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The Sierpinski smoothie: blending area and perimeter

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

Abstract

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.

Keywords

Conceptual blending Fractal Infinite processes Paradox Student thinking 

Notes

Funding information

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

References

  1. Alexander, J. C. (2011). Blending in mathematics. Semiotica, 2011(187), 1–48.CrossRefGoogle Scholar
  2. Apkarian, N., Rasmussen, C., Tabach, M., & Dreyfus, T. (2018). Conceptual blending: The case of the Sierpinski triangle area and perimeter. In A. Weinberg, C. Rasmussen, J. Rabin, M. Wawro, & S. Brown (Eds.), Proceedings of the 21st Annual Conference on Research in Undergraduate Mathematics Education (pp. 541–548). San Diego, CA.Google Scholar
  3. Dubinsky, E., Weller, K., McDonald, M. A., & Brown, A. (2005). Some historical issues and paradoxes regarding the concept of infinity: An APOS-based analysis: Parts 1&2. Educational Studies in Mathematics, 58, 335–359 60, 253–266.CrossRefGoogle Scholar
  4. Edwards, L. D. (2009). Gestures and conceptual integration in mathematical talk. Educational Studies in Mathematics, 70, 127–141.CrossRefGoogle Scholar
  5. Ely, R. (2011). Envisioning the infinite by projecting finite properties. Journal of Mathematical Behavior, 30, 1–18.CrossRefGoogle Scholar
  6. Fauconnier, G., & Turner, M. (2002). The way we think. New York: Basic Books.Google Scholar
  7. Gerson, H. & Walter, J. (2008). How blending illuminates understandings of calculus. In Electronic Proceedings for the Eleventh Special Interest Group of the Mathematical Association of America on Research in Undergraduate Mathematics. Retrieved from http://rume.org/crume2008/Proceedings/Gerson%20LONG.pdf. Accessed 13 June 2017.
  8. Jordan, B., & Henderson, A. (1995). Interaction analysis: Foundations and practice. The Journal of the Learning Sciences, 4(1), 39–103.CrossRefGoogle Scholar
  9. Lakoff, G., & Núñez, R. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. New York: Basic Books.Google Scholar
  10. Larsen, S., Marrongelle, K., Bressoud, D., & Graham, K. (2017). Understanding the concepts of calculus: Frameworks and roadmaps emerging from educational research. In J. Cai (Ed.), Compendium for Research in Mathematics Education (pp. 526–550). National Council of Teachers of Mathematics.Google Scholar
  11. Mamolo, A., & Zazkis, R. (2008). Paradoxes as a window to infinity. Research in Mathematics Education, 10(2), 167–182.CrossRefGoogle Scholar
  12. Núñez, R. (2005). Creating mathematical infinities: Metaphor, blending, and the beauty of transfinite cardinals. Journal of Pragmatics, 37(10), 1717–1741.CrossRefGoogle Scholar
  13. Radu, I., & Weber, K. (2011). Refinements in mathematics undergraduate students’ reasoning on completed infinite iterative processes. Educational Studies in Mathematics, 78, 165–180.CrossRefGoogle Scholar
  14. Rasmussen, C., Apkarian, N., Tabach, M., & Dreyfus, T. (in review). Ways in which engaging in someone else’s reasoning is productive.Google Scholar
  15. Rasmussen, C., & Wawro, M. (2017). Post-calculus research in undergraduate mathematics education. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 551–581). Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  16. Sacristán, A. I. (2001). Students’ shifting conceptions of the infinite through computer explorations of fractals and other visual models. In M. van den Heuvel-Panhuizen (Ed.), Proceedings of the 25th International Conference for the Psychology of Mathematics Education (Vol. 4, pp. 129–136). Utrecht, The Netherland: PME.Google Scholar
  17. Strauss, A., & Corbin, J. (1998). Basics of qualitative research: Techniques and procedures for developing grounded theory (2nd ed.). Thousand Oaks, CA: Sage.Google Scholar
  18. Tabach, M., Apkarian, N., Dreyfus, T., & Rasmussen, C. (2017). Can a region have no area but infinite perimeter? In B. Kaur, W. K. Ho, T. L. Toh, & B. H. Choy (Eds.), Proceedings of the 41st Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 241–248). Singapore: PME.Google Scholar
  19. Wijeratne, C., & Zazkis, R. (2015). On painter’s paradox: Contextual and mathematical approaches to infinity. International Journal of Research in Undergraduate Mathematics Education, 1, 163–186.CrossRefGoogle Scholar
  20. Yoon, C., Thomas, M. O. J., & Dreyfus, T. (2011). Grounded blends and mathematical gesture spaces: Developing mathematical understandings via gestures. Educational Studies in Mathematics, 78, 371–393.CrossRefGoogle Scholar
  21. Zandieh, M., Roh, K. H., & Knapp, J. (2014). Conceptual blending: Student reasoning when proving “conditional implies conditional” statements. Journal of Mathematical Behavior, 33, 209–229.CrossRefGoogle Scholar

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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