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.
Our specialties include analytic number theory, the Langlands program, the geometry of locally symmetric spaces, arithmetic geometry and the study of algebraic cycles. For example, attempts to prove “Fermat’s Last Theorem” resulted in the development of large areas of algebra over the course of three centuries, and its recent proof involved a profound unifying force in modern mathematics known as the Langlands program. In addition, conjectures in number theory have had an impressive track record of stimulating major advances even outside the subject. Yet other problems currently studied in number theory call upon deep methods from harmonic analysis.
A beautiful illustration is given by the use of complex analysis to prove the “Prime Number Theorem,” which gives an asymptotic formula for the distribution of prime numbers.
Many problems in number theory, while simple to state, have proofs that involve apparently unrelated areas of mathematics. Topics studied by number theorists include the problem of determining the distribution of prime numbers within the integers and the structure and number of solutions of systems of polynomial equations with integer coefficients. Number theory is the study of the integers (e.g.
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