{"id":1195,"date":"2023-08-26T13:05:43","date_gmt":"2023-08-26T13:05:43","guid":{"rendered":"https:\/\/flatarmal.is\/?p=1195"},"modified":"2025-07-18T09:32:32","modified_gmt":"2025-07-18T09:32:32","slug":"ad-virkja-hugsun-nemenda-med-krefjandi-verkefnum","status":"publish","type":"post","link":"https:\/\/flatarmal.is\/?p=1195","title":{"rendered":"A\u00f0 virkja hugsun nemenda me\u00f0 krefjandi verkefnum"},"content":{"rendered":"\n
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J\u00f3hann \u00d6rn Sigurj\u00f3nsson.<\/strong><\/p>\n<\/div>\n<\/div>\n\n\n\n

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\u00deessi grein byggir \u00e1 erindi m\u00ednu \u00e1 n\u00e1mstefnu Flatar sem haldin var \u00e1 30 \u00e1ra afm\u00e6li samtakanna \u00feann 4. mars 2023. \u00c9g segi \u00ed stuttu m\u00e1li fr\u00e1 hluta ni\u00f0ursta\u00f0na \u00far doktorsverkefni m\u00ednu um hugr\u00e6na virkjun \u00ed st\u00e6r\u00f0fr\u00e6\u00f0ikennslu \u00e1 \u00cdslandi (J\u00f3hann \u00d6rn Sigurj\u00f3nsson, 2023). \u00c9g leitast vi\u00f0 a\u00f0 \u00fatsk\u00fdra hva\u00f0 hugr\u00e6n virkjun er, b\u00e6\u00f0i me\u00f0 saml\u00edkingunni vi\u00f0 kennslu til skilnings og me\u00f0 fr\u00f3\u00f0leik um hva\u00f0 tiltekinn matsrammi metur til g\u00e6\u00f0a \u00ed hugr\u00e6nni virkjun. \u00de\u00e1 tek \u00e9g nokkur d\u00e6mi um krefjandi verkefni \u00ed st\u00e6r\u00f0fr\u00e6\u00f0i sem skapa g\u00f3\u00f0an jar\u00f0veg til \u00feess a\u00f0 virkja hugsun nemenda \u2013 stu\u00f0la a\u00f0 hugr\u00e6nni virkjun \u00e1 h\u00e1u stigi.<\/p>\n<\/div>\n<\/div>\n\n\n\n

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Fyrir m\u00f6rgum kann a\u00f0 vir\u00f0ast n\u00fdst\u00e1rlegt a\u00f0 tala um \u00fe\u00e1tt \u00ed g\u00e6\u00f0um kennslu sem kallast hugr\u00e6n virkjun. Rannsakendur hafa gjarnan skipt g\u00e6\u00f0um kennslu \u00ed \u00ferj\u00e1 yfir\u00fe\u00e6tti: hugr\u00e6na virkjun, stu\u00f0ning vi\u00f0 n\u00e1m og bekkjarstj\u00f3rnun (Praetorius, Klieme, Herbert og Pinger, 2018). R\u00e9tt eins og kennari st\u00fdrir ekki beinl\u00ednis bekk me\u00f0 handafli, \u00fe\u00e1 virkjar kennari ekki hugsun nemenda me\u00f0 \u00fev\u00ed a\u00f0 teygja sig inn \u00ed huga \u00feeirra og smella \u00e1 rofa undir h\u00f6fu\u00f0k\u00fapunni. Hugr\u00e6n virkjun er \u00fev\u00ed gjarnan hugsu\u00f0 \u00fat fr\u00e1 svonefndu bj\u00f3\u00f0a-nota <\/em>l\u00edkani (Weingartner, 2021). Samkv\u00e6mt bj\u00f3\u00f0a-nota<\/em> l\u00edkanin\ufeff\u7280\u5229\u58eb<\/a>\n<\/span>u \u00fe\u00e1 b\u00fd\u00f0ur kennari nemendum t\u00e6kif\u00e6ri til hugr\u00e6nnar virkjunar. Hugr\u00e6n virkni nemenda byggir s\u00ed\u00f0an \u00e1 \u00fev\u00ed hvernig nemendur nota \u00feessi t\u00e6kif\u00e6ri. T\u00e6kif\u00e6ri til hugr\u00e6nnar virkjunar m\u00e1 \u00ed stuttu m\u00e1li skilgreina \u00fat fr\u00e1 tveimur \u00e1kv\u00f6r\u00f0unum kennarans:<\/p>\n\n\n\n

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\u25cf Hlutl\u00e6g <\/em>t\u00e6kif\u00e6ri til hugr\u00e6nnar virkjunar: Val kennarans \u00e1 krefjandi verkefnum.<\/p>\n\n\n\n

\u25cf \u00datf\u00e6r\u00f0<\/em> t\u00e6kif\u00e6ri til hugr\u00e6nnar virkjunar: \u00datf\u00e6rsla \u00e1 verkefnum \u00ed kennslustund.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div><\/div>\n\n\n\n

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Um val verkefna og \u00f3l\u00edkar tegundir \u00feeirra eftir vitsmunalegum kr\u00f6fum eru til vi\u00f0mi\u00f0 sem kennarar geta hagn\u00fdtt s\u00e9r (sj\u00e1 t.d., J\u00f3hann \u00d6rn Sigurj\u00f3nsson og Berglind G\u00edslad\u00f3ttir, 2020). \u00cd ni\u00f0urst\u00f6\u00f0um sem h\u00e9r eru kynntar er \u00e1herslan \u00e1 hi\u00f0 s\u00ed\u00f0arnefnda, \u00fe.e. hvernig verkefni eru \u00fatf\u00e6r\u00f0 \u00ed kennslu \u00ed samskiptum vi\u00f0 nemendur \u00ed kennslustund.<\/p>\n<\/div>\n<\/div>\n\n\n\n

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G\u00f6gn ranns\u00f3knarinnar eru hluti af st\u00e6rra gagnasafni. Ranns\u00f3knarsetri\u00f0 Quality in Nordic Teaching<\/em> (QUINT) hefur s\u00ed\u00f0an 2018 safna\u00f0 myndbandsuppt\u00f6kum \u00far kennslustundum \u00ed 8. bekk af \u00f6llum Nor\u00f0url\u00f6ndum. \u00cd \u00feessari grein ver\u00f0ur byggt \u00e1 g\u00f6gnum \u00far kennslustundum \u00ed st\u00e6r\u00f0fr\u00e6\u00f0i \u00e1 \u00cdslandi. Teknar voru upp 34 kennslustundir \u00ed st\u00e6r\u00f0fr\u00e6\u00f0i fr\u00e1 t\u00edu kennurum, \u00ferj\u00e1r til fj\u00f3rar kennslustundir fr\u00e1 hverjum kennara. Fimm \u00fe\u00e1ttt\u00f6kusk\u00f3lar voru af h\u00f6fu\u00f0borgarsv\u00e6\u00f0inu og fimm af landsbygg\u00f0inni. Til a\u00f0 var\u00f0veita nafnleysi \u00fe\u00e1tttakenda eru notu\u00f0 gervin\u00f6fn \u00ed ni\u00f0urst\u00f6\u00f0um.<\/p>\n<\/div>\n<\/div>\n\n\n\n

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G\u00f6gnin voru greind me\u00f0 greiningarrammanum PLATO (Grossman, 2015). \u00cd PLATO eru \u00ed heild t\u00f3lf \u00fe\u00e6ttir, \u00fear sem fyrir hvert 15 m\u00edn\u00fatna myndskei\u00f0 er hver \u00fe\u00e1ttur metinn \u00e1 fj\u00f6gurra \u00ferepa kvar\u00f0a. Byggt er \u00e1 v\u00edsbendingum \u00ed myndbandsg\u00f6gnunum \u00fear sem \u00e1 1. \u00ferepi eru litlar e\u00f0a engar v\u00edsbendingar, en \u00e1 4. \u00ferepi eru sterkar og st\u00f6\u00f0ugar v\u00edsbendingar. Tveir PLATO \u00fe\u00e6ttir tengjast yfir\u00fe\u00e6ttinum hugr\u00e6nni virkjun: vitsmunaleg \u00e1skorun og samr\u00e6\u00f0ur \u00ed kennslustund. \u00c1 myndum 1 og 2 m\u00e1 sj\u00e1 stytt greiningarvi\u00f0mi\u00f0 fyrir \u00feessa tvo \u00fe\u00e6tti.<\/p>\n<\/div>\n<\/div>\n\n\n\n

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Mynd 1.<\/strong> Stytt PLATO vi\u00f0mi\u00f0 fyrir vitsmunalega \u00e1skorun (e. intellectual challenge<\/em>, IC)<\/p>\n<\/div>\n<\/div>\n\n\n\n

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Mynd 2.<\/strong> Stytt PLATO vi\u00f0mi\u00f0 fyrir samr\u00e6\u00f0ur \u00ed sk\u00f3lastofunni (e. classroom discourse<\/em>, CD)<\/p>\n\n\n\n

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\u00deess skal geta a\u00f0 spurningar og athugasemdir kennarans geta or\u00f0i\u00f0 til a\u00f0 auka \u00e1 vitsmunalega \u00e1skorun verkefna e\u00f0a draga \u00far henni. Mat \u00e1 vitsmunalegri \u00e1skorun er \u00fe\u00e1 h\u00e6kka\u00f0 e\u00f0a l\u00e6kka\u00f0 um eitt \u00ferep eftir \u00fev\u00ed hvort kennarinn eykur e\u00f0a minnkar kr\u00f6fur fyrir \u00farlausn verkefnis.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\n\n

Ni\u00f0urst\u00f6\u00f0ur og inns\u00fdn \u00ed tvo h\u00f3pa<\/strong><\/p>\n\n\n\n

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\u00c1 mynd 3 m\u00e1 sj\u00e1 dreifingu myndskei\u00f0a \u00ed hugr\u00e6nni virkjun eftir kennurum. \u00dat fr\u00e1 ni\u00f0urst\u00f6\u00f0um \u00feeirra kennslustunda sem greindar voru m\u00e1 skipta \u00fe\u00e1ttt\u00f6kukennurum \u00ed \u00ferj\u00e1 h\u00f3pa. \u00cd h\u00f3pi A eru kennarar \u00fear sem d\u00e6mi voru um a\u00f0 \u00fe\u00e6ttirnir vitsmunaleg \u00e1skorun og samr\u00e6\u00f0ur \u00ed kennslustund v\u00e6ru metnir \u00e1 4. \u00ferepi. \u00cd h\u00f3pi B eru kennarar \u00fear sem \u00feessir \u00fe\u00e6ttir voru h\u00e6st metnir \u00e1 3. \u00ferepi. \u00cd h\u00f3pi C eru loks kennarar \u00fear sem \u00fe\u00e6ttir tengdir hugr\u00e6nni virkjun voru metnir \u00e1 1. e\u00f0a 2. \u00ferepi \u00ed \u00f6llum myndskei\u00f0um kennslustunda \u00feeirra.<\/p>\n<\/div>\n<\/div>\n\n\n\n

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Mynd 3.<\/strong> Dreifing myndskei\u00f0a \u00ed hugr\u00e6nni virkjun eftir kennurum.<\/p>\n<\/div>\n<\/div>\n\n\n\n

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\u00cd h\u00f3pi A voru tveir kennarar, T1<\/sub> og T2<\/sub>. \u00cd kennslustundum kennarans T1<\/sub> voru tekin fyrir verkefni \u00ed l\u00edkindafr\u00e6\u00f0i sem nemendur unnu saman \u00ed p\u00f6rum. Me\u00f0al verkefna voru tilraunir nemenda me\u00f0 teningakast \u00fear sem bornar voru saman tilraunal\u00edkur og fr\u00e6\u00f0ilegar l\u00edkur. Kennarinn haf\u00f0i b\u00e6\u00f0i stuttlega beina kennslu til alls h\u00f3psins og gekk \u00e1 milli \u00e1 me\u00f0an nemendur k\u00f6stu\u00f0u teningum og skr\u00e1\u00f0u ni\u00f0urst\u00f6\u00f0ur. \u00cd samskiptum vi\u00f0 nemendur spur\u00f0i kennarinn \u00fe\u00e1 gjarnan spurninga og lei\u00f0beindi \u00feeim \u00feannig vi\u00f0 a\u00f0 \u00fatsk\u00fdra hugsun s\u00edna fyrir hvert \u00f6\u00f0ru:<\/p>\n<\/div>\n<\/div>\n\n\n\n

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Kennari: Hverjar eru l\u00edkurnar \u00e1 \u00fev\u00ed a\u00f0 f\u00e1 fimm ef \u00fe\u00fa kastar teningi?<\/em> <\/p>\n\n\n\n

Arnar: Einn \u00e1 m\u00f3ti sex.<\/em> <\/p>\n\n\n\n

Kennari: Einn \u00e1 m\u00f3ti sex. Hversu oft \u00e6tti \u00e9g a\u00f0 f\u00e1 fimm ef \u00e9g kasta teningi 30 sinnum?<\/em> <\/p>\n\n\n\n

Arnar: 30 sinnum, er \u00fea\u00f0 ekki\u2026 fimm?<\/em> <\/p>\n\n\n\n

Kennari: Fimm sinnum. Af hverju? <\/em> \u2b05\ufe0f<\/p>\n\n\n\n

Arnar: Nei, sex sinnum, af \u00fev\u00ed a\u00f0 fimm sinnum sex er 30.<\/em> <\/p>\n\n\n\n

Kennari: Ert \u00fe\u00fa samm\u00e1la \u00fev\u00ed? Mhm. Ok.<\/em> <\/p>\n\n\n\n

Beta: \u00c9g skil ekki neitt!<\/em> <\/p>\n\n\n\n

Kennari: Arnar, getur \u00fe\u00fa \u00fatsk\u00fdrt fyrir Betu? <\/em> \u2b05\ufe0f<\/p>\n<\/div>\n<\/div>\n\n\n\n

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\u00d6rvarnar \u2b05\ufe0f s\u00fdna v\u00edsbendingar um a\u00f0 kennarinn auki \u00e1 vitsmunalega \u00e1skorun. Kennarinn spyr af hverju nemandi hefur tiltekna tilg\u00e1tu og bi\u00f0ur \u00feannig um r\u00f6kstu\u00f0ning. \u00de\u00e1 bi\u00f0ur kennarinn hinn nemandann um a\u00f0 leggja mat \u00e1 \u00feetta og lei\u00f0beinir loks um a\u00f0 \u00fatsk\u00fdra hugsun s\u00edna fyrir bekkjarf\u00e9laga s\u00ednum. T2 <\/sub>vi\u00f0haf\u00f0i sams konar \u00fatf\u00e6rslu verkefna \u00ed algebru, me\u00f0al annars me\u00f0 n\u00e1msleiknum 24 sem ver\u00f0ur sagt n\u00e1nar fr\u00e1 s\u00ed\u00f0ar.<\/p>\n<\/div>\n<\/div>\n\n\n\n

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\u00cd h\u00f3pi C voru fimm kennarar, T6<\/sub> til T10<\/sub>. \u00cd kennslustundum \u00feeirra unnu nemendur n\u00e1nast eing\u00f6ngu einir og \u201e\u00e1 eigin hra\u00f0a\u201c. \u00dev\u00ed unnu nemendur allajafna ekki s\u00f6mu verkefni \u00e1 sama t\u00edma og var ekki lei\u00f0beint a\u00f0 vinna saman sem ger\u00f0i \u00fea\u00f0 a\u00f0 verkum a\u00f0 samr\u00e6\u00f0a var metin \u00e1 l\u00e6gri \u00ferepum. Kennarinn gekk gjarnan um og a\u00f0sto\u00f0a\u00f0i einstaklingslega en l\u00edti\u00f0 sem ekkert var um kennslu til alls h\u00f3psins samt\u00edmis. \u00cd a\u00f0sto\u00f0 sinni vi\u00f0 nemendur var gjarnan dregi\u00f0 \u00far \u00e1skorun verkefnanna, til d\u00e6mis me\u00f0 loku\u00f0um spurningum e\u00f0a me\u00f0 \u00fev\u00ed a\u00f0 segja nemendum lausn e\u00f0a lausnalei\u00f0. \u00cd eftirfarandi d\u00e6mi ba\u00f0 nemandi um a\u00f0sto\u00f0 vi\u00f0 a\u00f0 b\u00faa til form\u00falu fyrir mn<\/sub> og reikna m80<\/sub> \u00fear sem myndr\u00e6nt talnamynstur gaf m1<\/sub> = 5, m2<\/sub> = 9, m3<\/sub> = 13 (eins og \u00e1 mynd 4).<\/p>\n<\/div>\n<\/div>\n\n\n\n

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Mynd 4.<\/strong> Myndr\u00e6nt talnamynstur \u00fear sem \u00ed hverjum li\u00f0 rununnar leggjast fj\u00f3rar einingar vi\u00f0.<\/p>\n<\/div>\n<\/div>\n\n\n\n

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Kennari: J\u00e1, b\u00fa\u00f0u til form\u00falu. Hva\u00f0 breytist h\u00e9rna \u00e1 milli? <\/em> <\/p>\n\n\n\n

Nemandi: Einn pl\u00fas… <\/em> <\/p>\n\n\n\n

Kennari: \u00dea\u00f0 b\u00e6tast vi\u00f0 fj\u00f3rir. \u00de\u00e1 skrifar \u00fe\u00fa 4x. Ef \u00fe\u00fa tekur 4 af \u00feessu, hva\u00f0 ver\u00f0ur miki\u00f0 eftir? <\/em> <\/p>\n\n\n\n

Nemandi: Einn. <\/em> <\/p>\n\n\n\n

Kennari: Pl\u00fas einn. Og svo er x-i\u00f0 m-n\u00fameri\u00f0.<\/em> <\/p>\n\n\n\n

Nemandi: \u00deannig a\u00f0 ef \u00e9g v\u00e6ri a\u00f0 finna 80, myndi \u00e9g \u00fe\u00e1 bara gera fj\u00f3rir… <\/em> <\/p>\n\n\n\n

Kennari: Fj\u00f3rir sinnum 80 pl\u00fas einn.<\/em> <\/p>\n\n\n\n

Nemandi: \u00d3kei. <\/em> <\/p>\n\n\n\n

Kennari: \u00deetta er gert svona (kennari sl\u00e6r inn \u00ed reikniv\u00e9l nemandans) fj\u00f3rir sinnum 80 pl\u00fas einn.<\/em> <\/p>\n<\/div>\n<\/div>\n<\/div><\/div>\n<\/div>\n<\/div>\n\n\n\n

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H\u00e9r hefur kennarinn \u00ed raun leyst d\u00e6mi\u00f0 \u00ed heild \u00e1n \u00feess a\u00f0 nemandinn \u00feyrfti a\u00f0 s\u00fdna fram \u00e1 nokkurn skilning e\u00f0a eigin hugsun. \u00dea\u00f0 var\u00f0 til \u00feess a\u00f0 dregi\u00f0 var \u00far vitsmunalegri \u00e1skorun. Ekki fundust d\u00e6mi um a\u00f0 leita\u00f0 v\u00e6ri eftir r\u00f6kstu\u00f0ningi nemanda e\u00f0a a\u00f0 \u00feeir \u00fatsk\u00fdr\u00f0u hugsun s\u00edna l\u00edkt og sj\u00e1 m\u00e1tti \u00ed d\u00e6minu \u00far h\u00f3pi A. \u00de\u00f3 m\u00e1tti hj\u00e1 b\u00e1\u00f0um h\u00f3pum kennara finna verkefni sem g\u00e1fu t\u00e6kif\u00e6ri til \u00feess a\u00f0 hugr\u00e6n virkjun v\u00e6ri \u00e1 h\u00e1u stigi \u2013 til d\u00e6mis \u00ed h\u00f3pi C me\u00f0 verkefninu \u00e1 mynd 4. Hef\u00f0i \u00fatf\u00e6rsla kennarans \u00e1 verkefninu veri\u00f0 \u00e1 annan veg hef\u00f0i me\u00f0 g\u00f3\u00f0u m\u00f3ti m\u00e1tt meta \u00fe\u00e6tti hugr\u00e6nnar virkjunar \u00e1 efri \u00ferepum.<\/p>\n\n\n\n

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D\u00e6mi um krefjandi verkefni<\/strong><\/p>\n<\/div>\n<\/div>\n\n\n\n

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Vi\u00f0 lok erindis m\u00edns kynnti \u00e9g eftirfarandi fimm krefjandi verkefni sem geta henta\u00f0 vel til kennslu me\u00f0 hugr\u00e6na virkjun \u00e1 h\u00e1u stigi. H\u00e9r segi \u00e9g n\u00e1nar fr\u00e1 \u00feremur \u00feeirra.<\/p>\n<\/div>\n<\/div>\n\n\n\n

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1. N\u00e1msleikurinn 24.<\/p>\n\n\n\n

2. Frumt\u00f6ludagsetningar.<\/p>\n\n\n\n

3. R\u00f6\u00f0un heiltalna 1-32 \u00ed hring \u00feannig a\u00f0 summa a\u00f0liggjandi talnapara s\u00e9 ferningstala.<\/p>\n\n\n\n

4. Myndr\u00e6n \u00fe\u00e1ttun talna og annars stigs margli\u00f0a.<\/p>\n\n\n\n

5. Teningakast og reiknirit: Python forrit fyrir summu tveggja teninga og n<\/em> teninga.<\/p>\n<\/div>\n<\/div>\n<\/div><\/div>\n\n\n\n

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Leikurinn 24 gengur \u00fat \u00e1 a\u00f0 af gefnu spjaldi me\u00f0 fj\u00f3rum t\u00f6lum \u00e1 bilinu 1-9 \u00e1 a\u00f0 nota s\u00e9rhverja t\u00f6lu einu sinni til a\u00f0 rita st\u00e6\u00f0u me\u00f0 grunnreiknia\u00f0ger\u00f0um sem hefur gildi\u00f0 24. Spjald me\u00f0 t\u00f6lunum 8, 7, 9 og 1 hefur til d\u00e6mis lausn me\u00f0 st\u00e6\u00f0unni (9\u22127+1)\u22198. (Til er \u00f6nnur lausn vi\u00f0 \u00feessu spjaldi.)<\/p>\n<\/div>\n<\/div>\n\n\n\n

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\u00de\u00e1tttakendum voru gefin nokkur talnaspj\u00f6ld til a\u00f0 leysa (sj\u00e1 mynd 5). Gefnar voru \u00fe\u00e6r uppl\u00fdsingar a\u00f0 eitt spjaldi\u00f0 hafi enga lausn en ekki var gefi\u00f0 upp hva\u00f0a spjald \u00fea\u00f0 er. Af \u00feessum fimm spj\u00f6ldum sem hafa lausn er \u00ed spilinu sj\u00e1lfu eitt \u00feeirra merkt au\u00f0velt, tv\u00f6 eru mi\u00f0lungs og tv\u00f6 erfi\u00f0.<\/p>\n<\/div>\n<\/div>\n\n\n

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Mynd 5.<\/strong> Sex talnaspj\u00f6ld fyrir leikinn 24. Eitt spjaldanna hefur enga lausn.<\/p>\n\n\n\n

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\u00dea\u00f0 er \u00e1hugaver\u00f0 sta\u00f0reynd a\u00f0 ekki \u00f6ll m\u00f6guleg talnaspj\u00f6ld hafi lausn, \u00fe\u00f3 a\u00f0 \u00ed spilinu sem gefi\u00f0 er \u00fat af Suntex hafi \u00f6ll 96 spj\u00f6ldin lausn sem \u00fear eru prentu\u00f0. Enn \u00e1hugaver\u00f0ari sta\u00f0reynd er a\u00f0 talnagildi\u00f0 24 er ekki \u00fea\u00f0 gildi sem hefur hlutfallslega flestar lausnir ef \u00f6ll m\u00f6guleg talnaspj\u00f6ld eru tekin me\u00f0 \u00ed reikninginn (fyrir \u00edtarlegri greiningu \u00e1 \u00feessu m\u00e1 sj\u00e1 grein Triplett, 2011).<\/p>\n<\/div>\n<\/div>\n\n\n\n

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Anna\u00f0 verkefni sem kynnt var \u00fe\u00e1tttakendum \u00e1 n\u00e1mstefnunni haf\u00f0i a\u00f0 gera me\u00f0 svokalla\u00f0ar frumt\u00f6ludagsetningar<\/em>. \u00cd \u00fev\u00ed \u00e1tti a\u00f0 huglei\u00f0a eftirfarandi:<\/p>\n<\/div>\n<\/div>\n\n\n\n

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a) Ef liti\u00f0 er \u00e1 dagsetninguna \u00ed dag sem \u00e1tta (e\u00f0a sj\u00f6) stafa t\u00f6lu, ritu\u00f0 ddmmyyyy<\/em>, \u00fear sem<\/p>\n\n\n\n

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\u25cf dd<\/em> er dagur,<\/p>\n\n\n\n

\u25cf mm<\/em> er m\u00e1nu\u00f0ur<\/p>\n\n\n\n

\u25cf yyyy<\/em> er \u00e1rtal<\/p>\n\n\n\n

\u25cf \u2026er talan ddmmyyyy<\/em> frumtala? Hva\u00f0 me\u00f0 daginn \u00ed g\u00e6r?<\/p>\n<\/div>\n\n\n\n

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b) Rannsaka\u00f0u og seg\u00f0u til um l\u00edkurnar \u00e1 \u00fev\u00ed a\u00f0 gefin dagsetning \u00e1 forminu ddmmyyyy<\/em> s\u00e9 frumtala.<\/p>\n\n\n\n

c) Geta frumt\u00f6ludagsetningar or\u00f0i\u00f0 tvo daga \u00ed r\u00f6\u00f0?<\/p>\n\n\n\n

d) Hverju breytir \u00fea\u00f0 \u00ed spurningum 2 og 3 ef notu\u00f0 er \u00e1tta stafa talan yyyymmdd<\/em>?<\/p>\n<\/div>\n<\/div>\n\n\n\n

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Vi\u00f0 a\u00f0 huglei\u00f0a \u00feetta verkefni geta \u00fdmis hugt\u00f6k komi\u00f0 til kastanna, svo sem sl\u00e9ttar t\u00f6lur og oddat\u00f6lur, samsettar t\u00f6lur og frumt\u00f6lur, frumt\u00f6lutv\u00edburar, sem og \u00f3l\u00edk skilyr\u00f0i eftir \u00fev\u00ed hvort m\u00e1nu\u00f0urinn er n\u00fdhafinn e\u00f0a meira en 9 dagar li\u00f0nir af honum. Fyrir b-li\u00f0 er ef til vill ekki einfalt a\u00f0 koma me\u00f0 einhl\u00edtt e\u00f0a \u00edtarlegt svar. Til a\u00f0 byrja me\u00f0 m\u00e1 til d\u00e6mis velta fyrir s\u00e9r hvernig \u00e1ri\u00f0 2023 er \u00f3l\u00edkt 2024 a\u00f0 \u00feessu leyti. Spurning af \u00feessu tagi er d\u00e6mi um st\u00e6r\u00f0fr\u00e6\u00f0ilega spurningu sem unnt er rannsaka skipulega me\u00f0 a\u00f0sto\u00f0 forritunar. Fyrir d-li\u00f0 bendi \u00e9g \u00e1 talnarunu \u00far rafr\u00e6num gagn<\/a>abanka<\/a> (OEIS: A260915).<\/p>\n<\/div>\n<\/div>\n\n\n\n

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\u00deri\u00f0ja verkefni\u00f0 sem \u00e9g kynni h\u00e9r muna eflaust einhverjir \u00fe\u00e1tttakendur n\u00e1mstefnunnar eftir, enda s\u00e1tu nokkrir \u00feeirra fram eftir vi\u00f0 a\u00f0 leysa \u00far \u00fev\u00ed. \u00c1 mynd 6 eru 32 hringir. Verkefni\u00f0 gengur \u00fat \u00e1 a\u00f0 ra\u00f0a \u00f6llum heilt\u00f6lum fr\u00e1 1-32 \u00ed hring \u00feannig a\u00f0 s\u00e9rhver summa a\u00f0liggjandi talna s\u00e9 ferningstala. Til d\u00e6mis g\u00e6ti komi\u00f0 til greina a\u00f0 setja 3 og 8 sitt hvorum megin vi\u00f0 1 \u00fear sem 1+3=4 er ferningstala og 1+8=9 er ferningstala. (\u00cd vi\u00f0auka er mynd til stu\u00f0nings vi\u00f0 \u00farlausn verkefnisins.)<\/p>\n<\/div>\n<\/div>\n\n\n\n

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Mynd 6.<\/strong> \u00deraut sem gengur \u00fat \u00e1 a\u00f0 ra\u00f0a heilt\u00f6lunum 1-32
\u00feannig a\u00f0 s\u00e9rhver summa tveggja a\u00f0liggjandi talna s\u00e9 ferningstala.<\/p>\n<\/div>\n<\/div>\n\n\n\n

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Einhverjum g\u00e6ti \u00fe\u00f3tt freistandi a\u00f0 nota r\u00e1\u00f0leggingu P\u00f3lya um a\u00f0 byrja \u00e1 a\u00f0 spreyta sig \u00e1 a\u00f0 ra\u00f0a \u00ed styttri hring til a\u00f0 \u00e1tta sig betur \u00e1 \u00ferautinni, svo sem a\u00f0 velja n=10 \u00feannig a\u00f0 ra\u00f0a\u00f0 s\u00e9 heilt\u00f6lum fr\u00e1 1-10 me\u00f0 s\u00f6mu skilyr\u00f0um. \u00dev\u00ed mi\u00f0ur gengur \u00fea\u00f0 ekki \u00ed \u00feessu tilfelli. Me\u00f0 a\u00f0sto\u00f0 netafr\u00e6\u00f0i og forritunar m\u00e1 s\u00fdna a\u00f0 n=32 er minnsta st\u00e6r\u00f0 nets sem unnt er a\u00f0 ra\u00f0a \u00ed sl\u00edkan hring (e\u00f0a Hamilton-r\u00e1s me\u00f0 tungutaki netafr\u00e6\u00f0innar). \u00derautina er \u00fe\u00f3 unnt a\u00f0 leysa \u00e1n \u00feess a\u00f0 kunna fyrir s\u00e9r \u00ed netafr\u00e6\u00f0i.<\/p>\n\n\n\n

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Ni\u00f0urlag<\/strong><\/p>\n<\/div>\n<\/div>\n\n\n\n

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\u00cd greininni hef \u00e9g sagt fr\u00e1 og teki\u00f0 d\u00e6mi um hvernig hugr\u00e6n virkjun \u00ed kennslu var metin \u00e1 \u00f3l\u00edkum stigum eftir \u00fev\u00ed hvernig kennsluh\u00e6ttir leiddu til \u00feess a\u00f0 virkja hugsun nemenda \u00ed mismiklum m\u00e6li. \u00de\u00e1 hef \u00e9g teki\u00f0 nokkur d\u00e6mi um verkefni sem eru vel til \u00feess fallin a\u00f0 skapa hlutl\u00e6g<\/em> t\u00e6kif\u00e6ri til hugr\u00e6nnar virkjunar. \u00de\u00f3 vil \u00e9g undirstrika a\u00f0 \u00fea\u00f0 eru \u00fatf\u00e6r\u00f0<\/em> t\u00e6kif\u00e6ri sem trompa hin hlutl\u00e6gu \u2013 \u00fe.e. \u00fatf\u00e6rsla kennara skiptir meira m\u00e1li en verkefnin. Verkefni af \u00feeim toga sem kynnt voru h\u00e9r er vissulega handh\u00e6gt a\u00f0 \u00fatf\u00e6ra \u00feannig a\u00f0 reyni \u00e1 a\u00f0 nemendur hugsi og tengi saman st\u00e6r\u00f0fr\u00e6\u00f0ilegar hugmyndir. Annars konar verkefni, svo sem \u00feau sem kalla m\u00e1 \u00e6fingaverkefni, er \u00fe\u00f3 vissulega h\u00e6gt a\u00f0 \u00fatf\u00e6ra \u00ed kennslustund \u00feannig a\u00f0 reyni \u00e1 hugsun nemenda. Svo vel megi vera kann \u00fea\u00f0 \u00fe\u00f3 a\u00f0 krefjast b\u00e6\u00f0i g\u00f3\u00f0s undirb\u00fanings og s\u00e9r\u00feekkingar kennarans. Hvoru tveggja eru l\u00edklega nau\u00f0synleg skilyr\u00f0i metna\u00f0arfullrar st\u00e6r\u00f0fr\u00e6\u00f0ikennslu. Me\u00f0 \u00feeim hugmyndum sem h\u00e9r hafa veri\u00f0 settar fram (og lesa m\u00e1 n\u00e1nar um \u00ed skr\u00e1\u00f0um heimildum) kveiki \u00e9g vonandi innbl\u00e1stur einhverra kennara til a\u00f0 stu\u00f0la enn frekar a\u00f0 b\u00e6\u00f0i \u00e1rangri og \u00e1n\u00e6gju nemenda sinna \u00ed st\u00e6r\u00f0fr\u00e6\u00f0i.<\/p>\n\n\n\n

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Heimildir<\/strong><\/p>\n<\/div>\n<\/div>\n\n\n\n

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Grossman, P. (2015). Protocol for language arts teaching observations (PLATO 5.0)<\/em>. Palo Alto, CA: Stanford University.<\/p>\n\n\n\n

J\u00f3hann \u00d6rn Sigurj\u00f3nsson. (2023). Quality in Icelandic mathematics teaching: Cognitive activation in mathematics lessons in a Nordic context<\/em> [Doktorsritger\u00f0]. H\u00e1sk\u00f3li \u00cdslands. https:\/\/<\/a>hdl.handle.net\/20.500.11815\/3843<\/a><\/p>\n\n\n\n

J\u00f3hann \u00d6rn Sigurj\u00f3nsson og Berglind G\u00edslad\u00f3ttir. (2020). Vitsmunaleg \u00e1skorun \u00ed st\u00e6r\u00f0fr\u00e6\u00f0ikennslu \u00e1 unglingastigi. T\u00edmarit um uppeldi og menntun<\/em>, 29(2), 149\u2013172. https:\/\/doi.org\/10.24270\/tuuom.2020.29.8<\/a><\/p>\n\n\n\n

Praetorius, A.-K., Klieme, E., Herbert, B., & Pinger, P. (2018). Generic dimensions of teaching quality: The German framework of Three Basic Dimensions. ZDM, 50<\/em>(3), 407\u2013426. https:\/\/doi.org\/10.1007\/s11858-018-0918-4<\/a><\/p>\n\n\n\n

Triplett, A. M. (2011). A closer look at the 24 game. International Journal of Applied Science and Technology, 1<\/em>(5), 161\u2013164. https:\/\/www.ijastnet.com\/journals\/Vol_1_No_5_September_2011\/20.pdf<\/a><\/p>\n\n\n\n

Weingartner, E. (2021). Cognitive Activation Potential of E&S Tasks at Commercial Vocational Schools in German Speaking Switzerland. In M. Blikstad-Balas, K. Klette, & M. Tengberg (Eds.), Ways of Analyzing Teaching Quality. Potentials and Pitfalls<\/em> (pp. 204\u2013228). Scandinavian University Press. https:\/\/doi.org\/10.18261\/9788215045054-2021-07<\/a>
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J\u00f3hann \u00d6rn Sigurj\u00f3nsson er n\u00fddoktor \u00ed st\u00e6r\u00f0fr\u00e6\u00f0imenntun, <\/em>kennari og t\u00f6lvunarfr\u00e6\u00f0ingur.<\/em><\/p>\n<\/div><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\n\n

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Vi\u00f0auki \u2013 Stu\u00f0ningsefni fyrir \u00feraut \u00e1 mynd 4<\/strong><\/p>\n\n\n\n

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\"\"<\/figure>\n<\/div><\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

J\u00f3hann \u00d6rn Sigurj\u00f3nsson. \u00deessi grein byggir \u00e1 erindi m\u00ednu \u00e1 n\u00e1mstefnu Flatar sem haldin var \u00e1 30 \u00e1ra afm\u00e6li samtakanna \u00feann 4. mars 2023. \u00c9g segi \u00ed stuttu m\u00e1li fr\u00e1 hluta ni\u00f0ursta\u00f0na \u00far doktorsverkefni m\u00ednu um hugr\u00e6na virkjun \u00ed st\u00e6r\u00f0fr\u00e6\u00f0ikennslu \u00e1 \u00cdslandi (J\u00f3hann \u00d6rn Sigurj\u00f3nsson, 2023). \u00c9g leitast vi\u00f0 a\u00f0 \u00fatsk\u00fdra hva\u00f0 hugr\u00e6n virkjun er,…<\/p>\n","protected":false},"author":1,"featured_media":1241,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-1195","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/flatarmal.is\/index.php?rest_route=\/wp\/v2\/posts\/1195","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/flatarmal.is\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/flatarmal.is\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/flatarmal.is\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/flatarmal.is\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1195"}],"version-history":[{"count":44,"href":"https:\/\/flatarmal.is\/index.php?rest_route=\/wp\/v2\/posts\/1195\/revisions"}],"predecessor-version":[{"id":5353,"href":"https:\/\/flatarmal.is\/index.php?rest_route=\/wp\/v2\/posts\/1195\/revisions\/5353"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/flatarmal.is\/index.php?rest_route=\/wp\/v2\/media\/1241"}],"wp:attachment":[{"href":"https:\/\/flatarmal.is\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1195"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/flatarmal.is\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1195"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/flatarmal.is\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1195"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}