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Broadening the Scope of Research on Mathematical Problem Solving pp 219–242Cite as

Solving a Task with Infinitely Many Solutions: Convergent and Divergent Thinking in Mathematical Creativity

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Part of the book series: Research in Mathematics Education ((RME))

Abstract

Problem-solving has the potential to elicit creative mathematical thinking, especially when requesting to solve the problem in different ways. The current study focuses on the mathematical creativity elicited by such a problem, where there were infinitely many possible outcomes. Participants were adults with a range of mathematical backgrounds and resources. Their solutions were evaluated to identify both divergent and convergent thinking, taking into consideration fluency and flexibility. Findings show that the task was a suitable vehicle for eliciting mathematical creativity. For some participants, divergent thinking was coupled with convergent thinking, resulting in the identification of sets of infinitely many solutions to the task. Methodological issues relating to the analysis of mathematical creativity are discussed.

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Notes

  1. 1.

    It was clear for participants in this study that “different” polygons referred to noncongruent polygons.

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Authors and Affiliations

  1. Tel-Aviv University, Tel Aviv, Israel

    Michal Tabach & Esther Levenson

Authors
  1. Michal Tabach
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  2. Esther Levenson
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Correspondence to Michal Tabach .

Editor information

Editors and Affiliations

  1. Universidade do Algarve and UIDEF Instituto de Educação, Universidade de Lisboa, Lisbon, Portugal

    Nélia Amado

  2. Universidade do Algarve and UIDEF Instituto de Educação, Universidade de Lisboa, Lisbon, Portugal

    Susana Carreira

  3. School of Education, University of Southampton, Southampton, UK

    Keith Jones

Appendix

Appendix

Claim: The cardinality of the set of all polygons in the plane is א. (א denotes the cardinality of the continuum.)

Lemma: For any given natural number n ≥ 3, the cardinality of the set of all n-gons in the plane is א.

The claim follows from the Lemma and the fact that א0× א= א.

Proof of the Lemma:

The set of all n-gons in the plane is at most as large as the set of all n-tuples of points in the plane because different n-gons, when placed in the plane, determine different n-tuples of points. But the set of n-tuples of points in the plane is equivalent to the set of vectors in 2n-dimensional space (every point has two coordinates), which is of cardinality (א)2n=א.

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Tabach, M., Levenson, E. (2018). Solving a Task with Infinitely Many Solutions: Convergent and Divergent Thinking in Mathematical Creativity. In: Amado, N., Carreira, S., Jones, K. (eds) Broadening the Scope of Research on Mathematical Problem Solving. Research in Mathematics Education. Springer, Cham. https://doi.org/10.1007/978-3-319-99861-9_10

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  • DOI: https://doi.org/10.1007/978-3-319-99861-9_10

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  • Online ISBN: 978-3-319-99861-9

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