On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin View Full Text


Ontology type: schema:ScholarlyArticle     


Article Info

DATE

1991-02

AUTHORS

Anna Sfard

ABSTRACT

This paper presents a theoretical framework for investigating the role of algorithms in mathematical thinking. In the study, a combined ontological-psychological outlook is applied. An analysis of different mathematical definitions and representations brings us to the conclusion that abstract notions, such as number or function, can be conceived in two fundamentally different ways: structurally-as objects, and operationally-as processes. These two approaches, although ostensibly incompatible, are in fact complementary. It will be shown that the processes of learning and of problem-solving consist in an intricate interplay between operational and structural conceptions of the same notions. On the grounds of historical examples and in the light of cognitive schema theory we conjecture that the operational conception is, for most people, the first step in the acquisition of new mathematical notions. Thorough analysis of the stages in concept formation leads us to the conclusion that transition from computational operations to abstract objects is a long and inherently difficult process, accomplished in three steps: interiorization, condensation, and reification. In this paper, special attention is given to the complex phenomenon of reification, which seems inherently so difficult that at certain levels it may remain practically out of reach for certain students. More... »

PAGES

1-36

References to SciGraph publications

  • 1985. I Want to be a Mathematician, An Automathography in NONE
  • 1981-08. Concepts associated with the equality symbol in EDUCATIONAL STUDIES IN MATHEMATICS
  • Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1007/bf00302715

    DOI

    http://dx.doi.org/10.1007/bf00302715

    DIMENSIONS

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