Besides the mathematical symbolic language, geometric representations can also be used: in figurate numbers, “numbers are classified according to their geometric representation as sets of dots” ( Weaver, 1974, p. There are different ways of communicating facts and ideas in mathematics. 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. Working in this representational system has to be learned and practiced as another kind of knowledge is necessary for working with figurate numbers. 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. The considerations taken from both theoretical perspectives will help to partly explain such phenomena. 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. 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”). 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.
2Department of Mathematics, University of Paderborn, Paderborn, Germany.1Department of Mathematics, University of Rostock, Rostock, Germany.
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