Educational Studies in Mathematics

, Volume 93, Issue 2, pp 137–153 | Cite as

Categorizing and promoting reversibility of mathematical concepts

  • Martin A. Simon
  • Melike Kara
  • Nicora Placa
  • Hakan Sandir


Reversibility of concepts, a key aspect of mathematical development, is often problematic for learners. In this theoretical paper, we present a typology we have developed for categorizing the different reverse concepts that can be related to a particular initial concept and explicate the relationship among these different reverse concepts. We discuss uses of the typology and how reversibility can be understood as the result of reflective abstraction. Finally, we describe two strategies for promoting reversibility that have distinct uses in terms of the types of reverse concepts they engender. We share empirical results which led to our postulation of these strategies and discuss their theoretical basis.


Reversibility Mathematical concepts Reflective abstraction Task design 



This work was supported by the National Science Foundation under Grant No. DRL-1020154. Any opinions, findings, and conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the National Science Foundation.


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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • Martin A. Simon
    • 1
  • Melike Kara
    • 1
  • Nicora Placa
    • 1
  • Hakan Sandir
    • 2
  1. 1.New York UniversityNew YorkUSA
  2. 2.Gazi ÜniversitesiMerkezTurkey

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