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New materialist ontologies in mathematics education: the body in/of mathematics


In this paper we study the mathematical body as an assemblage of human and non-human mathematical concepts. We argue that learners’ bodies are always in the process of becoming assemblages of diverse and dynamic materialities. Following the work of the historian of science Karen Barad, we argue that mathematical concepts must be considered dynamic material, and we suggest a “pedagogy of the concept” that animates concepts as both logical and ontological. We draw on the philosopher of mathematics Gilles Châtelet in order to pursue this argument, elaborating on the way that mathematical concepts partake of the mobility of the virtual, while learners, in engaging with this mobility, enter a material process of becoming. We show how the concept of virtuality allows us to look at mathematical concepts in school curriculum in new ways.

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  1. See for instance the 2012 special issue of the Journal of the Learning Sciences 21(2).

  2. Cutler and MacKenzie borrow the use of the term from Deleuze, who nicely leverages this mathematical concept to talk about processes of determination in which relations are more ontologically primitive than relata.

  3. The simultaneous functioning we are describing here might evoke for readers a comparison with the process/content distinction made in the NCTM Standards. An important difference, however, is that the same concept (parity, in this case) partakes of both the content and the process. Moreover, when functioning on the ontological level, parity has much greater precision and power than process strands such as representation, communication and visualization.

  4. In this he follows the scholastic tradition of contrasting extension—that being the interval actually travelled and its duration in time—with intension—that being its quickness, slowness or “lateness” (Châtelet, 2000, p. 38). As odd as this distinction might seem to modern readers, it is used by Châtelet to disrupt the privileging of position over motion, and to try and imagine motion as the ontogenetic force by which position (or extension) comes into being.

  5. In referring to “iteration,” Châtelet means the kind of mechanical repeated juxtaposition of numbers that characterizes addition and subtraction (and some conceptions of multiplication).


  • Barad, K. (2003). Posthumanist performativity: How matter comes to matter. Signs, 28(3), 801–831.

    Article  Google Scholar 

  • Barad, K. (2007). Meeting the universe halfway: Quantum physics and the entanglement of matter and meaning. Durham: Duke University Press.

    Google Scholar 

  • Bostock, D. (2009). The philosophy of mathematics: An introduction. Oxford: Wiley-Blackwell.

    Google Scholar 

  • Butler, J. (1993). Bodies that matter. New York: Routledge.

    Google Scholar 

  • Châtelet, G. (2000/1993). Les enjeux du mobile. Paris: Seuil. (Engl. transl., by R. Shore & M. Zagha: Figuring space: Philosophy, Mathematics and Physics). Dordrecht: Kluwer Academy Press, 2000.

  • Cheah, P. (2010). Non-dialectical materialism. In D. Coole & S. Frost (Eds.), New materialisms: Ontology, agency, and politics (pp. 70–91). London: Duke University Press.

    Google Scholar 

  • Coole, D., & Frost, S. (Eds.). (2010). New materialisms: Ontology, agency, and politics. London: Duke University Press.

    Google Scholar 

  • Cutler, A., & MacKenzie, I. (2011). Bodies of learning. In L. Guillaume & J. Hughes (Eds.), Deleuze and the body (pp. 53–72). Edinburgh: Edinburgh University Press.

  • Davis, B., & Simmt, E. (2003). Understanding learning systems: Mathematics education and complexity science. Journal for Research in Mathematics Education, 34(2), 137–167.

    Article  Google Scholar 

  • de Freitas, E., & Sinclair, N. (2012). Diagram, gesture, agency: Theorizing embodiment in the mathematics classroom. Educational Studies in Mathematics, 80(1–2), 133–152.

    Google Scholar 

  • Deleuze, G. (1994). Difference and repetition. (Trans. Paul Patton). New York: Columbia University Press.

  • Ellsworth, E. (2005). Places of learning: Media, architecture, pedagogy. New York: Routledge.

    Google Scholar 

  • Gallese, V., & Lakoff, G. (2005). The brain’s concepts: The role of the sensory-motor system in conceptual knowledge. Cognitive Neuropsychology, 21, 1–25.

    Google Scholar 

  • Grosz, E. (1994). Volatile bodies: Towards a corporeal feminism. Indiana: Indiana University Press.

    Google Scholar 

  • Grosz, E. (2001). Architecture from the outside: Essays on virtual and real space. Cambridge: MIT Press.

    Google Scholar 

  • Hall, R., & Nemirovsky, R. (2011). Histories of modal engagements with mathematical concepts: A theory memo. Retrieved from

  • Harroway, D. (2008). When species meet. Minnesota: University of Minnesota Press.

    Google Scholar 

  • Hwang, S., & Roth, W. M. (2011). Scientific and mathematical bodies: The interface of culture and mind. Rotterdam: Sense.

  • Lakoff, G., & Núñez, R. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. New York: Basic Books.

    Google Scholar 

  • Latour, B. (2005). Reassembling the social: An introduction to actor-network-theory. New York: Oxford University Press.

    Google Scholar 

  • Malafouris, L. (2008). Between drains, bodies and things: Tectonoetic awareness and extended self. Philosophical Transactions of the Royal Society B, 363, 1993–2002.

    Article  Google Scholar 

  • Massumi, B. (2002). Parables for the virtual: Movement, affect, sensation. Durham: Duke University Press.

    Google Scholar 

  • Maturana, H. R., & Varela, F. J. (1987). The tree of knowledge: The biological roots of human understanding. Boston: Shambhala.

    Google Scholar 

  • Mendick, H. (2006). Masculinities in mathematics. Maidenhead: Open University Press.

    Google Scholar 

  • Nemirovsky, R., & Ferrara, F. (2009). Mathematical imagination and embodied cognition. Educational Studies of Mathematics, 70(2), 159–174.

    Article  Google Scholar 

  • Núñez, R., Edwards, L., & Matos, J. F. (1998). Embodied cognition as grounding for situatedness and context in mathematics education. Educational Studies in Mathematics, (39), pp. 45–65.

    Google Scholar 

  • Pickering, A. (1995). The mangle of practice: Time, agency and science. Chicago: University of Chicago Press.

    Book  Google Scholar 

  • Radford, L. (2008). The ethics of being and knowing: Towards a cultural theory of learning. In L. Radord, G. Schubring & F. Seeger (Eds.), Semiotics in mathematics education: Epistemology, history, classroom and culture (pp. 215–234). Sense Publishing.

  • Radford, L. (2009). Why do gestures matter? Sensuous cognition and the palpability of mathematical meanings. Educational Studies in Mathematics, 70(3), 111–126.

    Article  Google Scholar 

  • Radford, L. (2012). Towards an embodied, cultural, material conception of mathematics cognition. 12th International Congress on Mathematical Education.

  • Roth, W.-M. (2010). Incarnation: Radicalizing embodiment of mathematics. For the learning of Mathematics, 30(2), 8–17.

    Google Scholar 

  • Saxe, G. B., Shaughnessy, M. M., Shannon, A., Langer-Osuna, J. M., Chinn, R., & Gearhart, M. (2007). Learning about fractions as points on the number line. The learning of mathematics, 69th Yearbook of the The National Council of Teachers of Mathematics (pp. 221–236). Reston: NCTM.

  • Sfard, A. (2008). Thinking as communicating: Human development, the growth of discourses, and mathematizing. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • Sheets-Johnstone, M. (2012). Movement and mirror neurons: a challenging and choice conversation. Phenomenology and the cognitive sciences, 11(3), 385–401.

    Google Scholar 

  • Sinclair, N., de Freitas, E., & Ferrara, F. (2013). Virtual encounters: The murky and furtive world of mathematical inventiveness. ZDM, 45(4).

  • Tall, D. (2011). Crystalline concepts in long-term mathematical invention and discovery. For the Learning of Mathematics, 31(1), 3–8.

    Google Scholar 

  • Walkerdine, V. (1988). The Mastery of reason: Cognitive development and the production of rationality. New York: Routledge.

    Google Scholar 

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Correspondence to Elizabeth de Freitas.

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de Freitas, E., Sinclair, N. New materialist ontologies in mathematics education: the body in/of mathematics. Educ Stud Math 83, 453–470 (2013).

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  • Embodiment
  • Materialism
  • Concept
  • Virtual