-202112135932", "language": "", "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Pro gradu -tutkielma", "language": "fi", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Master\u2019s thesis", "language": "en", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Matemaattis-luonnontieteellinen tiedekunta", "language": "fi", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Faculty of Sciences", "language": "en", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Matematiikan ja tilastotieteen laitos", "language": "fi", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Department of Mathematics and Statistics", "language": "en", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "Jyv\u00e4skyl\u00e4n yliopisto", "language": "fi", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": "en", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Matematiikan opettajankoulutus", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Teacher education programme in Mathematics", "language": "en", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "yvv.contractresearch.funding", "value": "0", "language": "", "element": "contractresearch", "qualifier": "funding", "schema": "yvv"}, {"key": "dc.type.coar", "value": "http://purl.org/coar/resource_type/c_bdcc", "language": null, "element": "type", "qualifier": "coar", "schema": "dc"}, {"key": "dc.rights.accesslevel", "value": "openAccess", "language": null, "element": "rights", "qualifier": "accesslevel", "schema": "dc"}, {"key": "dc.type.publication", "value": "masterThesis", "language": null, "element": "type", "qualifier": "publication", "schema": "dc"}, {"key": "dc.subject.oppiainekoodi", "value": "4041", "language": "", "element": "subject", "qualifier": "oppiainekoodi", "schema": "dc"}, {"key": "dc.subject.yso", "value": "matematiikka", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "reaaliluvut", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "lukio", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.format.content", "value": "fulltext", "language": null, "element": "format", "qualifier": "content", "schema": "dc"}, {"key": "dc.rights.url", "value": "https://rightsstatements.org/page/InC/1.0/", "language": null, "element": "rights", "qualifier": "url", "schema": "dc"}, {"key": "dc.type.okm", "value": "G2", "language": null, "element": "type", "qualifier": "okm", "schema": "dc"}]
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