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[{"key": "dc.contributor.advisor", "value": "Lehrb\u00e4ck, Juha", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.advisor", "value": "Kaartinen, Sinikka", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Koukka, Riikka", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2019-11-29T06:53:55Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2019-11-29T06:53:55Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2019", "language": "", "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/66564", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "Opetuskokeilun idea l\u00e4hti liikkeelle halusta irrottautua perinteisest\u00e4 ik\u00e4sidonnaisesta koulusysteemist\u00e4 ja l\u00f6yt\u00e4\u00e4 uusia n\u00e4k\u00f6kulmia opetukseen eri ik\u00e4isten oppilaiden v\u00e4lisest\u00e4 vuorovaikutuksesta. Tutkimusteht\u00e4v\u00e4n\u00e4 oli selvitt\u00e4\u00e4, kuinka eri ik\u00e4iset oppilaat voivat tukea toistensa matematiikan oppimista. Tutkimukseen osallistui kahdeksan luokkien 4\u20139 oppilasta, joista muodostettiin kaksi mahdollisimman ik\u00e4sekoitteista ryhm\u00e4\u00e4. Pienryhm\u00e4t tekiv\u00e4t teht\u00e4vi\u00e4 kahdessa ty\u00f6skentelysessiossa, jotka \u00e4\u00e4nitettiin, ja \u00e4\u00e4nitteist\u00e4 valittiin tutkimusteht\u00e4v\u00e4n kannalta mielenkiintoisimmat katkelmat analysoitaviksi.\n\nKeskusteluissa nousi vahvasti esiin oppilaiden keskuudessa vallitseva k\u00e4sitys siit\u00e4, ett\u00e4 matematiikan osaaminen on tiukasti sidoksissa oppilaan ik\u00e4\u00e4n. T\u00e4m\u00e4n ik\u00e4hierarkian seurauksena ryhmiss\u00e4 muodostui melko selke\u00e4 jako, jossa vanhin oppilas ohjasi muuta ryhm\u00e4\u00e4. Toisaalta toisessa tutkimusryhm\u00e4ss\u00e4 esiintyi my\u00f6s hyvin selke\u00e4 poikkeama t\u00e4st\u00e4 ik\u00e4jaottelusta. Ryhm\u00e4n toiseksi nuorin j\u00e4sen toimi yhdenvertaisena ohjaajana ryhm\u00e4n vanhimman j\u00e4senen kanssa, ja h\u00e4nen osaamiseensa luotettiin. Oppilaiden v\u00e4linen keskustelu ryhm\u00e4ty\u00f6skentelyn aikana oli vuorovaikutteista, ja oppilaat vaativat kohtalaisen rohkeasti lis\u00e4perusteluja toistensa vastauksiin. Oppilaat kokivat erityisesti arkip\u00e4iv\u00e4n esimerkit toimivana keinona asioiden selitt\u00e4misess\u00e4. Uusia k\u00e4sitteit\u00e4 l\u00e4hestyttiin usein visuaalisesta n\u00e4k\u00f6kulmasta, joka on konkreettinen reitti l\u00e4hesty\u00e4 uutta asiaa. Haastavammissa teht\u00e4viss\u00e4 oppilaiden vuorovaikutusta ja ymm\u00e4rryst\u00e4 heikensi se, ettei omia p\u00e4\u00e4telmi\u00e4 perusteltu riitt\u00e4v\u00e4sti. N\u00e4iss\u00e4 tilanteissa opettajan ohjaus johdattelevien kysymysten kautta osoittautui t\u00e4rke\u00e4ksi.\n\nOpetuskokeilun yhdeksi p\u00e4\u00e4aiheeksi valikoitui yht\u00e4l\u00f6iden ratkaiseminen, joka on matematiikan opetussuunnitelman t\u00e4rkeimpi\u00e4 osa-alueista. Yht\u00e4l\u00f6iden ratkomiseen syvennyt\u00e4\u00e4n t\u00e4ss\u00e4 pro gradussa historiakatsauksen kautta. Yht\u00e4l\u00f6iden ja k\u00e4yrien v\u00e4linen yhteys ymm\u00e4rrettiin vasta uuden matematiikan aikana 1600-luvulla. Havainto siit\u00e4, ett\u00e4 kukin yht\u00e4l\u00f6 m\u00e4\u00e4rittelee k\u00e4yr\u00e4n toi matemaatikot uusien kysymysten \u00e4\u00e4relle. Yleist\u00e4 konstruktiota n. asteen yht\u00e4l\u00f6ille etsittiin aktiivisesti 1750-luvulle saakka. Yritykset eiv\u00e4t tuottaneet toivottua lopputulosta, mutta etsint\u00f6jen aikana onnistuttiin ennustamaan, ett\u00e4 kaksi k\u00e4yr\u00e4\u00e4, jotka ovat m. ja n. astetta, leikkaavat nm pisteess\u00e4. Nyky\u00e4\u00e4n t\u00e4m\u00e4 periaate tunnetaan B\u00e9zout \u0301n lauseena. Jotta lause p\u00e4tee, t\u00e4ytyy huomioon ottaa my\u00f6s yht\u00e4l\u00f6parin kompleksiarvoiset ratkaisut sek\u00e4 \u00e4\u00e4rett\u00f6myyden pisteet. \u00c4\u00e4rett\u00f6myyden pisteiden tarkastelussa avuksi tulivat homogeeniset koordinaatit: Projektiotason pisteet voidaan esitt\u00e4\u00e4 kolmiulotteisen avaruuden suorina, jotka kulkevat origon kautta. Kun valitaan mik\u00e4 tahansa tavallinen taso T, joka ei kulje origon kautta, tason T suhteen \u00e4\u00e4rett\u00f6myydess\u00e4 sijaitsevat pisteet voidaan tulkita homogeenisesti tason T kanssa yhdensuuntaisina origon kautta kulkevina suorina.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2019-11-29T06:53:54Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2019-11-29T06:53:55Z (GMT). No. of bitstreams: 0\n Previous issue date: 2019", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "85", "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": "B\u00e9zout\u2019n lause", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "homogeeniset koordinaatit", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "ryhm\u00e4ty\u00f6skentely", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "kielent\u00e4minen", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "matematiikkakuva", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "matematisointi", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "yht\u00e4l\u00f6n ratkaiseminen", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Matematiikkaa yhdess\u00e4 : ik\u00e4sidonnaisuuden ylitt\u00e4v\u00e4 opetuskokeilu perusopetuksessa", "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-201911295050", "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", 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