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Vol.11, No.4, November 2022. ISSN: 2217-8309 eISSN: 2217-8333
TEM Journal
TECHNOLOGY, EDUCATION, MANAGEMENT, INFORMATICS Association for Information Communication Technology Education and Science |
Gridpoint Method for Proving Combinatorial Identities
Mária Kmetová
© 2022 Mária Kmetová, published by UIKTEN. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. (CC BY-NC-ND 4.0)
Citation Information: TEM Journal. Volume 11, Issue 4, Pages 1634-1639, ISSN 2217-8309, DOI: 10.18421/TEM114-26, November 2022.
Received: 08 August 2022. Revised: 19 September 2022.
Abstract:
Proofs are an important part of mathematical understanding. The three basic methods of proving combinatorial identities are mathematical induction, algebraic calculation, and combinatorial proofs. The last two of them are usually based on socalled double counting, which means counting the number of elements in one group with two different methods. In this article, we show an approach that uses gridpoints (points with integer coordinates) to calculate the number of elements of a set expressed by the left and the right side of a combinatorial identity. The gridpoint method for combinatorial calculation is known from [1] and [2]. This article presents the advantages of gridpoint approach in two aspects. The first one is simplifying the proofs in some cases, and the second one, to show students a way for independent work in invention (or reinvention) of combinatorial identities using the gridpoint method combining integer coordinates in an appropriate way. Finally, we discuss the acceptance of proofs by the gridpoint method as explanatory proofs.
Keywords – –mathematical education, combinatorial identities, explanatory proof, generalisation, gridpoint method. |
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