Using Figurate Numbers in Elementary Number Theory – Discussing a ‘Useful’ Heuristic From the Perspectives of Semiotics and Cognitive Psychology

Frontiers in Psychology

article

Leander Kempen¹  Rolf Biehler² 

[1] Department of Mathematics, University of Rostock;Department of Mathematics, University of Paderborn

关键词: figurate numbers;    mathematical proof;    proof that explain;    diagrammatic reasoning;    generic proof;   

DOI  :  10.3389/fpsyg.2020.01180

学科分类：社会科学、人文和艺术（综合）

来源: Frontiers

【 摘 要 】

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.

【 授权许可】

CC BY