Mathematical reasoning competency and digital tools

Rikke Maagaard Gregersen

doi:10.7146/aul.593

Mathematical reasoning competency and digital tools

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Publiceret 26 september 2025
copyright Ophavsret (c) 2025 Rikke Maagaard Gregersen (Forfatter)
Dette værk er under følgende licens Creative Commons Del på samme vilkår (ShareAlike) (by-sa).
DOI: https://doi.org/10.7146/aul.593
ISBN-13 (15) : 978-87-7507-584-3
Serier Ph.D-afhandlinger
Kategorier Faculty of Arts Danmarks institut for Pædagogik og Uddannelse (DPU)

Bidragsydere

Rikke Maagaard Gregersen, Aarhus Universitet, https://orcid.org/0000-0001-7641-8428

Nøgleord

Competency, digital tools, Mathematical reasoning, GeoGebra

Synopsis

I de seneste årtier er to fokusområder løbende blevet diskuteret inden for forskningen i matematikdidaktik. Det første fokusområde italesætter at mestre og lære matematik som besiddelse og udvikling af matematisk kompetence i stedet for viden og læring af matematiske fakta og færdigheder (Niss, 2016). Det andet område udforsker brugen af digitale teknologier i undervisning og læring af matematik (Artigue & Trouche, 2021). Den danske KOM-rapport skitserer otte matematiske kompetencer, der beskriver aktiviteter relateret til at udøve og håndtere matematik (Niss & Højgaard, 2019). Blandt de otte kompetencer er matematisk ræsonnementskompetence, som er omdrejningspunkt for dette studie. Ræsonnementskompetence indebærer at analysere eller producere argumenter for at begrunde matematiske påstande (Niss & Højgaard, 2019). Kompetencerne, sammen med brugen af digitale værktøjer, er en del af de danske matematiklæreplaner på både grundskole- og gymnasieniveau (Danmarks Evalueringsinstitut [EVA], 2009).
Moderne digitale matematikværktøjer, såsom dynamiske geometri programmer og computer algebra systemer (CAS), integrerer i stigende grad funktionalitet fra hinanden (Freiman, 2014; Sutherland & Rojano, 2014), hvilket giver nye muligheder men også øger programmernes kompleksitet. Dette studie, rapporteret i denne afhandling, har både praktiske og teoretiske formål. Praktisk undersøges, hvordan integrationen af geometri og algebra i digitale matematikværktøjer kan støtte elevers ræsonnementsprocesser i udskolingens matematikundervisning. Studiet lægger vægt på at gøre det muligt for eleverne at udøve deres matematiske ræsonnementskompetence snarere end som sådan at udvikle elevernes kompetence. Teoretisk søger studiet at fremme bæredygtig teoretisk udvikling ved at forbinde KOM med international matematikdidaktiskforskning. Dette opnås ved at anvende et netværksperspektiv (Prediger, Bikner-Ahsbahs et al., 2008) på teoriudvikling.
Projektets metodiske udgangspunkt er designbaseret forskning (Bakker, 2018; Cobb et al., 2003; Gravemeijer & Prediger, 2019; McKenney & Reeves, 2014). Det har guidet indsamling og analyse af data samt design af opgaver, der fordrer elevers udøvelse af matematiske ræsonnementskompetence i brugen af digitale teknologier med fokus på variable som et generaliseret tal. I den henseende er der udviklet og designet en ”microworld” (Hoyles, 1993) med variable punkter med tilhørende opgavesekvenser. Den er udviklet med henblik på at lade eleverne udøve matematiske ræsonnementskompetencer gennem deres undersøgelse af grundlæggende algebraiske udtryk og strukturelle implikationer i de variable punkters dynamiske egenskaber.
Udover KOM anvender projektet andre teoretiske perspektiver, såsom den instrumentelle tilgang til matematikundervisning (IAME) (Guin & Trouche, 1998) og dens opfattelse af kognitive skemaer (Vergnaud, 1998b) i relation til skema-teknik-dualiteten (Drijvers et al., 2013), samt Toulmins (2003) argumentationsmodel. Disse perspektiver bruges til at analysere, beskrive og forklare empiriske data, særligt i forhold til elevernes brug af digitale værktøjer og deres matematiske ræsonnementskompetence.
Afhandlingen består af seks forskningsartikler og denne tilhørende rapport, der beskriver den teoretiske baggrund, metodologi samt bidrager med yderligere analyser og resultater. Paper 1 er et litteraturstudie, der identificerer potentielle værktøjer i GeoGebra, et dynamisk geometri- og algebraprogram, til elevers ræsonnementer af variablen som et generaliseret tal. Paper 1 har informeret studiets efterfølgende designprocesser og produkter. Papers 2 til 5 analyserer empiriske data fra elever, der arbejder med forskellige opgaver, og udviklingen af et analytisk værktøj til instrumented justification. Artikel 2 introducerer den indledende teoretiske udvikling i at forbinde KOM og IAME, som genfortolkes gennem Toulmins model, hvilket fører til udviklingen af et analytisk værktøj. Artikel 3 forfiner dette værktøj, instrumented justification, og beskriver elevers ræsonnementsprocesser ved brug af digitale værktøjer. Artikel 4 fokuserer på elevers mål i brugen af digitale værktøjer i deres undersøgelser og løsning af en opgave. Opgaven videreudvikles på baggrund heraf, og opgavens potentiale og udfordringer for udøvelse af ræsonnementskompetence undersøges. Artikel 5 fokuserer på kognitive skemaer i det analytiske værktøj i analyser af Vergnauds (1998) skema-bestanddele og uddyber, hvordan elevers konceptuelle viden integreres i deres instrumenterede ræsonnementsprocesser. Artikel 6 behandler begrebet forgrunds- og baggrundsteori i et netværksperspektiv på teoriudvikling.
Afhandlingen bidrager med tre designprincipper for opgavedesign, der understøtter elevers udøvelse af ræsonnementskompetence, en ”microworld” til at udforske og begrunde variable punkters dynamiske bevægelse og tilhørende opgaver. Den uddyber også ræsonnementskompetence i elevers instrumenterede ræsonnementsprocesser og skema-teknik-dualiteten og giver forslag til at understøtte disse processer i klasseværelset gennem opgavedesigns. Derudover identificerer den en hybridopfattelse mellem kontinuerlige og diskrete forståelser af variable i forudsigelsen af variable punkters bevægelsesmønstre i dynamiske geometri- og algebraprogrammer og foreslår teoretiske forbindelser mellem KOM og IAME som potentialer for yderligere teoretisk udvikling.

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Gregersen, R. M. (2025). Mathematical reasoning competency and digital tools. Aarhus Universitet. https://doi.org/10.7146/aul.593

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