Educational Studies in Mathematics

, Volume 101, Issue 1, pp 19–34 | Cite as

The Sierpinski smoothie: blending area and perimeter

  • Naneh ApkarianEmail author
  • Michal Tabach
  • Tommy Dreyfus
  • Chris Rasmussen


This study furthers the theory of conceptual blending as a useful tool for revealing the structure and process of student reasoning in relation to the Sierpinski triangle (ST). We use conceptual blending to investigate students’ reasoning, revealing how students engage with the ST and coordinate their understandings of its area and perimeter. Our analysis of ten individual interviews with mathematics education masters’ student documents diverse ways in which students reason about this situation through the constituent processes of blending: composition, completion, and elaboration. This reveals that students who share basic understandings of the area and perimeter of the ST recruit idiosyncratic ideas to engage with and resolve the paradox of a figure with infinite perimeter and zero area.


Conceptual blending Fractal Infinite processes Paradox Student thinking 


Funding information

This research was supported by the Israel Science Foundation grant no. 438/15.


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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Center for Research on Instructional Change in Postsecondary EducationWestern Michigan UniversityKalamazooUSA
  2. 2.School of EducationTel Aviv UniversityTel Aviv – YafoIsrael
  3. 3.Department of Mathematics and StatisticsSan Diego State UniversitySan DiegoUSA

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