Volume 18 · Number 2 · Pages 259–276
Random Walks as a Royal Road to E-STEAM in Math Education

Amaranta Valdés-Zorrilla, Daniela Díaz-Rojas, Leslie Jiménez & Jorge Soto-Andrade

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Abstract

Context: Millions of learners worldwide experience mathematics nowadays as an inescapable tool of cognitive abuse and punitive selection. Most traditional teaching thwarts natural human cognitive resources. Problem: We would like to contribute to alleviating the aforementioned cognitive abuse, sharing the insights afforded by our exploration of enactive and metaphorical approaches to learning and teaching, inspired by E (embodied, enactive, extended, embedded, ecological)-cognition. We aim at understanding mathematical thinking processes and practicing an experimental epistemology of mathematics, not just prescribing actions to be undertaken in the classroom Method: Our theoretical scope is E-cognition. Our main research method is based on enactivism (enaction à la Varela) including metaphorical analysis, participant observation, and semi-structured interviews. Moreover, we discuss illustrative examples of learning activities related to random walks and STEAM (Science, Technology, Engineering, Art & Mathematics), particularly physics and art (dance and choreography. Results: Embodiment, enacting and metaphorising make a dramatic difference in mathematics learning processes. Learning activities related to random walks and deterministic dynamical systems enacted through dance and choreography can play a significant antidotal and remedial role against cognitive abuse in the teaching of mathematics. Beneficial insights are triggered, for students, teachers, and mathematics educators. Implications: We suggest new horizons for research and practice in mathematics education informed by E-cognition and metaphorisation, with an antidotal and therapeutic effect against cognitive abuse in teaching. Further research is commendable on the often-stressful transition process from an abusive and repressive education to a more open enactivist education, which could use micro-phenomenological interviews among other techniques. It could involve scaling up our experimentation, particularly with prospective and in-service teachers. Limitations are related to the small number of students and teachers hitherto involved. Constructivist content: Our research aims at developing a radically enactivist mathematics education inspired by Varela’s enaction.

Erratum: In the last sentence of §111 "so fostering their metacognitive abilities" should have been deleted.

Key words: Bayes, choreography, cognitive bullying, cognitive abuse, dance, dynamical systems, embodiment, enaction, metaphor, random walks, Francisco Varela

Citation

Valdés-Zorrilla A., Díaz-Rojas D., Jiménez L. & Soto-Andrade J. (2023) Random walks as a royal road to e-steam in math education. Constructivist Foundations 18(2): 259–276. https://constructivist.info/18/2/259

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References

Abrahamson D., Dutton E. & Bakker A. (2021) Towards an enactivist mathematics pedagogy. In: Stolz S. A. (ed.) The body, embodiment, and education: An interdisciplinary approach. Routledge, New York: 156–182. https://cepa.info/7085
Abrahamson D., Gutiérrez J. F. & Baddorf A. K. (2012) Try to see it my way: The discursive function of idiosyncratic mathematical metaphor. Mathematical Thinking and Learning 14(1): 55–80. ▸︎ Google︎ Scholar
Ashcraft M. H. (2002) Math anxiety: Personal, educational, and cognitive consequences. Current Directions in Psychological Science 11(5): 181–185. ▸︎ Google︎ Scholar
Bachelier L. (1900) Théorie de la spéculation [Theory of speculation]. Annales de l’École Normale Supérieure 17: 21–86. ▸︎ Google︎ Scholar
Becker J. P. & Shimada S. (1997) The open-ended approach: A new proposal for teaching mathematics. National Council of Teachers of Mathematics, Reston, VA. ▸︎ Google︎ Scholar
Beiser F. C. (2022) Johann Friedrich Herbart: Grandfather of analytic philosophy. Oxford University Press, Oxford. ▸︎ Google︎ Scholar
Borges J. L. (1997) The garden of forking paths. In: González Echavarría R. (ed.) The Oxford book of Latin American short stories. Oxford University Press, Oxford: 211–220. Spanish original published in 1941. ▸︎ Google︎ Scholar
Brousseau G. (1965) Les Mathématiques du cours préparatoire [The mathematics of the preparatory course]. Dunod, Paris. ▸︎ Google︎ Scholar
Brousseau G. (2002) Theory of didactical situations in mathematics. Kluwer, Dordrecht. ▸︎ Google︎ Scholar
Bruner J. (1966) Toward a theory of instruction. Harvard University Press, Cambridge MA. ▸︎ Google︎ Scholar
Bryman A. (2016) Social research methods. Fourth edition. Oxford University Press, Oxford. ▸︎ Google︎ Scholar
Chiu M. (2000) Metaphorical reasoning: Origins, uses, development and interaction in mathematics. Educational Journal 28(1): 13–46. ▸︎ Google︎ Scholar
Davis B., Sumara D. & Luce-Kapler R. (2015) Engaging minds: Cultures of education and practices of teaching. Taylor & Francis, New York NY. ▸︎ Google︎ Scholar
Dewey J. (1997) How we think. Dover, Mineola NY. Originally published in 1910. ▸︎ Google︎ Scholar
Dorier J. L. (2017) La didactique des mathématiques: Une épistémologie expérimentale? [Didactics of mathematics: an experimental epistemology?] In: Bächtold M., Durand-Guerrier V. & Munier V. (ed.) Epistémologie et didactique. Presses Universitaires de Franche-Comté, Besançon: 33–44. ▸︎ Google︎ Scholar
English A. R. (2013) Discontinuity in learning: Dewey, Herbart and education as transformation. Cambridge University Press, Cambridge UK. ▸︎ Google︎ Scholar
English L. (ed.) (1997) Mathematical reasoning: Analogies, metaphors, and images. Lawrence Erlbaum, Mahwah NJ. ▸︎ Google︎ Scholar
Freire P. & Faúndez A. (1985) Por uma pedagogia da pergunta [Towards a question-based pedagogy]. Paz e Terra, Rio de Janeiro. ▸︎ Google︎ Scholar
Fuchs T. (2017) Ecology of the brain. Oxford University Press, Oxford. ▸︎ Google︎ Scholar
Gallagher S. & Lindgren R. (2015) Enactive metaphors: Learning through full body engagement. Educational Psychology Review 27: 391–404. https://cepa.info/684
Gerofsky S. (2017) Mathematics and movement. In: Jao L. & Radakovic N. (eds.) Transdisciplinarity in mathematics education. Springer, Berlin: 239–254. ▸︎ Google︎ Scholar
Gigerenzer G. (2011) What are natural frequencies? Doctors need to find better ways to communicate risk to patients. BMJ 343: D6386. ▸︎ Google︎ Scholar
Hall E. T. (1959) The silent language. Fawcett, Greenwich CT. ▸︎ Google︎ Scholar
Harlow D., Shenker S. H., Stanford D. & Susskind L. (2012) Tree-like structure of eternal inflation: A solvable model. Physical Review D 85: 063516. ▸︎ Google︎ Scholar
Herbart J. F. (1804) Pestalozzi’s Idee eines ABC der Anschauung als ein Cyklus von Vorübungen im Auffassen der Gestalten [Pestalozzi’s idea of an ABC of intuition as preliminary exercises towards an apprehension of forms]. Johann Friedrich Röwer, Göttingen. https://doi.org/10.11588/diglit.18805
Hofe R. vom & Reyes-Santander P. (2021) Nociones básicas: Un enfoque didáctico para promover la comprensión del contenido en clase de matemáticas [Basic notions: A didactic approach to foster the understanding of content in mathematics lessons]. In: vom Hofe R., Puraivan Huenumán E., Ramos-Rodríguez E., Reyes-Santander P., Soto-Andrade J. & Vargas Díaz C. L. (eds.) Matemática enactiva. Graó, Barcelona: 27–60. ▸︎ Google︎ Scholar
Hofe R. vom (1995) Grundvorstellungen mathematischer Inhalte [Basic notions for mathematical contents]. Spektrum, Heidelberg. ▸︎ Google︎ Scholar
Hutto D. & Abrahamson D. (2022) Embodied, enactive education: Conservative versus radical approaches. In: Macrine, S L. & Fugate J. M. B. (eds.) Movement matters: How embodied cognition informs teaching and learning. MIT Press, Cambridge MA: 39–52. https://cepa.info/7989
Ingram J. & Elliot V. (2020) Research methods for classroom discourse. Bloomsbury, London. ▸︎ Google︎ Scholar
Isoda M & Katagiri S. (2012) Mathematical thinking. World Scientific, Singapore. ▸︎ Google︎ Scholar
Kidron I., Lenfant A., Bikner-Ahsbahs A., Artigue M. & Dreyfus T. (2008) Toward networking three theoretical approaches. ZDM 40(2): 247–264. ▸︎ Google︎ Scholar
Kuzniak A., Nechache A. & Drouhard J.-P. (2016) Understanding the development of mathematical work in the context of the classroom. ZDM Mathematics Education 48(6): 861–874. ▸︎ Google︎ Scholar
Lakoff G. & Johnson M. (2003) Metaphors we live by. The University of Chicago Press, Chicago. Originally published in 1980. ▸︎ Google︎ Scholar
Lakoff G. & Núñez R. E. (2000) Where mathematics comes from: How the embodied mind brings mathematics into being. Basic Books, New York, NY. ▸︎ Google︎ Scholar
Makarenko A. S. (1951) The road to life: An epic of education. Progress Publishers, Moscow. ▸︎ Google︎ Scholar
Manin Y. I. (2007) Mathematics as metaphor: Selected essays. American Mathematical Society, Providence RI. ▸︎ Google︎ Scholar
Mason J. (2002) Researching your own practice: The discipline of noticing. Routledge, London. ▸︎ Google︎ Scholar
Mason J. (2020) Questioning in mathematics education. In: Lerman S. (ed.) Encyclopedia of mathematics education. Second edition. Springer, Cham: 705–711. ▸︎ Google︎ Scholar
Milner S. J., Duque A. C. & Gerofsky S. (2019) Dancing Euclidean proofs. In: Goldstine S., McKenna D. & Fenyvesi K. (eds.) Proceedings of Bridges 2019: Mathematics, art, music, architecture, education, culture. Tessellations Publishing, Phoenix AZ: 239–246. https://archive.bridgesmathart.org/2019/bridges2019-239.pdf
Nathan M. J. (2022) Foundations of embodied learning: A paradigm for education. First edition. Routledge, New York NY. ▸︎ Google︎ Scholar
Newen A., De Bruin L. & Gallagher S. (eds.) (2018) The Oxford handbook of 4E cognition. Oxford University Press, Oxford. ▸︎ Google︎ Scholar
Pearson K. (1905) The problem of the random walk. Nature 72(294): 318–342. ▸︎ Google︎ Scholar
Petitmengin C., Remillieux A. & Valenzuela-Moguillansky C. (2019) Discovering the structures of lived experience: Towards a micro-phenomenological analysis method. Phenomenology and the Cognitive Sciences 18: 691–730. https://cepa.info/6664
Powles J. G. (1978) Brownian motion – June 1827 (for teachers). Physics Education 13(5): 310–312. ▸︎ 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, Charlotte NC. ▸︎ Google︎ Scholar
Punch K. F. & Oncea A. (2014) Introduction to research methods in education. Second edition. SAGE, Newcastle upon Tyne. ▸︎ Google︎ Scholar
Raichlen D. A., Wood B. M., Gordon A. D., Mabulla A. Z., Marlowe F. W. & Pontzer H. (2014) Evidence of Lévy walk foraging patterns in human hunter-gatherers. PNAS 111(2): 728–733. https://doi.org/10.1073/pnas.1318616111
Reddy M. J. (1979) The conduit metaphor: A case of frame conflict in our language about language. In: Ortony A. (ed.) Metaphor and thought. Cambridge University Press, Cambridge UK: 284–310. ▸︎ Google︎ Scholar
Reid D. A. & Mgombelo J. (2015) Key concepts in enactivist theory and methodology. ZDM 47(2): 171–183. https://cepa.info/2520
Reid D. A. (1996) Enactivism as a methodology. In: Puig L. & Gutiérrez A. (eds.) Proceedings of the Twentieth Annual Conference of the International Group for the Psychology of Mathematics Education (PME-20), Volume 4. PME, Valencia: 203–210. https://cepa.info/2519
Samuel P. (1979) Mathématiques, mathématiciens et société [Mathematics, mathematicians and society]. Publications mathématiques d’Orsay 86–74.16. http://sites.mathdoc.fr/PMO/PDF/S_SAMUEL-141.pdf
Seth A. (2021) Being you: A new science of consciousness. Faber, London. ▸︎ Google︎ Scholar
Sfard A. (1994) Reification as the birth of metaphor. For the Learning of Mathematics 14(1): 44–54. ▸︎ 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, Milton Park: 39–50. ▸︎ Google︎ Scholar
Soto-Andrade J. & Shulman A. (2021) A random walk in stochastic dance. LINK 2021 Conference Proceedings 2(1) https://doi.org/10.24135/link2021.v2i1.71
Soto-Andrade J. (2013) Metaphoric random walks: A royal road to stochastic thinking. In: Ubuz B., Haser C. & Mariotti M. A. (eds.) Proceedings of Eighth Congress of the European Society for Research in Mathematics Education (ERME 8) ERME: 890–900. ▸︎ Google︎ Scholar
Soto-Andrade J. (2015) Une voie royale vers la pensée stochastique: Les marches aléatoires comme pousses d’apprentissage [A royal road towards stochastic thinking: Random walks as learning sprouts]. Statistique et Enseignement 6(2): 3–24. ▸︎ Google︎ Scholar
Soto-Andrade J. (2017) Enactivistic metaphoric approach to problem solving. In: Stein M. (ed.) A life’s time for mathematics education and problem solving: Festschrift for the occasion of András Ambrus 75th birthday. Verlag für wissenschaftliche Texte und Medien, Münster: 393–408. ▸︎ Google︎ Scholar
Soto-Andrade J. (2018) Enactive metaphorising in the learning of mathematics. In: Kaiser G., Forgasz H., Graven M., Kuzniak A., Simmt E. & Xu B. (eds.) Invited lectures from the 13th International Congress on Mathematical Education (ICME 13) Springer, Cham: 619–637. https://cepa.info/6219
Soto-Andrade J. (2020) Metaphors in mathematics education. In: Lerman S. (ed.) Encyclopedia of mathematics education. Second Edition. Springer, Cham: 447–453. ▸︎ Google︎ Scholar
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. Springer, Cham: 125–143. https://cepa.info/6889
Stewart J., Gapenne O. & Di Paolo E. A. (eds.) (2010) Enaction: Toward a new paradigm for cognitive science. MIT Press, Cambridge MA. Reviewed in. https://constructivist.info/7/1/81
Tall D. (2013) How humans learn to think mathematically: Exploring the three worlds of mathematics. Cambridge University Press, New York NY. ▸︎ Google︎ Scholar
Thom R. (1994) Spectre bord d’un centre obscur [Border spectrum of a dark centre]. In: Porte M. (ed.) Passion des formes. Editions E. N. S., Fontenay-Saint Cloud: 13–24. ▸︎ Google︎ Scholar
Thurston W. P. (1994) On proof and progress in mathematics. Bulletin of the American Mathematical Society 30(2): 161–177. ▸︎ Google︎ Scholar
Varela F. J. (1984) The creative circle: Sketches on the natural history of circularity. In: Watzlawick P. (ed.) The invented reality. W. W. Norton, New York NY: 309–323. https://cepa.info/2089
Varela F. J. (1987) Lying down a path in walking. In: Thompson W. I. (ed.) Gaia: A way of knowing, Lindisfarne Press, Barrington MA: 48–64. https://cepa.info/2069
Varela F. J., Thompson E. & Rosch E. (1991) The embodied mind: Cognitive science and human experience. Cambridge. MIT Press, Cambridge MA. ▸︎ Google︎ Scholar
Watson A. (2021) Care in mathematics education. Palgrave Macmillan, London. ▸︎ Google︎ Scholar
Weissglass J. (1979) Exploring elementary mathematics. W. H. Freeman, New York NY. ▸︎ Google︎ Scholar
Wilhem R. (1956) I Ging, das Buch der Wandlungen [Yi Jing, the book of transformations]. Eugen Diederichs Verlag, Cologne. ▸︎ Google︎ Scholar
Wittmann E. (2013) Fundamental ideas of mathematics as the natural source of teaching and learning. Quaderni di Ricerca in Didattica 23(Supplement 1): 98–110. ▸︎ Google︎ Scholar

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