{"id":2411,"date":"2024-02-17T17:30:39","date_gmt":"2024-02-17T17:30:39","guid":{"rendered":"https:\/\/flatarmal.is\/?p=2411"},"modified":"2024-02-17T18:20:05","modified_gmt":"2024-02-17T18:20:05","slug":"sveigjanleiki-og-adlogunarhaefni-i-reikniadferdum-skiptir-mali-hvernig-nemandinn-reiknar","status":"publish","type":"post","link":"https:\/\/flatarmal.is\/?p=2411","title":{"rendered":"Sveigjanleiki og a\u00f0l\u00f6gunarh\u00e6fni \u00ed reiknia\u00f0fer\u00f0um: Skiptir m\u00e1li hvernig nemandinn reiknar?\u00a0"},"content":{"rendered":"\n

L\u00f3a Bj\u00f6rk J\u00f3elsd\u00f3ttir.<\/strong><\/p>\n\n\n\n

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S\u00ed\u00f0an \u00e1 t\u00edunda \u00e1ratug s\u00ed\u00f0ustu aldar hefur veri\u00f0 v\u00ed\u00f0t\u00e6k s\u00e1tt me\u00f0al s\u00e9rfr\u00e6\u00f0inga \u00ed st\u00e6r\u00f0fr\u00e6\u00f0imenntun um a\u00f0 a\u00f0l\u00f6gunarh\u00e6fni (e. adaptivity) og sveigjanleiki (e. flexibility) s\u00e9u mikilv\u00e6g markmi\u00f0 \u00ed st\u00e6r\u00f0fr\u00e6\u00f0ikennslu (Baroody, 2003; Hickendorff o.fl., 2018; Xu o.fl., 2017). \u00cd l\u00f6ndum eins og Bandar\u00edkjunum, \u00c1stral\u00edu, Singap\u00far (Rittle-Johnson o.fl., 2012), Hollandi (Hickendorff, 2018), Belg\u00edu (Torbeyns o.fl., 2018) og einnig Danm\u00f6rku (B\u00f8rne- og Undervisningsministeriet, 2019a) er sl\u00edk h\u00e6fni skrifu\u00f0 inn \u00ed st\u00e6r\u00f0fr\u00e6\u00f0in\u00e1mskr\u00e1r landanna. \u00cd d\u00f6nsku n\u00e1mskr\u00e1nni, F\u00e6lles M\u00e5l<\/em> fr\u00e1 2019 er l\u00f6g\u00f0 \u00e1hersla \u00e1 mikilv\u00e6gi \u00feess a\u00f0 st\u00e6r\u00f0fr\u00e6\u00f0ikennarinn hvetji og sty\u00f0ji nemendur s\u00edna \u00e1 \u00feann h\u00e1tt a\u00f0 \u00feeir \u00fer\u00f3i reiknia\u00f0fer\u00f0ir s\u00ednar \u00e1 grundvelli talnaskilnings fremur en a\u00f0 l\u00e6ra a\u00f0 nota hef\u00f0bundin reiknirit og \u00e6fa notkun \u00feeirra (B\u00f8rne- og Undervisningsministeriet, 2019a, bls. 37). \u00deetta felur \u00ed s\u00e9r a\u00f0 nemendur \u00feurfa a\u00f0 \u00fer\u00f3a skilning \u00e1 m\u00f6rgum mismunandi a\u00f0fer\u00f0um til a\u00f0 leysa verkefnin.<\/p>\n<\/div>\n<\/div>\n\n\n\n

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Reiknia\u00f0fer\u00f0ir sem byggja \u00e1 me\u00f0fer\u00f0 talna e\u00f0a t\u00f6lustafa<\/strong><\/p>\n<\/div>\n<\/div>\n\n\n\n

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\u00cd \u00feessari grein er hugtaki\u00f0 reiknia\u00f0fer\u00f0ir nota\u00f0 sem \u00fe\u00fd\u00f0ing \u00e1 danska hugtakinu regnestrategi<\/em> sem skilgreint er sem r\u00f6\u00f0 a\u00f0ger\u00f0a \u00ed lausnarferlinu, sem byggja \u00e1 hvernig t\u00f6lurnar eru me\u00f0h\u00f6ndla\u00f0ar \u00ed hverju d\u00e6mi fyrir sig. Til d\u00e6mis vi\u00f0 lausn verkefnisins 482 + 218, munu \u00feeir sem miki\u00f0 hafa unni\u00f0 me\u00f0 100 vini sj\u00e1 a\u00f0 82 + 18 = 100 og a\u00f0 400 + 200 = 600, sem n\u00fdta m\u00e1 sem reiknia\u00f0fer\u00f0 \u00fear sem teki\u00f0 er mi\u00f0 af \u00feeim t\u00f6lum og reiknia\u00f0ger\u00f0 sem unni\u00f0 er me\u00f0. <\/p>\n\n\n\n

Reiknia\u00f0fer\u00f0ir geta \u00feannig b\u00e6\u00f0i veri\u00f0 talnatengdar reiknia\u00f0fer\u00f0ir og a\u00f0fer\u00f0ir sem einungis eru bundnar vi\u00f0 me\u00f0h\u00f6ndlun t\u00f6lustafa, eins og \u00ed hef\u00f0bundnu reikniriti. \u00dea\u00f0 er reginmunur \u00e1 talnatengdum reiknia\u00f0ger\u00f0um sem geta byggt \u00e1 mismunandi lei\u00f0um sem h\u00e6gt er a\u00f0 breyta og a\u00f0laga a\u00f0 einst\u00f6kum a\u00f0st\u00e6\u00f0um, allt eftir eiginleika \u00feeirra talna og \u00feeim reiknia\u00f0ger\u00f0um sem unni\u00f0 er me\u00f0 \u00e1samt reiknia\u00f0ger\u00f0um (sj\u00e1 t.d. Ostad, 1997; Siegler og Jenkins, 1989; Verschaffel o.fl., 2007) og a\u00f0fer\u00f0um sem byggja einungis \u00e1 me\u00f0fer\u00f0 t\u00f6lustafa [e. digit-based].\u00a0<\/p>\n\n\n\n

Hef\u00f0bundin reiknirit \u00ed samlagningu, fr\u00e1dr\u00e6tti, margf\u00f6ldun og deilingu eins og flestir hafa l\u00e6rt eru d\u00e6mi um a\u00f0fer\u00f0ir sem byggja \u00e1 me\u00f0h\u00f6ndlun t\u00f6lustafa. \u00deau eru bygg\u00f0 upp af f\u00f6stum og \u00f3breytanlegum skref-fyrir-skref a\u00f0fer\u00f0um (Siegler og Jenkins, 1989), \u00fear sem einmitt er a\u00f0eins unni\u00f0 me\u00f0 gildi hvers t\u00f6lustafs fyrir sig. Notkun hinna hef\u00f0bundnu reiknirita krefst \u00fev\u00ed ekki a\u00f0 liti\u00f0 s\u00e9 \u00e1 s\u00e6tisgildi \u00feeirra talna sem unni\u00f0 er me\u00f0, og h\u00e6gt a\u00f0 l\u00e6ra reikniriti\u00f0 utanb\u00f3kar og jafnvel \u00e1n skilnings \u00e1 \u00feeim reiknia\u00f0ger\u00f0um sem unni\u00f0 er me\u00f0.<\/p>\n\n\n\n

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Talnatengdu a\u00f0fer\u00f0irnar byggja aftur \u00e1 m\u00f3ti \u00e1 \u00fev\u00ed a\u00f0 skilja t\u00f6lur og eiginleika reiknia\u00f0ger\u00f0a og a\u00f0laga a\u00f0 \u00feeim a\u00f0st\u00e6\u00f0um sem unni\u00f0 er me\u00f0 hverju sinni. \u00dev\u00ed ber a\u00f0 l\u00edta \u00e1 t\u00f6lurnar og meta \u00e1\u00f0ur en a\u00f0fer\u00f0 vi\u00f0 h\u00e6fi er valin. A\u00f0fer\u00f0irnar krefjast skilnings \u00e1 \u00fev\u00ed hvernig h\u00e6gt er a\u00f0 skipta \u00ed einingar, tugi, hundru\u00f0 o.s.frv., til d\u00e6mis a\u00f0 384 = 300 + 80 + 4, og skilnings \u00e1 st\u00e6r\u00f0 talnanna og sta\u00f0setningu \u00e1 talnal\u00ednunni, til d\u00e6mis a\u00f0 199 er einum minni en 200 , \u00fe.e. 199 = 200 \u2013 1. D\u00e6mi um talnatengdar a\u00f0fer\u00f0ir og a\u00f0fer\u00f0ir sem byggja einungis \u00e1 t\u00f6lust\u00f6fum, m\u00e1 sj\u00e1 \u00ed t\u00f6flu 1.<\/p>\n<\/div>\n<\/div>\n\n\n\n

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1. Reiknia\u00f0fer\u00f0ir bygg\u00f0ar \u00e1 talna- og hugtakaskilningi<\/strong><\/p>\n<\/div>\n<\/div>\n\n\n\n

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Fl\u00fdtilei\u00f0ir<\/strong><\/p>\n<\/div>\n<\/div>\n\n\n\n

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A\u00f0rar a\u00f0fer\u00f0ir bygg\u00f0ar \u00e1 talna- og hugtakaskilningi<\/strong><\/p>\n<\/div>\n<\/div>\n\n\n\n

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2. A\u00f0fer\u00f0ir bygg\u00f0ar \u00e1 reikningi me\u00f0 t\u00f6lust\u00f6fum<\/strong><\/p>\n<\/div>\n<\/div>\n\n\n\n

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Hef\u00f0bundin reiknirit<\/strong><\/p>\n<\/div>\n<\/div>\n\n\n\n

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A\u00f0rar a\u00f0fer\u00f0ir sem byggja \u00e1 reikningi me\u00f0 t\u00f6lust\u00f6fum<\/strong><\/p>\n<\/div>\n<\/div>\n\n\n\n

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Tafla 1.<\/strong> D\u00e6mi um reiknia\u00f0fer\u00f0ir bygg\u00f0ar \u00e1 talna- og hugtakaskilningi [e. number-based strategies] og a\u00f0fer\u00f0ir er byggja \u00e1 reikningi me\u00f0 t\u00f6lust\u00f6fum.\u00a0<\/em><\/p>\n<\/div>\n<\/div>\n\n\n\n

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A\u00f0l\u00f6gunarh\u00e6fni og sveigjanleiki<\/strong><\/p>\n<\/div>\n<\/div>\n\n\n\n

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A\u00f0l\u00f6gunarh\u00e6fni \u00ed vali \u00e1 vi\u00f0eigandi a\u00f0fer\u00f0um, m\u00e1 sko\u00f0a fr\u00e1 \u00feremur mismunandi sj\u00f3narhornum (Verschaffel o.fl., 2007). Fyrsta sj\u00f3narhorni\u00f0, sem er megin\u00e1herslan \u00ed \u00feessari grein, er verkefnatengd a\u00f0l\u00f6gunarh\u00e6fni. H\u00e9r eru reiknia\u00f0fer\u00f0irnar valdar e\u00f0a a\u00f0laga\u00f0ar \u00feannig a\u00f0 \u00fe\u00e6r henti best \u00fev\u00ed verkefni (d\u00e6mi) sem unni\u00f0 er me\u00f0. \u00deannig er verkefni\u00f0 gert einfaldara, \u00ed formi f\u00e6rri skrefa og einfaldra \u00fatreikninga, \u00fe.e. me\u00f0 \u00fev\u00ed a\u00f0 nota svokalla\u00f0a fl\u00fdtilei\u00f0ir (e. shortcut) (Torbeyns o.fl., 2009; Xu o.fl., 2017). D\u00e6mi er notkun a\u00f0l\u00f6gunar a\u00f0 verkefnum sem hafa 9 einingar, t.d. 199 + 323 = 200 + 323 – 1 = 522 e\u00f0a vi\u00f0 \u00f3beina samlagningu (a\u00f0 fylla upp) fyrir fr\u00e1dr\u00e1tt me\u00f0 tveimur t\u00f6lum me\u00f0 litlum mun, t.d. 103 \u2013 98 = 2 + 3 = 5. Hin tv\u00f6 sj\u00f3narhornin \u00e1 a\u00f0l\u00f6gunarh\u00e6fni eru einstaklingsvitr\u00e6n (e. cognitive), \u00fe.e. \u00feegar val a\u00f0fer\u00f0ar tengist \u00fev\u00ed hva\u00f0 er hra\u00f0virkast og \u00f6ruggast fyrir einstaklinginn og loks menningarsj\u00f3narmi\u00f0, \u00fe.e. hva\u00f0 er vi\u00f0urkennt \u00ed \u00fev\u00ed (bekkjar)umhverfi sem nemandinn er \u00ed (Verschaffel o.fl., 2009).<\/p>\n<\/div>\n<\/div>\n\n\n\n

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D\u00f6nsk n\u00e1mskr\u00e1 og reiknia\u00f0fer\u00f0ir \u00ed s\u00f6gulegu samhengi<\/strong><\/p>\n<\/div>\n<\/div>\n\n\n\n

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Eins og \u00ed al\u00fej\u00f3\u00f0asamf\u00e9laginu er umr\u00e6\u00f0an um sveigjanleika \u00ed reiknia\u00f0fer\u00f0um ekki n\u00fd \u00ed d\u00f6nsku samhengi. Kennsla \u00ed sveigjanlegum reiknia\u00f0fer\u00f0um me\u00f0 \u00e1herslu \u00e1 talnatengdar a\u00f0fer\u00f0ir hefur veri\u00f0 hluti af markmi\u00f0um \u00ed st\u00e6r\u00f0fr\u00e6\u00f0ikennslu s\u00ed\u00f0an \u00e1ri\u00f0 2001 sem tilgreind eru \u00ed Klare m\u00e5l (Undervisningsministeriet, 2001).<\/p>\n\n\n\n

\u00dea\u00f0 leikur \u00fev\u00ed enginn vafi \u00e1 \u00fev\u00ed a\u00f0 r\u00e1\u00f0leggingar gefnar \u00fat af danska menntam\u00e1lar\u00e1\u00f0uneytinu undanfarna \u00e1ratugi hafa veri\u00f0 tengdar \u00fer\u00f3un talnatengdra reiknia\u00f0fer\u00f0a og sty\u00f0ur danskt n\u00e1msefni oftast vi\u00f0 \u00fe\u00e6r \u00e1herslur (J\u00f3elsd\u00f3ttir, 2023 bls. 112). \u00der\u00e1tt fyrir breyttar \u00e1herslur fr\u00e1 s\u00ed\u00f0ustu \u00f6ld, \u00feegar f\u00e6rni \u00ed notkun hef\u00f0bundinna reiknirita var allsr\u00e1\u00f0andi hefur skort \u00feekkingu \u00e1 \u00fev\u00ed hvernig danskir nemendur reikna, og hvort tengsl s\u00e9u \u00e1 milli vals \u00e1 reiknia\u00f0fer\u00f0um, \u00ed formi sveigjanleika og a\u00f0l\u00f6gunarh\u00e6fni, og st\u00e6r\u00f0fr\u00e6\u00f0ih\u00e6fni nemenda. Ef vi\u00f0 sko\u00f0um \u00feau fj\u00f6gur d\u00e6mi \u00ed reikningi me\u00f0 heilum t\u00f6lum \u00ed reiknia\u00f0ger\u00f0unum fj\u00f3rum sem hef\u00f0 er a\u00f0 birtist \u00ed lokapr\u00f3fi 9. bekkjar \u00ed st\u00e6r\u00f0fr\u00e6\u00f0i (10. bekkur \u00ed \u00edslensku sk\u00f3lakerfi), m\u00e1 sj\u00e1 \u00ed f\u00e6rnihluta pr\u00f3fsins fr\u00e1 ma\u00ed 2019, a\u00f0 30% n\u00e1 ekki a\u00f0 reikna r\u00e9tt fr\u00e1dr\u00e1ttard\u00e6mi\u00f0 701 – 149 og tilsvarandi 31% geta ekki leyst r\u00e9tt deilingard\u00e6mi\u00f0 7021 : 7 (B\u00f8rne- og Undervisningsministeriet, 2019b). B\u00e6\u00f0i verkefnin hafa \u00fe\u00e1 eiginleika a\u00f0 sty\u00f0ja vi\u00f0 notkun talnami\u00f0a\u00f0ra reiknia\u00f0fer\u00f0a (sj\u00e1 t\u00f6flu 1). H\u00e9r eru einnig hentugar fl\u00fdtilei\u00f0ir \u00ed bo\u00f0i, sem eiga \u00fea\u00f0 sameiginlegt a\u00f0 hafa f\u00e6kka\u00f0 \u00ferepum \u00ed \u00fatreikningi samanbori\u00f0 vi\u00f0 notkun hef\u00f0bundinna reiknirita. Ef vi\u00f0 sko\u00f0um n\u00e1nar d\u00e6miger\u00f0 r\u00f6ng sv\u00f6r kemur \u00e1kve\u00f0i\u00f0 mynstur \u00ed lj\u00f3s. Fyrir verkefni\u00f0 701 \u2013 149 eru fj\u00f6gur af fimm algengustu r\u00f6ngu sv\u00f6runum 462, 562, 648 og 652 e\u00f0a samanlagt t\u00e6pur helmingur allra rangra svara (8750 nemendur) og 13,6% allra lausna (B\u00f8rne- og Undervisningsministeriet, 2019b). \u00deessi r\u00f6ngu sv\u00f6r eiga \u00fea\u00f0 sameiginlegt a\u00f0 geta veri\u00f0 aflei\u00f0ing d\u00e6miger\u00f0ra villna \u00feegar hef\u00f0bundi\u00f0 reiknirit er nota\u00f0 (tafla 2).<\/p>\n<\/div>\n<\/div>\n\n\n\n

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Tafla 2.<\/em><\/strong> Fr\u00e1dr\u00e1ttur \u00e1 lokapr\u00f3fi eftir 9. bekk, ma\u00ed 2019. Yfirlit yfir fj\u00f6gur af fimm algengustu r\u00f6ngu sv\u00f6rum vi\u00f0 701 <\/em>– 149, \u00e1samt m\u00f6gulegri lei\u00f0 a\u00f0 r\u00f6ngu svari. N = 64 331.<\/em><\/p>\n<\/div>\n<\/div>\n\n\n\n

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Ni\u00f0urst\u00f6\u00f0urnar sty\u00f0ja tilg\u00e1tu um a\u00f0 \u00fer\u00e1tt fyrir tilm\u00e6li \u00ed lei\u00f0beiningum me\u00f0 n\u00e1mskr\u00e1 noti danskir \u200b\u200bnemendur talnatengdar a\u00f0fer\u00f0ir \u00ed takm\u00f6rku\u00f0um m\u00e6li, en noti hef\u00f0bundin reiknirit. Ni\u00f0urst\u00f6\u00f0ur \u00e1 greiningu \u00e1 deilingarverkefnum fr\u00e1 2011, 2016 og 2019, s\u00fdna hvernig einf\u00f6ld villa getur or\u00f0i\u00f0 \u00feegar nemendur muna ekki a\u00f0ger\u00f0ar\u00f6\u00f0ina \u00ed reikniritinu \u00feegar 0 er ein af t\u00f6lunum \u00ed d\u00e6minu. Eiginleikar verkefnisins sty\u00f0ja sem fyrr vi\u00f0 notkun fl\u00fdtilei\u00f0a og a\u00f0 h\u00e6gt s\u00e9 a\u00f0 einfalda lausnarferli\u00f0 me\u00f0 talnatengdri a\u00f0fer\u00f0, h\u00e9r t.d. me\u00f0 \u00fev\u00ed a\u00f0 brj\u00f3ta t\u00f6luna upp \u00e1 vi\u00f0eigandi h\u00e1tt; 700 \u2013 150 = 550, 550 + 2 = 552 og 7000 : 7 = 1000 og 21 : 7 = 3, 1000 + 3 = 1003.<\/p>\n<\/div>\n<\/div>\n\n\n\n

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Tengsl a\u00f0l\u00f6gunarh\u00e6fni og sveigjanleika \u00ed reiknia\u00f0fer\u00f0um og almennrar h\u00e6fni \u00ed st\u00e6r\u00f0fr\u00e6\u00f0i<\/strong><\/p>\n<\/div>\n<\/div>\n\n\n\n

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\u00cd b\u00e6\u00f0i al\u00fej\u00f3\u00f0legum og d\u00f6nskum ranns\u00f3knum hefur veri\u00f0 s\u00fdnt fram \u00e1 tengsl milli notkunar nemenda \u00e1 talnatengdum reiknia\u00f0fer\u00f0um snemma \u00ed n\u00e1msferlinu og s\u00ed\u00f0ari \u00fer\u00f3unar h\u00e6fni \u00ed st\u00e6r\u00f0fr\u00e6\u00f0i (Dowker, 2014; Ostad, 1997; Sunde, 2023; Vanbinst o.fl., 2014 ). Ni\u00f0urst\u00f6\u00f0ur ranns\u00f3kna Pernille B\u00f8dtker Sunde (2023)   s\u00fdna a\u00f0 nemendur \u00ed 1. bekk (2. bekkur \u00ed \u00edslensku sk\u00f3lakerfi) sem leystu einf\u00f6ld d\u00e6mi \u00fear sem lag\u00f0ar eru saman tv\u00e6r eins stafa t\u00f6lur [e. single digit addition] me\u00f0 \u00fev\u00ed a\u00f0 telja st\u00f3\u00f0u sig verr \u00ed 4. bekk b\u00e6\u00f0i \u00ed textaverkefnum, almennum brotum og j\u00f6fnum \u00ed samanbur\u00f0i vi\u00f0 nemendur sem notu\u00f0u \u00fer\u00f3a\u00f0ri a\u00f0fer\u00f0ir \u00ed 1. bekk t.d. leystu d\u00e6mi eins og 8 + 7 = 8 + 8 \u2013 1, 7 + 7 + 1, 8 + 2 + 5 e\u00f0a me\u00f0 \u00f6\u00f0rum svipu\u00f0um reiknia\u00f0fer\u00f0um sem byggja \u00e1 talna- og a\u00f0ger\u00f0arskilningi nemenda.<\/p>\n\n\n\n

\u00der\u00e1tt fyrir a\u00f0 s\u00e9rfr\u00e6\u00f0ingar \u00e1 \u00feessu svi\u00f0i s\u00e9u samm\u00e1la um mikilv\u00e6gi \u00feess a\u00f0 \u00fer\u00f3a margv\u00edslegar a\u00f0fer\u00f0ir, \u00fe.e. sveigjanleika \u00ed reiknia\u00f0fer\u00f0um og getu til a\u00f0 velja \u00e1kj\u00f3sanlegar a\u00f0fer\u00f0ir fyrir mismunandi a\u00f0st\u00e6\u00f0ur, \u00fe.e. a\u00f0l\u00f6gunarh\u00e6fni, eru takmarka\u00f0ar ranns\u00f3knir \u00e1 \u00feessu svi\u00f0i. Einnig eru ni\u00f0urst\u00f6\u00f0ur var\u00f0andi tengsl sveigjanleika \u00ed reiknia\u00f0fer\u00f0um og frammist\u00f6\u00f0u \u00ed st\u00e6r\u00f0fr\u00e6\u00f0i ekki sk\u00fdrar (Verschaffel, 2023). Fyrir meira en 30 \u00e1rum kynntu Hatano og Inagaki (1984) hugt\u00f6kin s\u00e9rfr\u00e6\u00f0ingar \u00ed a\u00f0l\u00f6gun og s\u00e9rfr\u00e6\u00f0ingar \u00ed verklagi (e. adaptive and rutine experts). \u00deeir bentu \u00e1 a\u00f0 markmi\u00f0 st\u00e6r\u00f0fr\u00e6\u00f0ikennslu hlyti a\u00f0 vera a\u00f0 \u00fer\u00f3a a\u00f0l\u00f6gunarh\u00e6fni, sem feli \u00ed s\u00e9r a\u00f0 nemendur geti a\u00f0laga\u00f0 a\u00f0fer\u00f0aval sitt a\u00f0 mismunandi og breyttum a\u00f0st\u00e6\u00f0um \u00fat fr\u00e1 skilningi s\u00ednum. \u00de\u00f3 a\u00f0 verklagss\u00e9rfr\u00e6\u00f0ingar, sem byggja \u00e1 mikilli \u00e6fingu, hafi f\u00e6rni til a\u00f0 leysa \u00e1kve\u00f0in verkefni b\u00e6\u00f0i flj\u00f3tt og \u00f6rugglega, skortir \u00fe\u00e1 f\u00e6rni til a\u00f0 a\u00f0lagast n\u00fdjum a\u00f0st\u00e6\u00f0um \u00e1 sveigjanlegan h\u00e1tt (Hatano og Inagaki, 1984).\u00a0<\/p>\n\n\n\n

\u00cd doktorsverkefni m\u00ednu hef \u00e9g rannsaka\u00f0 \u00fe\u00e6r reiknia\u00f0fer\u00f0ir sem danskir nemendur nota \u00ed samlagningu, fr\u00e1dr\u00e6tti og margf\u00f6ldun me\u00f0 heilum st\u00e6rri t\u00f6lum (2-3 t\u00f6lustafir), b\u00e6\u00f0i me\u00f0 tilliti til mismunandi aldursh\u00f3pa, kyns og annarra einstaklingsbundinna einkenna og tengsla milli a\u00f0l\u00f6gunarh\u00e6fni og sveigjanleika \u00ed reiknia\u00f0fer\u00f0um vi\u00f0 st\u00e6r\u00f0fr\u00e6\u00f0ih\u00e6fni nemandans.<\/p>\n<\/div>\n<\/div>\n\n\n\n

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Hvernig reikna danskir nemendur og skiptir m\u00e1li hvernig \u00feeir reikna?<\/strong><\/p>\n<\/div>\n<\/div>\n\n\n\n

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\u00de\u00e1tttakendur \u00ed doktorsranns\u00f3kn minni voru 2298 nemendur \u00ed 3., 6. og 8. bekk (sem er 4., 7. og 9. bekkur \u00e1 \u00cdslandi) og leystu \u00feeir \u201eTri-phase Flexibility Assessment\u201c<\/em> (TriFA), sem er matsverkefni \u00ed \u00feremur \u00ferepum (Xu et al, 2017, J\u00f3elsd\u00f3ttir og Andrews, 2023). TriFA var upphaflega \u00fer\u00f3a\u00f0 til a\u00f0 meta sveigjanleika \u00ed tengslum vi\u00f0 j\u00f6fnur en er h\u00e9r a\u00f0laga\u00f0 a\u00f0 sveigjanleika \u00ed reiknia\u00f0fer\u00f0um, me\u00f0 heilum t\u00f6lum. \u00cd \u00ferepi eitt, leysa nemendur d\u00e6min, (\u00e1tta d\u00e6mi \u00ed 3. bekk, n\u00edu d\u00e6mi \u00ed 6. bekk og t\u00f3lf verkefni \u00ed 8. bekk). Nemendur voru be\u00f0nir um a\u00f0 sko\u00f0a vel t\u00f6lurnar \u00e1\u00f0ur en \u00feeir leystu d\u00e6min. Ef \u00feeir v\u00f6ldu a\u00f0 leysa verkefni\u00f0 \u00ed huganum, voru \u00feeir be\u00f0nir um a\u00f0 skr\u00e1 me\u00f0 st\u00e6r\u00f0fr\u00e6\u00f0it\u00e1knum, ritu\u00f0um texta e\u00f0a teikningum hvernig \u00feeir leystu d\u00e6mi\u00f0, og ef nemandi leysti d\u00e6mi\u00f0 me\u00f0 bla\u00f0\/bl\u00fdant, \u00fe\u00e1 var nemandinn be\u00f0inn um a\u00f0 s\u00fdna \u00fatreikninga fyrir hvert skref \u00ed lausnarferlinu. Fyrsta \u00ferepi\u00f0 er n\u00fdtt til a\u00f0 meta a\u00f0l\u00f6gunarh\u00e6fni nemandans\u00a0 \u00fear sem notkun fl\u00fdtilei\u00f0a er metin sem t\u00e1kn um a\u00f0l\u00f6gun. \u00dea\u00f0 er mikilv\u00e6gt a\u00f0 geta \u00feess a\u00f0 \u00f6ll d\u00e6min \u00ed TriFA eru h\u00f6nnu\u00f0 til a\u00f0 draga fram notkun fl\u00fdtilei\u00f0a er byggja \u00e1 talna- og hugtakaskilningi nemandans t.d. 199 + 323, 103 \u2013 98 og\u00a012 x 15. \u00cd n\u00e6sta \u00ferepi fengu nemendur fyrirm\u00e6li um a\u00f0 leysa s\u00f6mu d\u00e6min aftur en n\u00fa me\u00f0 n\u00fdjum a\u00f0fer\u00f0um, eins m\u00f6rgum og nemandinn var f\u00e6r um, \u00fe\u00f3 a\u00f0 h\u00e1marki 4 n\u00fdjar a\u00f0fer\u00f0ir (2 \u00ed 3. bekk). Nemendur voru aftur hvattir til a\u00f0 sko\u00f0a vel t\u00f6lurnar \u00ed hverju d\u00e6mi fyrir sig og \u00feeim bent \u00e1 a\u00f0 m\u00f6gulegt er a\u00f0 leysa verkefnin \u00ed huganum og skr\u00e1 lei\u00f0ina e\u00f0a a\u00f0 nota skriflega \u00fatreikninga.<\/p>\n\n\n\n

Meginni\u00f0urst\u00f6\u00f0ur ranns\u00f3knarinnar eru birtar \u00ed J\u00f3elsd\u00f3ttir og Andrews (2023). \u00dear kemur fram a\u00f0 langflestir nemendur \u00e1 \u00f6llum \u00feremur aldursstigum s\u00fdna takmarka\u00f0a a\u00f0l\u00f6gunarh\u00e6fni en 70% nemenda \u00ed 3. bekk, 45% \u00ed 6. bekk og 59% \u00ed 8. bekk nota aldrei fl\u00fdtilei\u00f0ir til a\u00f0 leysa d\u00e6min, \u00fer\u00e1tt fyrir a\u00f0 d\u00e6min s\u00e9u h\u00f6nnu\u00f0 til a\u00f0 sty\u00f0ja vi\u00f0 sl\u00edkar a\u00f0fer\u00f0ir og nemendur v\u00e6ru s\u00e9rstaklega be\u00f0nir um a\u00f0 sko\u00f0a vel t\u00f6lurnar \u00ed hverju d\u00e6mi fyrir sig. Um 3% nemenda \u00e1 hverju aldursstigi n\u00fdttu fl\u00fdtilei\u00f0ir vi\u00f0 lausn flestra d\u00e6manna e\u00f0a 6-8 af 8 d\u00e6mum \u00ed 3. bekk, 7-9 af 9 d\u00e6mum \u00ed 6. bekk og 9-12 af 12 \u00ed 8. bekk. Ni\u00f0urst\u00f6\u00f0ur \u00far \u00ferepi 2 s\u00fdna sveigjanleika nemenda \u00fe.e. h\u00e6fni til a\u00f0 leysa verkefni me\u00f0 fleiri a\u00f0fer\u00f0um. \u00de\u00e6r ni\u00f0urst\u00f6\u00f0ur s\u00fdna a\u00f0 49% nemenda \u00ed 3. bekk, 24% nemenda \u00ed 6. bekk og 21% nemenda \u00ed 8. bekk eru ekki f\u00e6rir um a\u00f0 leysa eitt d\u00e6mi me\u00f0 fleiri en einni a\u00f0fer\u00f0, en 9% nemenda \u00ed 3. bekk, 14% nemenda \u00ed 6. bekk og 15% nemenda \u00ed 8. bekk leysa (6-8\/7-9\/9-12) d\u00e6mi me\u00f0 fleiri en einni a\u00f0fer\u00f0.<\/p>\n\n\n\n

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Af \u00feessum t\u00f6lum m\u00e1 sj\u00e1 a\u00f0 nemendur \u00ed 6. bekk s\u00fdna meiri a\u00f0l\u00f6gunarh\u00e6fni samanbori\u00f0 vi\u00f0 nemendur \u00ed 3. og 8. bekk og enginn markt\u00e6kur munur er \u00e1 sveigjanleika nemenda \u00ed 6. og 8. bekk. \u00deessar ni\u00f0urst\u00f6\u00f0ur benda til a\u00f0 a\u00f0l\u00f6gunarh\u00e6fni og sveigjanleiki aukist ekki me\u00f0 aldri nemenda, \u00fer\u00e1tt fyrir a\u00f0 forsendur byggi \u00e1 talna- og hugtakaskilningi sem \u00fer\u00f3ast me\u00f0 reynslu nemenda, ekki s\u00edst me\u00f0 \u00fe\u00e1ttt\u00f6ku \u00feeirra \u00ed \u00feeirri st\u00e6r\u00f0fr\u00e6\u00f0ikennslu sem \u00feau f\u00e1 \u00ed sk\u00f3lanum. \u00dea\u00f0 m\u00e1 \u00fev\u00ed \u00e6tla a\u00f0 \u00fe\u00e6r lei\u00f0beiningar sem fram koma \u00ed n\u00e1mskr\u00e1nni \u00ed st\u00e6r\u00f0fr\u00e6\u00f0i n\u00e1i ekki inn \u00ed kennslustofuna og a\u00f0 eldri nemendur n\u00fdti sjaldan a\u00f0fer\u00f0ir er byggja \u00e1 talnaskilningi en treysti \u00e1 hef\u00f0bundin reiknirit.\u00a0<\/p>\n\n\n\n

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\u00deegar liti\u00f0 er n\u00e1nar \u00e1 a\u00f0fer\u00f0ir nemenda vi\u00f0 a\u00f0 leysa einstaka d\u00e6mi kemur sk\u00fdrt fram a\u00f0 notkun hef\u00f0bundinna reiknirita er lang mest nota\u00f0a a\u00f0fer\u00f0in. Vi\u00f0 lausn verkefnanna 298 + 483 og 199 + 323 nota 86% nemenda \u00ed 8. bekk hef\u00f0bundi\u00f0 reiknirit, samanbori\u00f0 vi\u00f0 48% \u00ed 3. bekk og 69% \u00ed 6. bekk (J\u00f3elsd\u00f3ttir og Sunde, 2024). Einungis 7% nemenda \u00ed 3. bekk, 8% \u00ed 6. bekk og 5% \u00ed 8. bekk nota a\u00f0l\u00f6gun<\/em>, \u00fe.e leysa t.d. 199 + 323 me\u00f0 \u00fev\u00ed a\u00f0 hugsa 200 + 323 \u2013 1, e\u00f0a samb\u00e6rilegar \u00fatf\u00e6rslur sem n\u00fdta a\u00f0 199 liggur n\u00e1l\u00e6gt t\u00f6lunni 200 og a\u00f0 \u00fea\u00f0 er au\u00f0veldara a\u00f0 leggja 200 saman vi\u00f0 n\u00e6stu t\u00f6lu samanbori\u00f0 vi\u00f0 t\u00f6luna 199. H\u00e9r vekur s\u00e9rstaka athygli a\u00f0  l\u00e6gsta hlutfall \u00feeirra sem nota a\u00f0l\u00f6gun eru nemendur 8. bekkjar, \u00fear sem hef\u00f0bundin reiknirit vir\u00f0ast hafa teki\u00f0 yfirh\u00f6ndina, einnig \u00ed a\u00f0st\u00e6\u00f0um \u00fear sem m\u00f6guleiki er \u00e1 einf\u00f6ldum og hagkv\u00e6mum fl\u00fdtilei\u00f0um. <\/p>\n\n\n\n

En skiptir m\u00e1li hva\u00f0a a\u00f0fer\u00f0ir nemendur n\u00fdta? Til a\u00f0 gefa einhl\u00edtt svar vi\u00f0 \u00feessari spurningu skortir frekari ranns\u00f3knir sem s\u00fdna sk\u00fdrt fram \u00e1 tengsl aukins sveigjanleika og a\u00f0l\u00f6gunarh\u00e6fni vi\u00f0 betri st\u00e6r\u00f0fr\u00e6\u00f0ih\u00e6fni. \u00dea\u00f0 m\u00e1 \u00fe\u00f3 finna nokkrar v\u00edsbendingar \u00feegar n\u00e1nar eru sko\u00f0u\u00f0 tengsl \u00feeirra a\u00f0fer\u00f0a sem nota\u00f0ar eru vi\u00f0 annars vegar n\u00e1kv\u00e6mni \u00ed sv\u00f6rum (r\u00e9tt sv\u00f6r) og hins vegar vi\u00f0 \u00e1rangur nemenda t.d. \u00e1 samr\u00e6mdum pr\u00f3fum \u00ed st\u00e6r\u00f0fr\u00e6\u00f0i. \u00cd J\u00f3elsd\u00f3ttir og Andrews (2023) kemur fram a\u00f0 b\u00e6\u00f0i meiri sveigjanleiki og meiri a\u00f0l\u00f6gunarh\u00e6fni eru tengd aukinni n\u00e1kv\u00e6mni, \u00fe.e. a\u00f0 \u00feeir nemendur sem s\u00fdna meiri sveigjanleika (geta leyst sama d\u00e6mi me\u00f0 fleiri a\u00f0fer\u00f0um) reikna fleiri d\u00e6mi r\u00e9tt og\/e\u00f0a nemendur sem s\u00fdna meiri a\u00f0l\u00f6gunarh\u00e6fni (n\u00fdta oftar fl\u00fdtilei\u00f0ir) reikna einnig fleiri d\u00e6mi r\u00e9tt. Til d\u00e6mis leystu 75% nemenda \u00ed 3. bekk verkefnin 298 + 483 e\u00f0a 199 + 323 r\u00e9tt me\u00f0 hef\u00f0bundnu reikniriti samanbori\u00f0 vi\u00f0 88% af \u00feeim sem notu\u00f0u a\u00f0l\u00f6gun (t.d. 200 + 323 -1). Munurinn er enn sk\u00fdrari \u00feegar liti\u00f0 er \u00e1 lausn fr\u00e1dr\u00e1ttard\u00e6ma \u00fear sem st\u00e6rri h\u00f3pur nemenda \u00e1 \u00f6llum aldursstigum s\u00fdnir \u00f6r\u00f0ugleika vi\u00f0 a\u00f0 f\u00e1 r\u00e9ttar lausnir verkefnanna. \u00cd t\u00f6flu 3 m\u00e1 sj\u00e1 hlutfall nemenda sem n\u00fdtir hef\u00f0bundin reiknirit og a\u00f0l\u00f6gun vi\u00f0 lausn d\u00e6manna. H\u00e9r kemur fram a\u00f0 notkun a\u00f0l\u00f6gunar er \u00e1 \u00f6llum aldursstigum n\u00e1kv\u00e6mari a\u00f0fer\u00f0 mi\u00f0a\u00f0 vi\u00f0 hef\u00f0bundin reiknirit.<\/p>\n<\/div>\n<\/div>\n\n\n\n

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Tafla 3.<\/em><\/strong> Hlutfall nemenda sem leysir verkefnin 693 \u2013 499 e\u00f0a 673 \u2013 199 me\u00f0 hef\u00f0bundnu reikniriti e\u00f0a me\u00f0 a\u00f0l\u00f6gun og hlutfall r\u00e9ttra lausna fyrir hvora a\u00f0fer\u00f0 (J\u00f3elsd\u00f3ttir og Sunde, 2024).<\/em><\/p>\n<\/div>\n<\/div>\n\n\n\n

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Til a\u00f0 svara spurningunni m\u00e1 einnig sko\u00f0a n\u00e1nar a\u00f0fer\u00f0ir \u00feeirra nemenda sem hafa n\u00e1\u00f0 mikilli f\u00e6rni \u00ed a\u00f0 reikna r\u00e9tt, \u201ereiknis\u00e9rfr\u00e6\u00f0inga\u201c (e. arithmetic experts). Ef \u00feessum h\u00f3pi er skipt \u00ed annars vegar \u00fe\u00e1 nemendur sem treysta \u00e1 hef\u00f0bundin reiknirit, s\u00e9rfr\u00e6\u00f0inga \u00ed verklagi, og hins vegar s\u00e9rfr\u00e6\u00f0ingana \u00ed a\u00f0l\u00f6gun, nemendur sem n\u00fdta fl\u00fdtilei\u00f0ir \u00ed a\u00f0 minnsta kosti 3 d\u00e6mum e\u00f0a meira og \u00e1rangur \u00feessara tveggja h\u00f3pa \u00e1 samr\u00e6mdum pr\u00f3fum \u00ed st\u00e6r\u00f0fr\u00e6\u00f0i (fr\u00e1 sama sk\u00f3la\u00e1ri \u00fe.e. 3., 6. og 8. bekk) er sko\u00f0a\u00f0ur, m\u00e1 sj\u00e1 markt\u00e6kan mun.  B\u00e1\u00f0ir h\u00f3par standa sig vel \u00ed samr\u00e6mdum pr\u00f3fum en me\u00f0altal s\u00e9rfr\u00e6\u00f0inga \u00ed verklagi er 0,56 samanbori\u00f0 vi\u00f0 1,05 hj\u00e1 s\u00e9rfr\u00e6\u00f0ingum \u00ed a\u00f0l\u00f6gun. H\u00e9r er mi\u00f0a\u00f0 vi\u00f0 sta\u00f0la\u00f0an kvar\u00f0a \u00fear sem me\u00f0alskor er 0 og sta\u00f0alfr\u00e1vik 1. Me\u00f0altal s\u00e9rfr\u00e6\u00f0inga \u00ed a\u00f0l\u00f6gun er \u00fev\u00ed u.\u00fe.b. h\u00e1lfu sta\u00f0alfr\u00e1viki h\u00e6rra en s\u00e9rfr\u00e6\u00f0inga \u00ed verklagi, sem er mikill munur \u00ed \u00feessu samhengi. <\/p>\n\n\n\n

\u00deessar ni\u00f0urst\u00f6\u00f0ur benda \u00fev\u00ed til a\u00f0 \u00fea\u00f0 s\u00e9 sterkt samhengi \u00e1 milli \u00e1rangurs \u00ed st\u00e6r\u00f0fr\u00e6\u00f0i og \u00feeirra a\u00f0fer\u00f0a sem nemendur nota vi\u00f0 reikning.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\n\n

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Hvernig m\u00e1 n\u00fdta \u00feessar ni\u00f0urst\u00f6\u00f0ur \u00ed \u00fer\u00f3un st\u00e6r\u00f0fr\u00e6\u00f0ikennslu?<\/strong><\/p>\n<\/div>\n<\/div>\n\n\n\n

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Fyrir \u00fea\u00f0 fyrsta s\u00fdna \u00feessar ni\u00f0urst\u00f6\u00f0ur a\u00f0 \u00fer\u00e1tt fyrir tilm\u00e6li n\u00e1mskr\u00e1r er takm\u00f6rku\u00f0 \u00e1hersla \u00e1 a\u00f0 nemendur \u00fer\u00f3i me\u00f0 s\u00e9r a\u00f0fer\u00f0ir sem byggja \u00e1 talna- og hugtakaskilningi og \u00fer\u00f3i me\u00f0 s\u00e9r sveigjanleika og a\u00f0l\u00f6gunarh\u00e6fni. Ekki s\u00edst er \u00e1hugaver\u00f0 s\u00fa ni\u00f0ursta\u00f0a a\u00f0 nemendur \u00ed 8. bekk noti fl\u00fdtilei\u00f0ir, bygg\u00f0ar \u00e1 talnaskilningi \u00ed minna m\u00e6li en nemendur 6. bekkjar. Eldri ranns\u00f3knir t.d. \u00fe\u00fdsk ranns\u00f3kn fr\u00e1 2001 (Selter, 2001) s\u00fdna a\u00f0 nemendur treysta \u00e1 hef\u00f0bundin reiknirit fr\u00e1 \u00feeim degi sem \u00feau eru innleidd \u00ed st\u00e6r\u00f0fr\u00e6\u00f0ikennsluna, \u00fe.e. a\u00f0 a\u00f0rar a\u00f0fer\u00f0ir v\u00edkja fyrir hef\u00f0bundna reikniritinu. T\u00edminn sem n\u00fdttur er \u00ed a\u00f0 l\u00e6ra og \u00e6fa notkun hef\u00f0bundinna reiknirita, fer \u00ed a\u00f0 l\u00e6ra utanb\u00f3kar (oft \u00e1n frekari skilnings) reikniferli \u00ed skrefum sem framkv\u00e6ma \u00fearf \u00ed \u00e1kve\u00f0inni r\u00f6\u00f0 og einungis er unni\u00f0 me\u00f0 t\u00f6lustafi, \u00e1n s\u00e6tisgilda. Kostir hinna hef\u00f0bundnu reiknirita eru tengd \u00fev\u00ed a\u00f0 \u00feau eru eins fyrir allar t\u00f6lur innan hverrar reiknia\u00f0ger\u00f0ar og veitir \u00fea\u00f0 notendum \u00e1kve\u00f0i\u00f0 \u00f6ryggi. \u00dea\u00f0 ber \u00fe\u00f3 a\u00f0 nefna a\u00f0 me\u00f0 breyttum t\u00edmum og \u00e1herslum er a\u00f0gengi a\u00f0 reikniv\u00e9lum or\u00f0i\u00f0 svo gott a\u00f0 hlutverk reikniritanna t.d. \u00ed daglegu l\u00edfi er takmarka\u00f0 en \u00e1 sama t\u00edma hefur \u00fe\u00f6rfin \u00e1 g\u00f3\u00f0um talna- og hugtakaskilningi, \u00e1samt h\u00e6fni \u00ed lausnami\u00f0u\u00f0u ferli aukist. \u00d3kostir \u00feess a\u00f0 n\u00fdta t\u00edmann \u00ed kennslu og \u00fej\u00e1lfun \u00ed notkun reikniritanna er \u00fev\u00ed b\u00e6\u00f0i a\u00f0 l\u00edtill t\u00edmi er til r\u00e1\u00f0st\u00f6funar til a\u00f0 \u00fej\u00e1lfa talna- og hugtakaskilning vi\u00f0 \u00fer\u00f3un annarra reiknia\u00f0fer\u00f0a og a\u00f0, eins og fyrri ranns\u00f3knir s\u00fdna, er erfitt a\u00f0 f\u00e1 nemendur til a\u00f0 n\u00fdta talnaskilning sinn og \u00fer\u00f3a frekar sveigjanleika og a\u00f0l\u00f6gunarh\u00e6fni eftir a\u00f0 reikniritin hafa veri\u00f0 kynnt og \u00e6f\u00f0. \u00cd \u00feessu samhengi m\u00e1 \u00fev\u00ed huglei\u00f0a hva\u00f0a h\u00e6fni er mikilv\u00e6gast a\u00f0 nemendur hafi a\u00f0 loknum grunnsk\u00f3la og \u00fev\u00ed hvernig t\u00edminn er best n\u00fdttur til a\u00f0 n\u00e1 \u00feeim markmi\u00f0um sem unni\u00f0 er eftir.<\/p>\n\n\n\n

A\u00f0 l\u00e6ra hef\u00f0bundin reiknirit er \u00fea\u00f0 sem margir tengja vi\u00f0 sk\u00f3last\u00e6r\u00f0fr\u00e6\u00f0i, \u00fea\u00f0 gildir b\u00e6\u00f0i um kennara en ekki s\u00edst foreldra. A\u00f0 breyta grunnhugmyndum um hva\u00f0 er mikilv\u00e6gt \u00ed st\u00e6r\u00f0fr\u00e6\u00f0ikennslunni tekur t\u00edma og mikilv\u00e6gt er a\u00f0 rannsaka frekar b\u00e6\u00f0i samhengi \u00e1 milli aukinnar a\u00f0l\u00f6gunarh\u00e6fni og sveigjanleika og st\u00e6r\u00f0fr\u00e6\u00f0ih\u00e6fni og hvernig sty\u00f0ja eigi vi\u00f0 \u00feessa \u00fer\u00f3un \u00ed st\u00e6r\u00f0fr\u00e6\u00f0ikennslunni. \u00deetta er verkefni sem \u00fer\u00e1tt fyrir a\u00f0 hafa veri\u00f0 \u00ed umr\u00e6\u00f0unni n\u00fa \u00ed yfir 20 \u00e1r krefst enn frekari athygli \u00e1 komandi \u00e1rum.<\/p>\n<\/div>\n<\/div>\n\n\n\n

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Heimildaskr\u00e1<\/strong><\/p>\n<\/div>\n<\/div>\n\n\n\n

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Baroody, A. J. (2003). The development of adaptive expertise and flexibility: The integration of conceptual and procedural knowledge. In A. J. Baroody og A. Dowker (Eds.), The development of arithmetic concepts and skills. Constructing adaptive expertise<\/em> (bls. 1-33). Lawrence Erlbaum Associates Publishers. <\/p>\n\n\n\n

B\u00f8rne- og Undervisningsministeriet. (2019a) Matematik F\u00e6lles M\u00e5l 2019. <\/em>https:\/\/emu.dk\/sites\/default\/files\/2020-09\/GSK_F%C3%A6llesM%C3%A5l_Matematik.pdf<\/a><\/p>\n\n\n\n

B\u00f8rne- og Undervisningsministeriet. (2019b). Vejledning til folkeskolens pr\u00f8ver i faget matematik \u2013 9. klasse<\/em>. <\/p>\n\n\n\n

Dowker, A. (2014). Young children\u2019s use of derived fact strategies for addition and subtraction. Frontiers in Human Neuroscience<\/em>, 7(924), 1-9. https:\/\/doi.org\/10.3389\/fnhum.2013.00924<\/a>\u00a0<\/p>\n\n\n\n

Hatano, G., og Inagaki, K. (1984). Two courses of expertise. Research & Clinical Center for Child Development<\/em>, 82-83, 27\u201336. <\/p>\n\n\n\n

Hickendorff, M. (2018). Dutch sixth graders\u2019 use of shortcut strategies in solving multidigit arithmetic problems. European Journal of Psychology of Education<\/em>, 33(4), 577-594. https:\/\/doi.org\/10.1007\/s10212-017-0357-6<\/a>\u00a0\u00a0<\/p>\n\n\n\n

J\u00f3elsd\u00f3ttir, L. B. (2023). Essays on Adaptivity and Flexibility in Multidigit Arithmetic.<\/em> Aarhus University. Aarhus. https:\/\/pure.au.dk\/portal\/da\/publications\/essays-on-adaptivity-and-flexibility-in-multidigit-arithmetic(441b0429-68a7-45e1-ab6f-04dbddacefb6).html<\/a><\/p>\n\n\n\n

J\u00f3elsd\u00f3ttir, L. B. og Andrews, P. (2023). Danish third, sixth and eighth grade students\u2019 strategy adaptivity, strategy flexibility and accuracy when solving multidigit arithmetic tasks., Advance online publication. https:\/\/doi.org\/10.1007\/s10212-023-00786-2<\/a><\/p>\n\n\n\n

Ostad, S. (1997). Developmental differences in addition strategies: A comparison of mathematically disabled and mathematically normal children. British Journal of Educational Psychology and Aging<\/em>, 67(3), 345 357. https:\/\/doi.org\/10.1111\/j.2044-8279.1997.tb01249.x<\/a><\/p>\n\n\n\n

Rittle-Johnson, B., Star, J. R., og Durkin, K. (2012). Developing procedural flexibility: are novices prepared to learn from comparing procedures? British Journal of Educational Psychology<\/em>, 82, 436-455. https:\/\/doi.org\/10.1111\/j.2044-8279.2011.02037.x<\/a><\/p>\n\n\n\n

Selter, C. (2001). Addition and Subtraction of Three-Digit Numbers German Elementary Children’s success methods and strategies. Educational Studies in Mathematics<\/em>, 47(2), 145-173. https:\/\/doi.org\/10.1023\/A:1014521221809 <\/a><\/p>\n\n\n\n

Siegler, R. S., og Jenkins, E. (1989). How children discover new strategies.<\/em> Lawrence Erlbaum Associates, Inc. <\/p>\n\n\n\n

Sunde, P. B., De Smedt, B., Verschaffel, L., og Sunde, P. (2023). Grade one single-digit addition strategies as predictors of grade four achievement in mathematics. European Journal of Psychology of Education<\/em>, Advance online publication. https:\/\/doi.org\/10.1007\/s10212-023-00761-x<\/a><\/p>\n\n\n\n

Torbeyns, J., De Smedt, B., Ghesqui\u00e8re, P., og Verschaffel, L. (2009). Acquisition and use of shortcut strategies by traditionally schooled children. Educational Studies in Mathematics<\/em>, 71(1), 1-17. https:\/\/doi.org\/10.1007\/s10649-008-9155-z<\/a><\/p>\n\n\n\n

Torbeyns, J., Peters, G., De Smedt, B., Ghesqui\u00e8re, P., og Verschaffel, L. (2018). Subtraction by addition strategy use in children of varying mathematical achievement level: A choice\/no-choice study. Journal of Numerical Cognition<\/em>, 4(1), 215-234. https:\/\/doi.org\/10.5964\/jnc.v4i1.77<\/a><\/p>\n\n\n\n

Undervisningsministeriet. (2001). Klare M\u00e5l \u2013 Matematik – Fagh\u00e6fte 12<\/em>. Uddannelsesstyrelsens h\u00e5ndbogsserie del nr. 4 \u2013 2001.<\/p>\n\n\n\n

Vanbinst, K., Ghesqui\u00e8re, P., & De Smedt , B. (2014). Arithmetic strategy development and its domain-specific and domain-general cognitive correlates: A longitudinal study in children with persistent mathematical learning difficulties. Research in Developmental Disabilities<\/em>, 35, 3001\u20133013. https:\/\/doi.org\/10.1016\/j.ridd.2014.06.023<\/a><\/p>\n\n\n\n

Verschaffel, L., Greer, B., og De Corte, E. (2007). Whole number concepts and operations. In F. K. Lester Jr (Ed.), Second handbook of research on mathematics teaching and learning<\/em> (bls. 557-628). The National Council of Teachers of Mathematics. <\/p>\n\n\n\n

Verschaffel, L., Luwel, K., Torbeyns, J., og Van Dooren, W. (2009). Conceptualizing, investigating, and enhancing adaptive expertise in elementary mathematics education. European Journal of Psychology of Education<\/em>, 14(3), 335-359. https:\/\/doi.org\/10.1007\/BF03174765<\/a><\/p>\n\n\n\n

Xu, L., Liu, R. D., Star, J. R., Wang, J., Liu, Y., og Zhen, R. (2017). Measures of Potential Flexibility and Practical Flexibility in Equation Solving. Front Psychol,<\/em> 8, 1368. https:\/\/doi.org\/10.3389\/fpsyg.2017.01368<\/a><\/p>\n\n\n\n

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L\u00f3a Bj\u00f6rk J\u00f3elsd\u00f3ttir, lektor \u00ed st\u00e6r\u00f0fr\u00e6\u00f0imenntun, VIA University College, Aarhus Danm\u00f6rk<\/strong><\/p>\n<\/div><\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

L\u00f3a Bj\u00f6rk J\u00f3elsd\u00f3ttir. S\u00ed\u00f0an \u00e1 t\u00edunda \u00e1ratug s\u00ed\u00f0ustu aldar hefur veri\u00f0 v\u00ed\u00f0t\u00e6k s\u00e1tt me\u00f0al s\u00e9rfr\u00e6\u00f0inga \u00ed st\u00e6r\u00f0fr\u00e6\u00f0imenntun um a\u00f0 a\u00f0l\u00f6gunarh\u00e6fni (e. adaptivity) og sveigjanleiki (e. flexibility) s\u00e9u mikilv\u00e6g markmi\u00f0 \u00ed st\u00e6r\u00f0fr\u00e6\u00f0ikennslu (Baroody, 2003; Hickendorff o.fl., 2018; Xu o.fl., 2017). \u00cd l\u00f6ndum eins og Bandar\u00edkjunum, \u00c1stral\u00edu, Singap\u00far (Rittle-Johnson o.fl., 2012), Hollandi (Hickendorff, 2018), Belg\u00edu (Torbeyns o.fl.,…<\/p>\n","protected":false},"author":1,"featured_media":2488,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-2411","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\/2411","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=2411"}],"version-history":[{"count":60,"href":"https:\/\/flatarmal.is\/index.php?rest_route=\/wp\/v2\/posts\/2411\/revisions"}],"predecessor-version":[{"id":2499,"href":"https:\/\/flatarmal.is\/index.php?rest_route=\/wp\/v2\/posts\/2411\/revisions\/2499"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/flatarmal.is\/index.php?rest_route=\/wp\/v2\/media\/2488"}],"wp:attachment":[{"href":"https:\/\/flatarmal.is\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2411"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/flatarmal.is\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2411"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/flatarmal.is\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2411"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}