Using Figurate Numbers in Elementary Number Theory - Discussing a 'Useful' Heuristic From the Perspectives of Semiotics and Cognitive Psychology.

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  • Author(s): Kempen L;Kempen L; Biehler R; Biehler R
  • Source:
    Frontiers in psychology [Front Psychol] 2020 Jun 17; Vol. 11, pp. 1180. Date of Electronic Publication: 2020 Jun 17 (Print Publication: 2020).
  • Publication Type:
    Journal Article
  • Language:
    English
  • Additional Information
    • Source:
      Publisher: Frontiers Research Foundation Country of Publication: Switzerland NLM ID: 101550902 Publication Model: eCollection Cited Medium: Print ISSN: 1664-1078 (Print) Linking ISSN: 16641078 NLM ISO Abbreviation: Front Psychol Subsets: PubMed not MEDLINE
    • Publication Information:
      Original Publication: Pully, Switzerland : Frontiers Research Foundation
    • Abstract:
      The use of figurate numbers (e.g., in the context of elementary number theory) can be considered a heuristic in the field of problem solving or proving. In this paper, we want to discuss this heuristic from the perspectives of the semiotic theory of Peirce ("diagrammatic reasoning" and "collateral knowledge") and cognitive psychology ("schema theory" and "Gestalt psychology"). We will make use of several results taken from our research to illustrate first-year students' problems when dealing with figurate numbers in the context of proving. The considerations taken from both theoretical perspectives will help to partly explain such phenomena. It will be shown that the use of figurate numbers must not be considered to be any kind of help for learners or some way of 'easy' mathematics. Working in this representational system has to be learned and practiced as another kind of knowledge is necessary for working with figurate numbers. The named findings also touch upon the concept of 'proofs that explain.' Finally, we will highlight some implications for teaching and point to a number of demands for future research.
      (Copyright © 2020 Kempen and Biehler.)
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    • Contributed Indexing:
      Keywords: diagrammatic reasoning; figurate numbers; generic proof; mathematical proof; proof that explain
    • Publication Date:
      Date Created: 20200707 Latest Revision: 20231103
    • Publication Date:
      20240829
    • Accession Number:
      PMC7313546
    • Accession Number:
      10.3389/fpsyg.2020.01180
    • Accession Number:
      32625134