Vejen til differentiabilitet: Analyse af elevers begrebsdannelse i funktionsteori gennem skriftlige elevproduktioner p˚a gymnasialt mellemniveau
Oversat titel
The Way to Differentiability: Analysis of students' concept formation in function theory through written productions at secondary school level
Forfatter
Madsen, Kenneth
Semester
4. semester
Uddannelse
Udgivelsesår
2020
Afleveret
2020-06-02
Antal sider
86
Resumé
Matematikfaget er ofte beskrevet som presset af svage resultater og større spredning i elevers forudsætninger, mens elevers faglige forståelse står mindre i fokus. Dette speciale undersøger netop denne forståelse. Specialet kombinerer Anna Sfards teori om reifikation—overgangen fra at se matematiske handlinger som processer til at opfatte dem som genstande for tænkning—med Heinz Steinbrings epistemologiske trekant, som beskriver samspillet mellem matematiske objekter, symboler og mening i undervisningen. På den baggrund udvikles en udvidet reifikationsmodel, der begrebsliggør, hvordan elever udvikler begreber gennem et matematikforløb. Modellen illustreres med skriftlige elevarbejder fra et matematik C–B-hold i gymnasiet. Analysen viser, at brug af mange forskellige repræsentationer og bevidst vekslen mellem dem understøtter en mere strukturel og dyb forståelse. Samtidig peger empirien på en udfordring: Når elever først har reificeret en bestemt proces og repræsentationsform, holder de fast i den frem for at udforske alternative repræsentationer. I arbejdet med funktioner sker reifikationen primært via funktionens forskrift. Derfor opstår et helt funktionsbegreb ikke, hvilket er problematisk, fordi differentiabilitet ikke reificeres.
Mathematics in upper secondary education is often described as being under pressure from weak results and wider variation in students’ prior knowledge, while students’ conceptual understanding receives less attention. This thesis addresses that understanding. It combines Anna Sfard’s theory of reification—the transition from viewing mathematical activity as processes to treating ideas as objects of thought—with Heinz Steinbring’s epistemological triangle, which describes the interplay between mathematical objects, symbols, and meaning in classroom practice. On this basis, it proposes an extended reification model to conceptualize how students’ concepts develop over a teaching sequence. The model is illustrated with students’ written work from a Mathematics C–B class in upper secondary school. The analysis shows that using a broad range of representations and deliberately switching among them supports deeper, more structural understanding. At the same time, the data reveal a challenge: once students have reified a particular process and form of representation, they tend to stick with it instead of exploring alternatives. For functions, reification occurs mainly through the function’s formula. As a result, a full concept of function does not emerge, which is problematic because differentiability is not reified.
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