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Mathematical pathologies as pathways into creativity

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Abstract

In this paper, the role of mathematical pathologies as a means of fostering creativity in the classroom is discussed. In particular, it delves into what constitutes a mathematical pathology, examines historical mathematical pathologies as well as pathologies in contemporary classrooms, and indicates how the Lakatosian heuristic can be used to formulate problems that illustrate mathematical pathologies. We discuss the relationship between mathematical pathologies and their role in fostering creativity. The paper concludes with remarks on mathematical pathologies from the perspective of creativity studies at large.

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Notes

  1. http://mathworld.wolfram.com/Pathological.html.

  2. More generally, see http://mathworld.wolfram.com/Prime-GeneratingPolynomial.html on Prime-Generating Polynomials.

  3. This famous conjecture, and its generalization as Thurston’s Geometrization Conjecture, were finally proved in the positive by Grigori Perelman in a trio of papers posted to the arXiv in 2002 and 2003 (arXiv:math.DG/0211159; arXiv:math.DG/0303109; arXiv:math.DG/0307245).

  4. The following URL gives a calculator developed by one of the students to generate anomalous fractions http://www.chasemaier.com/project/Anomalous-Cancellation-Calculator.

  5. Note that for the product of a pair of two digit whole numbers, any three partial products uniquely determine the fourth partial product; thus, for example, any one of the four entries 800, 40, 160, or 8 could be removed from the interior of Fig. 5 without yielding additional solutions.

  6. For completeness, we note that it is possible to construct a 4-Venn diagram using ellipses. However, as we move into N-Venn diagrams for N > 5 the constructions are no longer viable.

  7. As mentioned earlier, one may ask for an alternative solution, or, at least, articulation of a solution. In this case, it is possible to observe that each of the N primes can be multiplied by 2, or not; this gives 2 N scenarios. N of these scenarios involve multiplication of just one prime by a 2, each of which will not yield a solution, and 1 of these scenarios will involve no 2 s, which will not yield a solution; thus, we find a total of 2 N – N − 1 possible solutions.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Montana, Missoula, USA

    Bharath Sriraman

  2. Department of Mathematics, The Hewitt School, New York City, USA

    Benjamin Dickman

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  1. Bharath Sriraman
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Correspondence to Bharath Sriraman.

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Sriraman, B., Dickman, B. Mathematical pathologies as pathways into creativity. ZDM Mathematics Education 49, 137–145 (2017). https://doi.org/10.1007/s11858-016-0822-8

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