Volume 16 · Number 3 · Pages 265–274
Enactive Metaphorizing in the Mathematical Experience

Daniela Díaz-Rojas, Jorge Soto-Andrade & Ronnie Videla-Reyes

Log in download the full text in PDF

> Citation > Similar > References > Add Comment

Abstract

Context: How can an enactive approach to the teaching and learning of mathematics be implemented, which fosters mathematical thinking, making intensive use of metaphorizing and taking into account the learner’s experience? Method: Using in-person and remote ethnographic participant observation, we observe students engaged in mathematical activities suggested by our theoretical approach. We focus on their idiosyncratic metaphorizing and affective reactions while tackling mathematical problems, which we interpret from our theoretical perspective. We use these observations to illustrate our theoretical approach. Results: Our didactic examples show that alternative pathways are possible to access mathematical thinking, which bifurcate from the metaphors prevailing in most of our classrooms, like teaching as “transmission of knowledge” and learning as “climbing a staircase.” Our participant observations suggest that enacting and metaphorizing may indeed afford a new and more meaningful kind of experience for mathematics learners. Implications: Our observations highlight the relevance of leaving the learners room to ask questions, co-construct their problems, explore, and so on, instead of just learning in a prescriptive way the method to solve each type of problem. Consequently, one kind of solution to the current grim situation regarding mathematics teaching and learning would be to aim at relaxing the prevailing didactic contract that thwarts natural sense-making mechanisms of our species. Our conclusions suggest a possible re-shaping of traditional teaching practice, although we refrain from trying to implement this in a prescriptive way. A limitation of our didactic experience might be that it exhibits just a couple of illustrative examples of the application of our theoretical perspective, which show that some non-traditional learning pathways are possible. A full fledged ethnomethodological and micro-phenomenological study would be commendable. Constructivist content: We adhere to the enactive approach to cognition initiated by Francisco Varela, and to the embodied perspective as developed by Shaun Gallagher. We emphasize the cognitive role of metaphorization as a key neural mechanism evolved in humans, deeply intertwined with enaction and most relevant in our “hallucinatory construction of reality,” in the sense of Anil Seth.

Key words: Metaphor, enaction, embodiment, mathematical experience, participant observation, emotional anesthesia, negative emotions, Francisco Varela, Shaun Gallagher

Citation

Díaz-Rojas D., Soto-Andrade J. & Videla-Reyes R. (2021) Enactive metaphorizing in the mathematical experience. Constructivist Foundations 16(3): 265–274. https://constructivist.info/16/3/265

Export article citation data: Plain Text · BibTex · EndNote · Reference Manager (RIS)

Similar articles

Vörös S. & Riegler A. (2017) A Plea for not Watering Down the Unseemly: Reconsidering Francisco Varela’s Contribution to Science
Vörös S., Froese T. & Riegler A. (2016) Epistemological Odyssey: Introduction to Special Issue on the Diversity of Enactivism and Neurophenomenology
Díaz-Rojas D., Soto-Andrade J. & Videla-Reyes R. (2021) Authors’ Response: Fathoming the Enactive Metaphorizing Elephant in the Dark…
Froese T., Gould C. & Barrett A. (2011) Re-Viewing from Within: A Commentary on First- and Second-Person Methods in the Science of Consciousness
Vörös S. & Bitbol M. (2017) Enacting Enaction: A Dialectic Between Knowing and Being

References

Abrahamson D., Gutiérrez J. & Baddorf A. (2012) Try to see it my way: The discursive function of idiosyncratic mathematical metaphor. Mathematical Thinking and Learning 14: 55–80. ▸︎ Google︎ Scholar
Araya R., Calfucura P., Jiménez A., Aguirre C., Palavicino M.-A., Lacourly N., Soto-Andrade J. & Dartnell P. (2010) The effect of analogies on learning to solve algebraic equations. Pedagogies: An International Journal 5(3): 216–232. ▸︎ Google︎ Scholar
Bergqvist E., Österholm M. C. & Sumpter L. (eds.) (2018) Proceedings of PME 42. PME, Umeå, Sweden. ▸︎ Google︎ Scholar
Brousseau G., Sarrazy B. & Novotná J. (2014) Didactic contract in mathematics education. In: Lerman S. (ed.) Encyclopedia of mathematics education. Springer, Berlin: 153–159. ▸︎ Google︎ Scholar
Chiu M. M. (2000) Metaphorical reasoning: Origins, uses, development and interactions in mathematics. Educational Journal 28(1): 13–46. ▸︎ Google︎ Scholar
Díaz-Rojas D. & Soto-Andrade J. (2015) Enactive metaphoric approaches to randomness. In: Krainer K. & Vondrová N. (eds.) Proceedings of the Ninth Congress of the European Society for Research in Mathematics Education (CERME 9) Charles University in Prague & ERME, Prague: 629–636. https://cepa.info/6844
English L. (ed.) (1997) Mathematical reasoning: Analogies, metaphors, and images. Lawrence Erlbaum, Mahwah NJ. ▸︎ Google︎ Scholar
Felmer P., Kilpatrick J. & Pekhonen E. (eds.) (2015) Posing and solving mathematical problems: Advances and new perspectives. Springer, New York NY. ▸︎ Google︎ Scholar
Font V., Bolite J. & Acevedo J. (2010) Metaphors in mathematics classrooms: Analyzing the dynamic process of teaching and learning of graph functions. Educational Studies in Mathematics 75: 131–152. ▸︎ Google︎ Scholar
Forgas J. P. (2008) Affect and cognition. Perspectives on Psychological Science 3(2): 94–101. ▸︎ Google︎ Scholar
Fuchs T. & Koch S. C. (2014) Embodied affectivity: On moving and being moved. Frontiers in Psychology 5: 508. ▸︎ Google︎ Scholar
Gallagher S. & Lindgren R. (2015) Enactive metaphors: Learning through full-body engagement. Educational Psychological Review 27: 391–404. https://cepa.info/6847
Gallagher S. & Zahavi D. (2012) The phenomenological mind. Routledge, London. https://cepa.info/4356
Gallagher S. (2005) How the body shapes the mind. Oxford University Press, Oxford. Reviewed in. https://constructivist.info/5/2/100
Gibbs R. (ed.) (2008) The Cambridge handbook of metaphor and thought. Cambridge University Press, Cambridge UK. ▸︎ Google︎ Scholar
Glasersfeld E. von (1990) Environment and communication. In: Steffe L. P. & Wood T. (eds.) Transforming children’s mathematics education. Erlbaum, Hillsdale NJ: 30–38. https://cepa.info/1290
Glasersfeld E. von (1991) Abstraction, re-presentation, and reflection: An interpretation of experience and Piaget’s approach. In: Steffe L. P. (ed.) Epistemological foundations of mathematical experience. Springer, New York NY. https://cepa.info/1418
Glasersfeld E. von (2005) Thirty years constructivism. Constructivist Foundations 1(1): 9–12. https://constructivist.info/1/1/009
Goldin G. A. (2014) Perspectives on emotion in mathematical engagement, learning, and problem solving. In: Pekrun R. & Linnenbrink-Garcia L. (eds.) International handbook of emotions in education. Routledge, New York NY: 391–414. ▸︎ Google︎ Scholar
Gould S. J. & Vrba E. (1982) Exaptation: A missing term in the science of form. Paleobiology 8(1): 4–15. ▸︎ Google︎ Scholar
Hannula M. S. (2015) Emotions in problem solving. In: Cho S. (eds.) Selected regular lectures from the 12th International Congress on Mathematical Education. Springer, Cham: 269–288. ▸︎ Google︎ Scholar
Knops A., Thirion B., Hubbard E., Michel V. & Dehaene S. (2009) Recruitment of an area involved in eye movements during mental arithmetic. Science 324(5934): 1583–1585. ▸︎ Google︎ Scholar
Lakoff G. & Nuñez R. (1997) The metaphorical nature of mathematics. In: English L. D. (ed.) Mathematical reasoning: Analogies, metaphors, and images. Lawrence Erlbaum, Mahwah NJ: 21–89. ▸︎ Google︎ Scholar
Lakoff G. & Núñez R. (2000) Where mathematics comes from. Basic Books, New York NY. ▸︎ Google︎ Scholar
Liljedahl P. (2015) Building thinking classrooms: Conditions for problem solving. In: Felmer P., Kilpatrick J. & Pekhonen E. (eds.) Posing and solving mathematical problems: Advances and new perspectives. Springer, New York NY: 361–386. ▸︎ Google︎ Scholar
Lindgren R. & Moshell J. M. (2011) Supporting children’s learning with body-based metaphors in a mixed reality environment. In: Proceedings of the Interaction Design and Children Conference. ACM, New York NY: 177–180. ▸︎ Google︎ Scholar
Machado A. (1988) Selected poems. Harvard University Press, Cambridge MA. ▸︎ Google︎ Scholar
Manin Y. (2007) Mathematics as metaphor. American Mathematical Society, Providence RI. ▸︎ Google︎ Scholar
Mason J. (2014) Questioning in mathematics education. In: Lerman S. (ed.) Encyclopedia of mathematics education. Springer-Verlag, Berlin: 513–519. ▸︎ Google︎ Scholar
McGowen M. C. & Tall D. O. (2010) Metaphor or Met-before? The effects of previous experience on the practice and theory of learning mathematics. Journal of Mathematical Behavior 29: 169–179. ▸︎ Google︎ Scholar
Merleau-Ponty M. (1962) Phenomenology of perception. Taylor & Francis, London. French original published in 1945. ▸︎ Google︎ Scholar
Merleau-Ponty M. (2013) Eye and mind: The primacy of perception. Edited by James Edie. Translated by Carleton Dallery. Northwestern University Press, Evanston. French original published in 1964. ▸︎ Google︎ Scholar
Nemirovsky R., Kelton M. & Rhodehamel B. (2013) Playing mathematical instruments: Emerging perceptuomotor integration with an interactive mathematics exhibit. Journal of Research in Mathematics Education 44(2): 372–415. ▸︎ Google︎ Scholar
Núñez R. (2008) A fresh look at the foundations of mathematics: Gesture and the psychological reality of conceptual metaphor. In: Cienki A. & Müller C (eds.) Gesture and metaphor. John Benjamins, Amsterdam: 93–114. ▸︎ Google︎ Scholar
O’Regan K. J. & Noë A. (2001) A sensorimotor account of vision and visual consciousness. Behavioral and Brain Sciences 24(05) 939–973. https://cepa.info/2285
Pekrun R. & Stephens E. J. (2010) Achievement emotions: A control-value approach. Social and Personality Psychology Compass 4(4): 238–255. ▸︎ Google︎ Scholar
Petitmengin C. (2006) Describing one’s subjective experience in the second person: An interview method for the science of consciousness. Phenomenology and the Cognitive Sciences 5: 229–269. https://cepa.info/2376
Piaget J. (1954) The construction of reality in the child. Ballantine, New York. Originally published in French as: Piaget J. (1937) La construction du réel chez l’enfant. Délachaux & Niestlé, Neuchâtel. ▸︎ Google︎ Scholar
Pólya G. (1945) How to solve it. Princeton University Press, Princeton NJ. ▸︎ Google︎ Scholar
Presmeg N. C. (1992) Prototypes, metaphors, metonymies and imaginative rationality in high school mathematics. Educational Studies in Mathematics 23(6): 595–610. ▸︎ Google︎ Scholar
Presmeg N. C. (1997) Reasoning with metaphors and metonymies in mathematics learning. In: English L. D. (ed.) Mathematical: Analogies, metaphors, and images. Lawrence Erlbaum Associates, Mahwah NJ: 267–279. ▸︎ Google︎ Scholar
Presmeg N. C. (1998) Metaphoric and metonymic signification in mathematics. Journal of Mathematical Behavior 17(1): 25–32. ▸︎ Google︎ Scholar
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): 161–167. https://constructivist.info/13/1/160
Proulx J. & Simmt E. (2015) Enactivism and mathematics education: Sources, meanings, and research. Information Age Publishing, Charlotte NC. ▸︎ Google︎ Scholar
Ricoeur P. (1997) La métaphore vive. Le Seuil, Paris. First edition published in 1975. English translation: Ricoeur P. (1978) The rule of metaphor: Multi-disciplinary studies of the creation of meaning in language. Translated by Robert Czerny with Kathleen McLaughlin and John Costello. Routledge & Kegan Paul, London. ▸︎ Google︎ Scholar
Seth A. (2019) Too many ghosts in the machine. Nature Machine Intelligence 1: 294–295. ▸︎ Google︎ Scholar
Sfard A. (1994) Reification as the birth of metaphor. For the Learning of Mathematics 141: 44–54. ▸︎ Google︎ Scholar
Sfard A. (1997) Commentary: On metaphorical roots of conceptual growth. In: English L. D. (ed.) Mathematical reasoning: Analogies, metaphors, and images. Lawrence Erlbaum Associates, London: 339–371. ▸︎ Google︎ Scholar
Sfard A. (2009) Metaphors in education. In: Daniels H., Lauder H. & Porter J. (eds.) Educational theories, cultures and learning: A critical perspective. Routledge, London: 39–50. ▸︎ Google︎ Scholar
Soto-Andrade J. (2014) Metaphors in mathematics education. In: Lerman S. (ed.) Encyclopedia of mathematics education. Springer-Verlag, Berlin: 447–453. ▸︎ Google︎ Scholar
Soto-Andrade J. (2018) Enactive metaphorising in the learning of mathematics. In: G. Kaiser G., Forgasz H., Graven M., Kuzniak A., Simmt E. & Xu B.(eds.) Invited Lectures from the 13th International Congress on Mathematical Education. Springer, Cham: 619–638. https://doi.org/10.1007/978–3-319–72170–5_34
Soto-Andrade J., Díaz-Rojas D. & Reyes-Santander P. (2018) Random walks in the didactics of probability: Enactive metaphoric learning sprouts. In: Batanero C. & Chernoff E. (eds.) Teaching and learning stochastics: Advances in probability education research. Springer, Cham: 125–143. https://cepa.info/6889
Steffe L. P. & Thompson P. W. (eds.) (2000) Radical constructivism in action: Building on the pioneering work of Ernst von Glasersfeld. Routledge, London. ▸︎ Google︎ Scholar
Stewart J. R., Gapenne O. & Di Paolo E. A. (eds.) (2010) Enaction: Toward a new paradigm for cognitive science. MIT Press, Cambridge MA. ▸︎ Google︎ Scholar
Thurston W. (1994) On proof and progress in mathematics. Bulletin of the American Mathematical Society 30(2): 161–177. ▸︎ Google︎ Scholar
Varela F. J. (1984) Living ways of sense-making: A middle path for neuroscience. In: Livingstone P. (ed.) Order and disorder: Proceedings of the Stanford International Symposium. Anma Libri, Stanford CA: 208–224. https://cepa.info/2068
Varela F. J. (1984) The creative circle. Sketches on the natural history of circularity. In: Watzlawick P. (ed.) The invented reality: Contributions to constructivism. W. W. Norton, New York NY: 309–325. https://cepa.info/2089
Varela F. J. (1987) Laying down a path in walking: A biologist’s look at a new biology. Cybernetic 2: 6–15. https://cepa.info/2069
Varela F. J. (1990) On the conceptual skeleton of current cognitive science. In: Luhman N., Maturana H. R., Namihi M., Volker R. & Varela F. J. (eds.) Beobachter – Konvergenz der Erkenntnistheorien? Fink, München: 13–23. https://cepa.info/2071
Varela F. J. (1992) Autopoiesis and a biology of intentionality. In: McMullin B. (ed.) Proceedings of the workshop Autopoiesis and Perception. Dublin City University, Dublin: 4–14. https://cepa.info/1274
Varela F. J. (1999) Ethical know-how: Action, wisdom, and cognition. Stanford University Press, Stanford CA. https://cepa.info/2119
Varela F. J. (1999) Steps to a science of interbeing: Unfolding the dharma implicit in modern cognitive science. In: Watson G., Bachelor S. & Claxton G. (eds.) The psychology of awakening. Rider, New York: 71–89. https://cepa.info/4118
Varela F. J., Thompson E. & Rosch E. (1991) The embodied mind: Cognitive science and human experience. MIT Press, Cambridge MA. ▸︎ Google︎ Scholar

Comments: 0

To stay informed about comments to this publication and post comments yourself, please log in first.