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Towards a theoretical synthesis of research in the early learning of algebra

Towards a theoretical synthesis of research in the early learning of algebra. James Aczel The Open University, UK j.c.aczel@open.ac.uk. Introduction. Can diverse research findings relating to students’ learning of algebra be synthesised? Maybe not… But here are some thoughts anyway ;-).

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Towards a theoretical synthesis of research in the early learning of algebra

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  1. Towards a theoretical synthesis of research in the early learning of algebra James Aczel The Open University, UK j.c.aczel@open.ac.uk

  2. Introduction Can diverse research findings relating to students’ learning of algebra be synthesised? Maybe not… But here are some thoughts anyway ;-)

  3. Popperian psychological perspective • Background – defence of democracy, critique of authoritarianism & positivism • An early philosophical advocate of intersubjectivity (World 3) • Against the “bucket theory of the mind” • role of linguistic & social conventions • centrality of creative conjectural processes • Against behaviourism

  4. Popperian psychological perspective (continued) • BVSR (Campbell, 1960) • psychological selection pressures • “concerns” - the fuel that powers learning • intertwining action, context & theory • “strategic theories” • problem-solving schema P1 TT P1 TT     EE EE   P2 P2 World 3 C1 ST C2 World 2

  5. Cognitive readiness and cognitive obstacles • A particular set of tasks • “Add 4 onto 3n”; “multiply n + 5 by 4” • “If cakes cost c pence each and buns b pence each, and 4 cakes are bought and 3 buns are bought, what does4c + 3b stand for?” • Categorising the treatment of letters • Küchemann - “letter evaluated”, “letter not used”, “letter as object”, “letter as specific unknown”, etc. • Why do these categories work? • unique hierarchy of difficulty – related to student maturation? • but presentation, language & experience matter (O’Reilly, 1990) • deep-rooted conceptions? • Popperian view: they are strategies devised by students for such tasks (tasks detached from the students’ mathematical concerns?)

  6. Interpretations, images, meanings and metaphors • Meta-algebraic theories • by-products • not static structures • not “underlying” • not “fundamental” • Value to researchers • Value to students • Should meta-algebraic theories be taught?

  7. The student-professor problem and reflection on meaning • Six times as many students as professors • 6S = P • Strategies described by Clement (1982) • «translate the situation syntactically» • «put the number next to the largest group» • «find an operation» • «find likely equations and test» • What concerns generate these strategies?

  8. The importance of concerns “Can we develop a school algebra culture in which pupils find a need for algebraic symbolism to explore their mathematical ideas?” (Sutherland, 1991)

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