Advertisement

Acknowledging the Ouroboros: An Enactivist and Metaphoric Approach to Problem Posing and Problem Solving

  • Jorge Soto-AndradeEmail author
  • Alexandra Yáñez-Aburto
Chapter
Part of the Research in Mathematics Education book series (RME)

Abstract

We are interested in exploring and developing an enactivist approach to problem posing and problem solving. We use here the term “enactivist approach” to refer to Varela’s radically nonrepresentationalist and pioneering “enactive approach to cognition” (Varela et al., The embodied mind: Cognitive science and human experience. Cambridge, MA: The MIT Press, 1991), to avoid confusion with the enactive mode of representation of Bruner, which is still compatible with a representationalist view of cognition. In this approach, problems are not standing “out there” waiting to be solved, by a solver equipped with a suitable toolbox of strategies. They are instead co-constructed through the interaction of a cognitive agent and a milieu, in a circular process well described by the metaphor of the Ouroboros (the snake eating its own tail). Also, cognition as enaction is metaphorized by Varela as “lying down a path in walking.” In this vein, we present here some paradigmatic examples of enactivist, and metaphorical, approaches to problem solving and problem posing, involving geometry, algebra, and probability, drawn from our didactical experimenting with a broad spectrum of learners, which includes humanities-inclined university students as well as prospective and in-service maths teachers. Our examples may be metaphorized as cognitive random walks in the classroom, stemming and unfolding from a situational seed.

Keywords

Enaction Metaphor Problem posing Problem solving 

Notes

Acknowledgments

Funding from PIA-CONICYT Basal Funds for Centres of Excellence Project FB0003, FIDOP 2016-60PAB, and DAAD Project 573 35022 D (Uni. Bielefeld - U. of Chile) is gratefully acknowledged.

References

  1. Alexander, R. R. (1982). Participant observation, ethnography, and their use in educational evaluation: A review of selected works. Studies in Art Education, 24(1), 63–69.CrossRefGoogle Scholar
  2. Artigue, M. (2009). Didactical design in mathematics education. In C. Winsløw (Ed.), Nordic research in mathematics education. Proceedings of NORMA 08. Rotterdam, Netherlands: Sense Publishers.Google Scholar
  3. Bachelard, G. (1938). La Formation de l’esprit scientifique. Paris, France: Librairie philosophique Vrin.Google Scholar
  4. Bauer, F. L. (1991). Sternpolygone und Hyperwürfel. In P. Hilton, F. Hirzebruch, & R. Remmert (Eds.), Miscellanea Mathematica (pp. 7–44). Berlin, Germany: Springer.CrossRefGoogle Scholar
  5. Brousseau, G. (1998). Théorie des situations didactiques. Grenoble, France: La pensée sauvage.Google Scholar
  6. Brousseau, G., Sarrazy, B., & Novotna, J. (2014). Didactic contract in mathematics education. In S. Lerman (Ed.), Encyclopedia of mathematics education (pp. 153–159). Berlin, Germany: Springer.Google Scholar
  7. Brown, L. (2015). Researching as an enactivist mathematics education researcher. ZDM Mathematics Education, 47, 185–196.CrossRefGoogle Scholar
  8. Bruner, J. (1966). Toward a theory of instruction. Harvard, MA: Harvard University Press.Google Scholar
  9. Bruner, J. S., & Kenny, H. J. (1965). Representation and mathematics learning. In Society for Research in Child Development, Cognitive development in children: Five monographs (pp. 485–494). Chicago, IL: University of Chicago Press.Google Scholar
  10. Csíkos, C., Rausch, A., & Szitányi, J. (Eds.). (2016). Proceedings of the 40th Conference of the International Group for the Psychology of Mathematics Education, Mathematics Education: How to solve it? (Vol. 1). Szeged, Hungary: IGPME.Google Scholar
  11. Dewey, J. (1997). How we think. Mineola, NY: Dover. (Original work published 1910).Google Scholar
  12. Díaz-Rojas, D., & Soto-Andrade, J. (2015). Enactive metaphoric approaches to randomness. In K. Krainer & N. Vondrová (Eds.), Proceedings of CERME9 (pp. 629–636). Prague, Czechia: Charles University & ERME.Google Scholar
  13. Diaz-Rojas, D., & Soto-Andrade, J. (2017). Enactive metaphors in mathematical problem solving. In T. Dooley & G. Gueudet (Eds.), Proceedings of CERME10 (pp. 3904–3911). Dublin, Ireland: DCU Institute of Education and ERME.Google Scholar
  14. Gallagher, S., & Lindgren, R. (2015). Enactive metaphors: Learning through full body engagement. Educational Psychology Review, 27, 391–404.CrossRefGoogle Scholar
  15. Gibbs, R. W. (Ed.). (2008). The Cambridge handbook of metaphor and thought. Cambridge, UK: Cambridge University Press.Google Scholar
  16. Glenberg, A. M. (2015). Few believe the world is flat: How embodiment is changing the scientific understanding of cognition. Canadian Journal of Experimental Psychology, 69, 165–171.CrossRefGoogle Scholar
  17. Goodchild, S. (2014). Enactivist theories. In S. Lerman (Ed.), Encyclopedia of mathematics education (pp. 209–214). Berlin, Germany: Springer-Verlag.Google Scholar
  18. Hosomizu, Y. (2008). Entrenando el pensamiento matemático. Edición Roja. Tsukuba, Japan: Tsukuba Incubation Lab.Google Scholar
  19. Isoda, M., & Katagiri, S. (2012). Mathematical thinking. Singapore: World Scientific.CrossRefGoogle Scholar
  20. Johnson, M., & Lakoff, G. (2003). Metaphors we live by. New York, NY: The University of Chicago Press.Google Scholar
  21. Johnston-Wilder, S. & Lee, C. (2010). Developing mathematical resilience. In: BERA Annual Conference 2010, 1–4 Sep 2010, University of Warwick.Google Scholar
  22. Krutetskii, V. A. (1976). The psychology of mathematical abilities in schoolchildren. Chicago, IL: University of Chicago Press.Google Scholar
  23. Lakoff, G., & Núñez, R. (2000). Where mathematics comes from. New York, NY: Basic Books.Google Scholar
  24. Libedinsky, N., & Soto-Andrade, J. (2015). On the role of corporeality, affect and metaphoring in problem solving. In P. Felmer, J. Kilpatrick, & E. Pehkonen (Eds.), Posing and solving mathematical problems: Advances and new perspectives (pp. 53–67). Berlin, Germany: Springer.Google Scholar
  25. Mac Lane, S. (1998). Categories for the working mathematician. Berlin, Germany: Springer.Google Scholar
  26. Machado, A. (1988). Selected poems. Cambridge, MA: Harvard University Press.Google Scholar
  27. Manin, Y. (2007). Mathematics as metaphor. Providence, RI: American Mathematical Society.Google Scholar
  28. Mason, J. (2002). Researching your own practice: The discipline of noticing. London, UK: Routledge.Google Scholar
  29. Mason, J. (2019). Pre-parative and post-parative play as key components of mathematical problem solving. This volume.Google Scholar
  30. Mason, J., Burton, L., & Stacey, K. (2003). Thinking mathematically. London, UK: Pearson.Google Scholar
  31. Maturana, H., & Varela, F. J. (1973). De Máquinas y Seres Vivos. Santiago, Chile: Editorial Universitaria.Google Scholar
  32. Maturana, H., & Varela, F. J. (1980). Autopoiesis and cognition: The realization of the living. Dordrecht, Netherlands: Reidel.CrossRefGoogle Scholar
  33. Maturana, H. R., & Guiloff, G. D. (1980). The quest for intelligence of intelligence. Journal of Social and Biological Structures, 3, 135–148.CrossRefGoogle Scholar
  34. NAEYC/NCMT. (2002). Learning paths and teaching strategies in early mathematics. USA: National Association for the Education of Young Children.Google Scholar
  35. National Research Council. (2009). Mathematics learning in early childhood: Paths toward excellence and equity. C. T. Cross, T. A. Woods, H. Schweingruber (Eds). Center for Education, Division of Behavioral and Social Sciences and Education. Washington DC: The National Academy Press.Google Scholar
  36. Proulx, J. (2013). Mental mathematics emergence of strategies, and the enactivist theory of cognition. Educational Studies in Mathematics, 84(3), 309–328.CrossRefGoogle Scholar
  37. Proulx, J., & Maheux, J.-F. (2017). From problem solving to problem posing, and from strategies to laying down a path in solving: Taking Varela’s ideas to Mathematics Education Research. Constructivist Foundations., 13(1), 160–167.Google Scholar
  38. Proulx, J., & Simmt, E. (2013). Enactivism in mathematics education: Moving toward a re-conceptualization of learning and knowledge. Education Sciences & Society, 4(1), 59–79.Google Scholar
  39. Reid, D. (1996). Enactivism as a methodology. In L. Puig & A. Gutierrez (Eds.), Proceedings of the twentieth annual conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 203–210). Valencia, Spain: PME.Google Scholar
  40. Reid, D. A., & Mgombelo, J. (2015). Survey of key concepts in enactivist theory and methodology. ZDM—The International Journal on Mathematics Education, 47(2), 171–183.CrossRefGoogle Scholar
  41. Sfard, A. (1997). Commentary: On metaphorical roots of conceptual growth. In L. English (Ed.), Mathematical reasoning: Analogies, metaphors, and images (pp. 339–371). London, UK: Erlbaum.Google Scholar
  42. Sfard, A. (2008). Thinking as communicating. Cambridge, UK: Cambridge University Press.CrossRefGoogle Scholar
  43. Sfard, A. (2009). Metaphors in education. In H. Daniels, H. Lauder, & J. Porter (Eds.), Educational theories, cultures and learning : a critical perspective (pp. 39–50). New York, NY: Routledge.Google Scholar
  44. Soto-Andrade, J. (2006). Un monde dans un grain de sable: Métaphores et analogies dans l’apprentissage des maths. Ann. Didactique Sciences Cogn., 11, 123–147.Google Scholar
  45. Soto-Andrade, J. (2007). Metaphors and cognitive styles in the teaching-learning of mathematics. In D. Pitta-Pantazi & J. Philippou (Eds.), Proceedings of CERME 5 (pp. 191–200). Larnaca, Cyprus: University of Cyprus. Retrieved from http://www.mathematik.uni-dortmund.de/~erme/CERME5b/Google Scholar
  46. Soto-Andrade, J. (2014). Metaphors in mathematics education. In S. Lerman (Ed.), Encyclopedia of mathematics education (pp. 447–453). Berlin, Germany: Springer-Verlag.Google Scholar
  47. Soto-Andrade, J. (2015). Une voie royale vers la pensée stochastique : les marches aléatoires comme pousses d’apprentissage. Statistique et Enseignement, 6(2), 3–24.Google Scholar
  48. Soto-Andrade, J. (2017). Enactivistic metaphoric approach to problem solving. In M. Stein (Ed.), A life’s time for mathematics education and problem solving. Festschrift for the occasion of András Ambrus 75th birthday (pp. 393–408). Münster, Germany: WTM (Verlag für wissenschaftliche Texte und Medien).Google Scholar
  49. Soto-Andrade, J. (2018). Enactive metaphorising in the learning of mathematics. In G. Kaiser, H. Forgasz, M. Graven, A. Kuzniak, E. Simmt, & B. Xu (Eds.), Invited lectures from the 13th international congress on mathematical education (pp. 619–638). Cham, Switzerland: Springer International Pub.CrossRefGoogle Scholar
  50. Soto-Andrade, J., Jaramillo, S., Gutiérrez, C., & Letelier, J. C. (2011). Ouroboros avatars: A mathematical exploration of self-reference and metabolic closure. In T. Lenaerts, M. Giacobini, H. Bersini, P. Bourgine, M. Dorigo, & R. Doursat (Eds.), Advances in artificial life ECAL 2011: Proceedings of the eleventh European conference on the synthesis and simulation of living systems (pp. 763–770). Cambridge, MA: The MIT Press.Google Scholar
  51. Soto-Andrade, J., & Varela, F. J. (1984). Self reference and fixed points. Acta Appl. Math., 2(1984), 1–19.CrossRefGoogle Scholar
  52. Soto-Andrade, J., & Varela, F. J. (1992). On mental rotations and cortical activity patterns. Biological Cybernetics, 64, 221–223.CrossRefGoogle Scholar
  53. Spradley, J. P. (1980). Participant observation. New York, NY: Holt, Rinehart & Winston.Google Scholar
  54. Stake, R. E. (1995). The art of case study research. Thousand Oaks, CA: Sage.Google Scholar
  55. Star, J. (2018). Flexibility in mathematical problem solving: The state of the field. In F.-J. Hsieh (Ed.), Proceedings of the 8th ICMI-East Asia Regional Conference on Mathematics Education (Vol. 2, pp. 15–25). EARCOME: Taipei, Taiwan.Google Scholar
  56. Towers, J., & Proulx, J. (2013). An enactivist perspective on teaching mathematics: Reconceptualising and expanding teaching actions. Mathematics Teacher Education and Development, 15(1), 5–28. http://cepa.info/4320Google Scholar
  57. Vallée-Tourangeau, F., Steffensen, S. V., Vallée-Tourangeau, G., & Sirota, M. (2016). Insight with hands and things. Acta Psychologica, 170, 195–205.  https://doi.org/10.1016/j.actpsy.2016.08.006CrossRefGoogle Scholar
  58. Varela, F. J. (1987). Lying down a path in walking. In W. I. Thompson (Ed.), Gaia: A way of knowing (pp. 48–64). Hudson, NY: Lindisfarne Press.Google Scholar
  59. Varela, F. J. (1992). Autopoiesis and a biology of intentionality. In B. McMullin (Ed.), Proceedings of the workshop Autopoiesis and Perception (pp. 4–14). Dublin, Ireland: Dublin City University.Google Scholar
  60. Varela, F. J. (1996). Invitation aux sciences cognitives. Paris, France: Éditions du Seuil.Google Scholar
  61. Varela, F. J. (1999). Ethical know-how: Action, wisdom, and cognition. Stanford, CA: Stanford University Press.Google Scholar
  62. Varela, F. J., Thompson, E., & Rosch, E. (1991). The embodied mind: Cognitive science and human experience. Cambridge, MA: The MIT Press.CrossRefGoogle Scholar
  63. von Glasersfeld, E. (1990). Environment and communication. In L. P. Stefe & T. Wood (Eds.), Transforming children’s mathematics education (pp. 30–38). Hillsdale, NJ: Erlsbaum. http://cepa.info/1290Google Scholar
  64. Watson, A. (2008). Adolescent learning and secondary mathematics. In P. Liljedahl, S. Oesterle, & C. Bernèche (Eds.), Proc. 2008 annual meeting of the CMESG (pp. 21–32). Burnaby, BC: CMESG.Google Scholar
  65. Wu, H. (2008). The critical foundations of algebra. Retrieved from https://math.berkeley.edu/~wu/Bilateral2008.pdf
  66. Yin, R. K. (2003). Case study research: Design and methods (3rd ed.). Thousand Oaks, CA: Sage.Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Mathematics DepartmentFaculty of Science and IEAE, University of ChileSantiagoChile
  2. 2.IEAE and PAB, University of ChileSantiagoChile

Personalised recommendations