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Mathematics Education Research Journal

, Volume 26, Issue 2, pp 325–352 | Cite as

The emerging and emergent present: a view on the indeterminate nature of mathematics lessons

  • Wolff-Michael Roth
  • Jean-François Maheux
Original Article
First Online:
  • 233 Downloads

Abstract

The notion of emergence has considerable currency in mathematics education. However, the notion tends to be used in a descriptive way rather than being theorized and developed as a phenomenon sui generis. The purpose of this article is to contribute to building a theory of emergence. After providing an exemplifying description and analysis of an episode from a second-grade mathematics classroom studying three-dimensional geometry, we discuss implications for theoretical and classroom praxis in mathematics education, especially for the curriculum planning and the preparation, training, and enhancement of teachers of mathematics.

Keywords

Emergence Indeterminacy Witness Intention Intuition Excess Sociality 
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Notes

Acknowledgments

This research was funded in part by a grant from the Social Sciences and Humanities Research Council of Canada. We thank the teachers and children for their participation. We are grateful to Mijung Kim and Lilian Pozzer-Ardenghi, who assisted in the collection of the data.

References

  1. Amerine, R., & Bilmes, J. (1990). Following instructions. In M. Lynch & S. Woolgar (Eds.), Representation in scientific practice (pp. 323–335). Cambridge: MIT Press.Google Scholar
  2. Bakhtin, M. M. (1981). Dialogical imagination. Austin: University of Texas Press.Google Scholar
  3. Black, L., & Williams, J. (2013). Contradiction and conflict between “leading identities”: becoming an engineer versus becoming a “good Muslim” woman. Educational Studies in Mathematics. doi: 10.1007/s10649-013-9481-7.Google Scholar
  4. Bourdieu, P. (1980). Le sens pratique. Paris: Les Éditions de Minuit.Google Scholar
  5. British Columbia Ministry of Education (2006). Principles of mathematics 10 to 12: Integrated resource package. Victoria, BC: Author.Google Scholar
  6. Cobb, P. (1999). Individual and collective mathematical development: the case of statistical data analysis. Mathematics Thinking and Learning, 1, 5–43.CrossRefGoogle Scholar
  7. Cobb, P., & Bauersfeld, H. (Eds.). (1995). Emergence of mathematical meaning: Interaction in classroom cultures. Hillsdale: Lawrence Erlbaum.Google Scholar
  8. Cobb, P., Yackel, E., & Wood, T. (2010). Young children's emotional acts while engaged in mathematical problem solving. In E. Yackel, A. Sfard, & K. Gravemeijer (Eds.), A journey in mathematics education research: Insights from the work of Paul Cobb (pp. 41–74). Dordrecht: Springer.CrossRefGoogle Scholar
  9. Davis, B. (2004). Inventions of teaching: a genealogy. Mahwah: Lawrence Erlbaum.Google Scholar
  10. Davis, B., & Simmt, E. (2003). Understanding learning systems: mathematics education and complexity science. Journal for Research in Mathematics Education, 34, 137–167.CrossRefGoogle Scholar
  11. Davis, B., Sumara, D., & Kieren, T. (1996). Co-emergence, cognition, curriculum. Journal of Curriculum Studies, 28, 151–169.CrossRefGoogle Scholar
  12. Derrida, J. (Ed., Transl.). (1962). Edmund Husserl: L'origine de la géométrie. Paris: Presses Universitaires de France.Google Scholar
  13. Dewey, J. (2008). Later works vol. 10: art as experience (J.-A. Boydston, Ed.). Carbondale: Southern Illinois University Press. (First published in 1934)Google Scholar
  14. Dewey, J., & Bentley, A. F. (1999). Knowing and the known. In R. Handy & E. E. Hardwood (Eds.), Useful procedures of inquiry (pp. 97–209). Great Barrington: Behavioral Research Council.Google Scholar
  15. Duranti, A. (1986). The audience as co-author: an introduction. Text, 6, 239–247.Google Scholar
  16. Durkheim, É. (1919). Les règles de la méthode sociologique. Paris: Felix Alcan.Google Scholar
  17. Font, V., Godino, J. D., & Gallardo, J. (2013). The emergence of objects from mathematical practices. Educational Studies in Mathematics, 82, 97–124.CrossRefGoogle Scholar
  18. Frost, R. (1966). The road not taken [1915; 1916]. In L. Herrig, H. Meller, & R. Sühnel (Eds.), British and American classical poems (p. 112). Braunschweig: Georg Westermann.Google Scholar
  19. Garfinkel, H. (2002). Ethnomethodology's program: working out Durkheim's aphorism. Lanham: Rowman & Littlefield.Google Scholar
  20. Hegedus, S. J., & Moreno-Armella, L. (2011). The emergence of mathematical structures. Educational Studies in Mathematics, 77, 369–388.CrossRefGoogle Scholar
  21. Heidegger, M. (1977). Sein und Zeit. Tübingen: Max Niemeyer.Google Scholar
  22. Heidegger, M. (2006). Gesamtausgabe I Abteilung: Veröffentlichte Schriften 1910–1976 Band 11: Identität und Differenz. Vittorio Klostermann: Frankfurt/M.Google Scholar
  23. Hershkowitz, R., & Schwarz, B. (1999). The emergent perspective in rich learning environments: some roles of tools and activities in the construction of sociomathematical norms. Educational Studies in Mathematics, 39, 149–166.CrossRefGoogle Scholar
  24. Husserl, E. (1976). Husserliana Band VI: Die Krisis der europäischen Wissenschaften und die transzendentale Phänomenologie. Eine Einleitung in die phänomenologische Philosophie. The Hague: Martinus Nijhoff.CrossRefGoogle Scholar
  25. Husserl, E. (1980). Vorlesungen zur Phänomenologie des inneren Zeitbewusstseins. Tübingen: Niemeyer.Google Scholar
  26. Il'enkov, E. V. (1977). Dialectical logic: essays on its history and theory. Moscow: Progress.Google Scholar
  27. Jacobs, J., & Morita, E. (2002). Japanese and American teachers' evaluations of videotaped mathematics lessons. Journal for Research in Mathematics Education, 33, 154–175.CrossRefGoogle Scholar
  28. Kant, I. (1956). Werke Bd. II: Kritik der reinen Vernunft. Wiesbaden: Insel.Google Scholar
  29. Kieren, T. (1995). Teaching in the middle: enactivist view of teaching and learning mathematics. Plenary lecture at the Queen's/Gage Canadian National Mathematics Leadership Conference for Teachers, Queens University, Kingston, Ontario.Google Scholar
  30. Lautner, A. (2012). Der Einsatz des Mathematikmaterials von Maria Montessori und dessen Auswirkung auf die Entwicklung des Zahlbegriffs und die Rechenleistung lernschwacher Schülerinnen und Schüler im ersten Schuljahr. Dissertation at the Ludwig-Maximilians University, Munich. http://d-nb.info/1031631089/34. Accessed 24 May 2013.
  31. Marion, J.-L. (1996). La croisée du visible. Paris: Presses Universitaires de France.Google Scholar
  32. Marion, J.-L. (1997). Étant donné: Essai d'une phénoménologie de la donation. Paris: Presses Universitaires de France.Google Scholar
  33. Marsico, G., & Valsiner, J. (2013). Mind the border! Experiencing the present and reconstructing the past through the boundaries. In R. Säljö, Å. Mäkitalo, & P. Linell (Eds.), Memory practices and learning: experiential, institutional, and sociocultural perspectives.Google Scholar
  34. Masciotra, D., Roth, W.-M., & Morel, D. (2008). Énaction. Apprendre et enseigner en situation. Brussels: de Boeck.Google Scholar
  35. McDermott, R. P., Gospodinoff, K., & Aron, J. (1978). Criteria for an ethnographically adequate description of concerted activities and their contexts. Semiotica, 24, 245–275.CrossRefGoogle Scholar
  36. Mead, G. H. (1932). The philosophy of the present. London: Open Court.Google Scholar
  37. Merleau-Ponty, M. (1945). Phénoménologie de la perception. Paris: Gallimard.Google Scholar
  38. Merleau-Ponty, M. (1964a). Le visible et l'invisible. Paris: Gallimard.Google Scholar
  39. Merleau-Ponty, M. (1964b). Sens et non-sens. Paris: Gallimard.Google Scholar
  40. Mikhailov, F. T. (2002). The “other within” for the psychologist. Journal of Russian and East European Psychology, 39(1), 6–31.CrossRefGoogle Scholar
  41. Nancy, J.-L. (1993). The birth to present. Stanford: Stanford University Press.Google Scholar
  42. Piaget, J. (1937). La construction du réel chez l'enfant. Paris: Delachaux et Niestlé.Google Scholar
  43. Protevi, J. (2009). Beyond autopoiesis: inflections of emergence and politics in the work of Francisco Varela. In B. Clark & M. B. Hansen (Eds.), Emergence and embodiment: new essays on second-order systems theory (pp. 94–112). Durham: Duke University Press.CrossRefGoogle Scholar
  44. Proulx, J. (2013). Mental mathematics, emergence of strategies, and the enactivist theory of cognition. Educational Studies in Mathematics. doi: 10.1007/s10649-013-9480-8.Google Scholar
  45. Ricœur, P. (1990). Soi-même comme un autre. Paris: Éditions du Seuil.Google Scholar
  46. Romano, C. (1998). L’événement et le monde. Paris: Presses Universitaires de France.Google Scholar
  47. Rorty, R. (1989). Contingency, irony, and solidarity. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  48. Roth, W.-M. (2001). Designing as distributed process. Learning and Instruction, 11, 211–239.CrossRefGoogle Scholar
  49. Roth, W.-M. (2002). Being and becoming in the classroom. Westport: Ablex/Greenwood.Google Scholar
  50. Roth, W.-M. (2005). Talking science: language and learning in science. Lanham: Rowman & Littlefield.Google Scholar
  51. Roth, W.-M. (2008). Bricolage, métissage, hybridity, heterogeneity, diaspora: concepts for thinking science education in the 21st century. Cultural Studies in Science Education, 3, 891–916.CrossRefGoogle Scholar
  52. Roth, W.-M. (2009). Radical uncertainty in scientific discovery work. Science, Technology & Human Values, 34, 313–336.CrossRefGoogle Scholar
  53. Roth, W.-M. (2010). Language, learning, context: talking the talk. London: Routledge.Google Scholar
  54. Roth, W.-M. (2012). Mathematical learning: the unseen and unforeseen. For the Learning of Mathematics, 32(3), 15–21.Google Scholar
  55. Roth, W.-M. (2013a). To event: towards a post-constructivist approach to theorizing and researching curriculum as event*-in-the-making. Curriculum Inquiry, 43, 388–417.CrossRefGoogle Scholar
  56. Roth, W.-M. (2013b). Toward a post-constructivist ethics in/of teaching and learning. Pedagogies: An International Journal, 8, 103–125.CrossRefGoogle Scholar
  57. Roth, W.-M. (2013). Working out the interstitial and syncopic nature of the human psyche: on the analysis of verbal data. Integrative Psychological and Behavioral Science, in press. Google Scholar
  58. Roth, W.-M., & Gardner, R. (2012). “They're gonna explain to us what makes a cube a cube?” Geometrical properties as contingent achievement of sequentially ordered child-centered mathematics lessons. Mathematics Education Research Journal, 24, 323–346.CrossRefGoogle Scholar
  59. Roth, W.-M., & Jornet, A. (2013a). Situated cognition. WIREs Cognitive Science, 4, 463–478.CrossRefGoogle Scholar
  60. Roth, W.-M., & Jornet, A. (2013b). Toward a theory of experience. Science Education. doi: 10.1002/sce.21085.Google Scholar
  61. Roth, W.-M., & Lawless, D. (2002). Scientific investigations, metaphorical gestures, and the emergence of abstract scientific concepts. Learning and Instruction, 12, 285–304.Google Scholar
  62. Roth, W.-M., & Radford, L. (2010). Re/thinking the zone of proximal development (symmetrically). Mind, Culture, and Activity, 17, 299–307.CrossRefGoogle Scholar
  63. Roth, W.-M., & Radford, L. (2011). A cultural-historical perspective on mathematics teaching and learning. Rotterdam: Sense.CrossRefGoogle Scholar
  64. Sensevy, G. (2011). Le sens du savoir: Éléments pour une théorie de l’action conointe en didactique. Brussels: de Boek.Google Scholar
  65. Sheets-Johnstone, M. (2011). The primacy of movement (expanded (2nd ed.). Amsterdam: John Benjamins.CrossRefGoogle Scholar
  66. Shimizu, Y. (2008). Exploring Japanese teachers' conception of mathematics lesson structure. Zeitschrift für Didaktik der Mathematik, 40, 941–950.CrossRefGoogle Scholar
  67. Suchman, L. A. (2007). Human–machine reconfigurations: plans and situated actions (2nd ed.). Cambridge: Cambridge University Press.Google Scholar
  68. Thom, R. (1981). Worüber soll man sich wundern. In K. Maurin, K. Michalksi, & E. Rudolph (Eds.), Offene Systeme II: Logik und Zeit (pp. 41–107). Stuttgart: Klett-Cotta.Google Scholar
  69. Tobin, K. (1987). The role of wait time in higher cognitive level learning. Review of Educational Research, 57, 69–85.CrossRefGoogle Scholar
  70. Tsu, L. (1972). Tao te ching (G.-F. Feng & J. English, Trans.). New York: Vintage.Google Scholar
  71. van Oers, B. (2010). Emergent mathematical thinking in the context of play. Educational Studies in Mathematics, 74, 23–37.CrossRefGoogle Scholar
  72. Varela, F. J. (1995). The emergent self. In J. Brockman (Ed.), The third culture: beyond the scientific revolution (pp. 209–222). New York: Simon & Schuster.Google Scholar
  73. Varela, F. J., & Depraz, N. (2000). At the source of time: valence and the constitutional dynamics of affect. In S. Gallagher, S. Watson, & P. Brun (Eds.), Ipseity and alterity: interdisciplinary approaches to intersubjectivity (pp. 153–174). Rouen: Université de Rouen.Google Scholar
  74. Veresov, N. (2010). Forgotten methodology: Vygotsky's case. In A. Toomela & J. Valsiner (Eds.), Methodological thinking in psychology: 60 years gone astray? (pp. 267–295). Charlotte: Information Age.Google Scholar
  75. Vološinov, V. N. (1930). Marksizm i filosofija jasyka: osnovnye problemy sociologičeskogo metoda v nauke o jazyke. Leningrad: Priboj.Google Scholar
  76. von Glasersfeld, E. (1983). On the concept of interpretation. Poetics, 12, 207–218.CrossRefGoogle Scholar
  77. von Weizsäcker, V. (1973). Der Gestaltkreis: Theorie der Einheit von Wahrnehmen und Bewegen. Frankfurt: Suhrkamp.Google Scholar
  78. Vygotskij, L. S. (2001). Lekcii po pedologii. Izhevsk: Udmurdskij University.Google Scholar
  79. Wagenschein, M. (1999). Verstehen Lehren: Genetisch – Sokratisch – Exemplarisch (5. Auflage) [Teaching understanding: genetically, Socratically, exemplarily, 5th ed.]. Beltz: Weinheim.Google Scholar

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© Mathematics Education Research Group of Australasia, Inc. 2014

Authors and Affiliations

  1. 1.Faculty of EducationUniversity of VictoriaVictoriaCanada
  2. 2.Département de MathématiqueUniversité du Québec à MontréalMontrealCanada

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