Skip to main content
Log in

Bringing forth mathematical concepts: signifying sensorimotor enactment in fields of promoted action

  • Original Article
  • Published:
ZDM Aims and scope Submit manuscript


Inspired by Enactivist philosophy yet in dialog with it, we ask what theory of embodied cognition might best serve in articulating implications of Enactivism for mathematics education. We offer a blend of Dynamical Systems Theory and Sociocultural Theory as an analytic lens on micro-processes of action-to-concept evolution. We also illustrate the methodological utility of design-research as an approach to such theory development. Building on constructs from ecological psychology, cultural anthropology, studies of motor-skill acquisition, and somatic awareness practices, we develop the notion of an “instrumented field of promoted action”. Children operating in this field first develop environmentally coupled motor-action coordinations. Next, we introduce into the field new artifacts. The children adopt the artifacts as frames of action and reference, yet in so doing they shift into disciplinary semiotic systems. We exemplify our thesis with two selected excerpts from our videography of Grade 4–6 volunteers participating in task-based clinical interviews centered on the Mathematical Imagery Trainer for Proportion. In particular, we present and analyze cases of either smooth or abrupt transformation in learners’ operatory schemes. We situate our design framework vis-à-vis seminal contributions to mathematics education research.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2

Similar content being viewed by others


  1. See Clancey (2008) for a survey of complementary intellectual antecedents to the situated/embodied/enactive paradigm, such as the cybernetics research of Gregory Bateson and the robotics work of Andy Clark.

  2. We acknowledge that Feldenkrais scholarship is unconventional as an academic perspective. Notwithstanding, we value its conjectures regarding the roles of embodiment and awareness with respect to learning. These conjectures are original and grounded in a practice that is empirically shown to be effective. Moreover, the conjectures parallel many of our own findings, some of which we arrived at prior to our exposure to Feldenkrais practice.

  3. For a brief video demonstration of the MIT-P, see

  4. Interestingly, dynamical-systems research into coordination of bimanual action (Kelso and Engstrøm 2006, p. 208) has demonstrated a dichotomy between “smooth” and “abrupt” transitions in the development of motor skill, analogous to our findings.

  5. For further empirical results from this line of work, see Abrahamson et al. (2014).


  • Abrahamson, D. (2009). Embodied design: Constructing means for constructing meaning. Educational Studies in Mathematics, 70(1), 27–47. [Electronic supplementary material at].

  • Abrahamson, D. (2012). Discovery reconceived: Product before process. For the Learning of Mathematics, 32(1), 8–15.

    Google Scholar 

  • Abrahamson, D. (2013). Toward a taxonomy of design genres: Fostering mathematical insight via perception-based and action-based experiences. In J. P. Hourcade, E. A. Miller & A. Egeland (Eds.), Proceedings of the 12th Annual Interaction Design and Children Conference (IDC 2013) (Vol. “Full Papers”, pp. 218–227). New York: The New School and Sesame Workshop.

  • Abrahamson, D., Lee, R. G., Negrete, A. G., & Gutiérrez, J. F. (2014). Coordinating visualizations of polysemous action: Values added for grounding proportion. ZDM - The international Journal on Mathematics Education, 46(1), 79–93.

    Article  Google Scholar 

  • Abrahamson, D., & Lindgren, R. (2014, in press). Embodiment and embodied design. In R. K. Sawyer (Ed.), The Cambridge handbook of the learning sciences (2nd ed.). Cambridge: Cambridge University Press.

  • Abrahamson, D., Trninic, D., Gutiérrez, J. F., Huth, J., & Lee, R. G. (2011). Hooks and shifts: A dialectical study of mediated discovery. Technology, Knowledge, and Learning, 16(1), 55–85.

    Google Scholar 

  • Allen, J. W. P., & Bickhard, M. H. (2013). Stepping off the pendulum: Why only an action-based approach can transcend the nativist–empiricist debate. Cognitive Development, 28(2), 96–133.

    Article  Google Scholar 

  • Antle, A. N., Corness, G., & Bevans, A. (2013). Balancing justice: Exploring embodied metaphor and whole body interaction for an abstract domain. International Journal of Arts and Technology, 6(4), 388–409.

    Article  Google Scholar 

  • Bamberger, J., & diSessa, A. A. (2003). Music as embodied mathematics: A study of a mutually informing affinity. International Journal of Computers for Mathematical Learning, 8(2), 123–160.

    Article  Google Scholar 

  • Barsalou, L. W. (2010). Grounded cognition: past, present, and future. Topics in Cognitive Science, 2(4), 716–724.

  • Bartolini Bussi, M. G., & Mariotti, M. A. (2008). Semiotic mediation in the mathematics classroom: Artefacts and signs after a Vygotskian perspective. In L. D. English, M. G. Bartolini Bussi, G. A. Jones, R. Lesh & D. Tirosh (Eds.), Handbook of international research in mathematics education (2nd revised ed., pp. 720–749). Mahwah: Lawrence Erlbaum Associates.

  • Becvar Weddle, L. A., & Hollan, J. D. (2010). Scaffolding embodied practices in professional education. Mind, Culture & Activity, 17(2), 119–148.

    Article  Google Scholar 

  • Bernstein, N. A. (1996). Dexterity and its development. In M. L. Latash & M. T. Turvey (Eds.). Mahwah: Lawrence Erlbaum Associates.

  • Chemero, A. (2009). Radical embodied cognitive science. Cambridge: MIT Press.

    Google Scholar 

  • Churchill, E. (2014). Skill learning, parsing, and narrated enactments: Decomposing and blending action at the potter‘s wheel. (Manuscript in preparation).

  • Clancey, W. J. (2008). Scientific antecedents of situated cognition. In P. Robbins & M. Aydede (Eds.), Cambridge handbook of situated cognition (pp. 11–34). New York: Cambridge University Press.

  • Cobb, P., Confrey, J., diSessa, A., Lehrer, R., & Schauble, L. (2003). Design experiments in educational research. Educational Researcher, 32(1), 9–13.

    Article  Google Scholar 

  • de Hevia, M. D., Izard, V., Coubart, A., Spelke, E. S., & Streri, A. (2014). Representations of space, time, and number in neonates. Proceedings of the National Academy of Sciences, 111(13), 4809–4813.

    Article  Google Scholar 

  • Edelman, G. M. (1987). Neural Darwinism: Theory of neuronal group selection. New York: Basic Books.

    Google Scholar 

  • Ernest, P. (2006). Reflections on theories of learning. ZDM - The international Journal on Mathematics Education, 38(1), 3–7.

    Google Scholar 

  • Fauconnier, G., & Turner, M. (2002). The way we think: Conceptual blending and the mind’s hidden complexities. New York: Basic Books.

    Google Scholar 

  • Gallese, V., & Lakoff, G. (2005). The brain’s concepts: The role of the sensory-motor system in conceptual knowledge. Cognitive Neuropsychology, 22(3–4), 455–479.

    Article  Google Scholar 

  • Ginsburg, C. (2010). The intelligence of moving bodies: A somatic view of life and its consequences. Santa Fe: AWAREing Press.

    Google Scholar 

  • Howison, M., Trninic, D., Reinholz, D., & Abrahamson, D. (2011). The Mathematical Imagery Trainer: From embodied interaction to conceptual learning. In G. Fitzpatrick, C. Gutwin, B. Begole, W. A. Kellogg & D. Tan (Eds.), Proceedings of the annual meeting of The Association for Computer Machinery Special Interest Group on Computer Human Interaction: “Human Factors in Computing Systems” (CHI 2011), Vancouver, May 712, 2011 (Vol. “Full Papers”, pp. 1989–1998). New York: ACM Press.

  • Hutchins, E. (2014). The cultural ecosystem of human cognition. Philosophical Psychology, 27(1), 34–49.

    Article  Google Scholar 

  • Hutto, D. D. (2013). Radically enactive cognition in our grasp. In Z. Radman (Ed.), The hand: An organ of the mind (pp. 227–252). Cambridge: MIT Press.

    Google Scholar 

  • Ingold, T. (2011). The perception of the environment: Essays on livelihood, dwelling, and skill (2nd ed.). New York: Routledge.

    Google Scholar 

  • Kelso, J. A. S., & Engstrøm, D. A. (2006). The complementary nature. Cambridge: MIT Press.

    Google Scholar 

  • Kieren, T. E., Pirie, S. E. B., & Gordon Calvert, L. (1999). Growing minds, growing mathematical understanding: Mathematical understanding, abstraction and interaction. In L. Burton (Ed.), Learning mathematics, from hierarchies to networks (pp. 209–231). London: Falmer Press.

    Google Scholar 

  • Kirsh, D., & Maglio, P. (1994). On distinguishing epistemic from pragmatic action. Cognitive Science, 18(4), 513–549.

    Article  Google Scholar 

  • Lundblad, I., Elert, J., & Gerdle, B. (1999). Randomized controlled trial of physiotherapy and Feldenkrais interventions in female workers with neck-shoulder complaints. Journal of Occupational Rehabilitation, 9(3), 179–194.

    Article  Google Scholar 

  • Maheux, J.-F., & Proulx, J. (2015). Doing|Mathematics: Analyzing data with/in an enactivist-inspired approach. ZDM - The international Journal on Mathematics Education, 47(2) (this issue) (pii:ZDMI-D-14-00004).

  • Maturana, H. (1987). Everything said is said by an observer. In W. Thompson (Ed.), Gaia: A way of knowing (pp. 65–82). Hudson: Lindisfarne Press.

    Google Scholar 

  • Maturana, H. R., & Varela, F. J. (1992). The tree of knowledge: The biological roots of human understanding. Boston: Shambhala Publications.

    Google Scholar 

  • Melser, D. (2004). The act of thinking. Cambridge: MIT Press.

    Google Scholar 

  • Noss, R., & Hoyles, C. (1996). Windows on mathematical meanings: Learning cultures and computers. Dordrecht: Kluwer.

    Book  Google Scholar 

  • Pirie, S. E. B., & Kieren, T. E. (1994). Growth in mathematical understanding: How can we characterize it and how can we represent it? Educational Studies in Mathematics, 26, 165–190.

    Article  Google Scholar 

  • Reed, E. S., & Bril, B. (1996). The primacy of action in development. In M. L. Latash & M. T. Turvey (Eds.), Dexterity and its development (pp. 431–451). Mahwah: Lawrence Erlbaum Associates.

    Google Scholar 

  • Roth, W.-M., & Thom, J. S. (2009). Bodily experience and mathematical conceptions: From classical views to a phenomenological reconceptualization. Educational Studies in Mathematics, 70(2), 175–189.

    Article  Google Scholar 

  • Salomon, G., Perkins, D. N., & Globerson, T. (1991). Partners in cognition: Extending human intelligences with intelligent technologies. Educational Researcher, 20(3), 2–9.

    Article  Google Scholar 

  • Schoenfeld, A. H., Smith, J. P., & Arcavi, A. (1991). Learning: The microgenetic analysis of one student’s evolving understanding of a complex subject matter domain. In R. Glaser (Ed.), Advances in instructional psychology (pp. 55–175). Hillsdale: Erlbaum.

    Google Scholar 

  • Schön, D. A. (1983). The reflective practitioner: How professionals think in action. New York: Basic Books.

    Google Scholar 

  • Schwartz, D. L., & Martin, T. (2006). Distributed learning and mutual adaptation. Pragmatics & Cognition, 14(2), 313–332.

    Article  Google Scholar 

  • Siegler, R. S. (2006). Microgenetic analyses of learning. In D. Kuhn & R. S. Siegler (Eds.), Handbook of child psychology (6th ed., Vol. 2, Cognition, perception, and language, pp. 464–510). Hoboken: Wiley.

  • Sinclair, N., de Freitas, E., & Ferrara, F. (2013). Virtual encounters: The murky and furtive world of mathematical inventiveness. ZDM - The international Journal on Mathematics Education, 45(2), 239–252.

  • Sriraman, B., & Lesh, R. (2007). Leaders in mathematical thinking & learning—a conversation with Zoltan P. Dienes. Mathematical Thinking and Learning, 9(1), 59–75.

    Article  Google Scholar 

  • Thelen, E., & Smith, L. B. (1994). A dynamic systems approach to the development of cognition and action. Cambridge: MIT Press.

    Google Scholar 

  • Trninic, D., & Abrahamson, D. (2013). Embodied interaction as designed mediation of conceptual performance. In D. Martinovic, V. Freiman, & Z. Karadag (Eds.), Visual mathematics and cyberlearning (Mathematics education in digital era) (Vol. 1, pp. 119–139). New York: Springer.

    Chapter  Google Scholar 

  • Varela, F. J. (1999). Ethical know-how: Action, wisdom, and cognition. Stanford: Stanford University Press.

    Google Scholar 

  • Varela, F. J., Thompson, E., & Rosch, E. (1991). The embodied mind: Cognitive science and human experience. Cambridge: MIT Press.

    Google Scholar 

  • Vérillon, P., & Rabardel, P. (1995). Cognition and artifacts: A contribution to the study of thought in relation to instrumented activity. European Journal of Psychology of Education, 10(1), 77–101.

    Article  Google Scholar 

  • von Glasersfeld, E. (1983). Learning as constructive activity. In J. C. Bergeron & N. Herscovics (Eds.), Proceedings of the 5th Annual Meeting of the North American Group for the Psychology of Mathematics Education (Vol. 1, pp. 41–69). Montreal: PME-NA.

    Google Scholar 

  • Vygotsky, L. S. (1978). Mind in society: The development of higher psychological processes. Cambridge: Harvard University Press. (Original work published 1930).

  • Wilensky, U. (1991). Abstract meditations on the concrete and concrete implications for mathematics education. In I. Harel & S. Papert (Eds.), Constructionism (pp. 193–204). Norwood: Ablex Publishing Corporation.

    Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Dor Abrahamson.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Abrahamson, D., Trninic, D. Bringing forth mathematical concepts: signifying sensorimotor enactment in fields of promoted action. ZDM Mathematics Education 47, 295–306 (2015).

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: