Skip to main content
SpringerLink
Log in

Mental mathematics, emergence of strategies, and the enactivist theory of cognition

  • Published:
Educational Studies in Mathematics Aims and scope Submit manuscript

Abstract

In this article, I present and build on the ideas of John Threlfall [(Educational Studies in Mathematics 50:29–47, 2002)] about strategy development in mental mathematics contexts. Focusing on the emergence of strategies rather than on issues of choice or flexibility of choice, I ground these ideas in the enactivist theory of cognition, particularly in issues of problem posing, for discussing the nature of the solving processes at play when solving mental mathematics problems. I complement this analysis and conceptualization by offering two examples about issues of emergence of strategies and of problem posing, in order to offer illustrations thereof, as well as to highlight the fruitfulness of this orientation for better understanding the processes at play in mental mathematics contexts.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

Notes

  1. In a subsequent article in ZDM, Threlfall (2009) continues the discussion of some of these aspects. When relevant, I refer to some of these additional arguments, often in footnotes.

  2. In his 2009 paper, Threlfall extends this comparison to include Torbeyns’ classification.

  3. As well, even if Threlfall does not discuss this issue, it could be argued that at the level of research itself the idea of offering classifications and categories is not a fruitful route for obtaining adequate “information” and understandings about students’ strategy processes in mental mathematics contexts. Threlfall’s view displaces the focus, as I have argued elsewhere (Proulx, 2012a,b), away from knowing categories of solving and more toward understanding the nature of the mathematical activity students are engaging with in mental mathematics contexts, for characterizing the kind of mathematics they generate in these.

  4. See e.g. the work of von Glasersfeld (1995); Morss (1990); Siegler (1996), or the review of Davis (2004a).

  5. Even if I do not go down this explanatory route (for simple matters of irrelevance for this article), there exist points of junction as well as points of rupture between Darwin’s theory of evolution and what Maturana and Varela offer. For more details on this, see Maturana and Mpodozis (1999).

  6. Referring to living systems, Davis (2004b) illustrates the idea of structural determinism by explaining that one cannot predict the behavior of a dog when kicked, since what it does does not depend on how it is kicked but on the dog’s structure: “The way that a complex system responds to a situation is determined by the system itself, not the situation”.

  7. This is an issue that Towers and Davis (2002) discuss in their article about the coupling, that is, mutual influence, of students’ activities.

  8. To avoid interpreting this notion in a causal manner, Kieren (e.g. 1995; Kieren et al. 1997) has often used the expressions “to occasion” or “occasioning” to depict Maturana’s idea of trigger. For the same reason, I use the expression in quotes, that is, “trigger”.

  9. Note that Varela’s distinction does not refer to Brown and Walter’s (e.g. 1983) or Silver’s (e.g. 1994) work on mathematical problem posing/solving.

  10. The problems and the strategies for solving them are thus not arbitrary. To paraphrase Nùñez, Edwards and Matos (1999), “they are motivated by our everyday experience—especially our bodily experience, which is biologically constrained” (p. 52).

  11. This said, as one reviewer noted, “tasks can be designed to be so constraining that there is an appearance of the task causing the student’s behaviour”, not leaving or opening much space of engagement. However, that causation, as Varela et al. (1991) explain, is to be seen strictly as an appearance, inferred after the fact by an external observer.

  12. Not that strategies are “new” in a sense that nothing similar has been attempted before, but are generated for the tasks faced, locally tailored to them, and thus reflect both the task and the solver.

  13. For those two examples, I make use of usual data material used in mental mathematics studies, that is, students’ oral explanations of their strategies and ways of solving. Obviously, this can raise methodological issues that are worth considering (for the data collection and analysis). However, those issues are not tackled here, as the focus of the article is elsewhere, related to conceptualizing/theorizing strategy development in mental mathematics.

  14. Those are not to be seen as exclusive and can be seen as overlapping. As well, they are not meant to represent an exhaustive list of possibilities for solving this task, as others can be developed in another context.

  15. The presence of the y = x line was used as a referent for helping students in writing their answers.

  16. Even if I do not discuss this, mainly due to space constraints but also for the direct relevance to the issues under consideration here, an argument could be made, and has been by others (see e.g. Kieren & Simmt, 2009; Maheux & Roth, 2011; Pirie & Kieren, 1994), that meanings and strategies engaged with are one and the same.

References

  • Alain. (1932). Propos sur l’éducation [Remarks on education]. Paris: Les presses universitaires de France.

    Google Scholar 

  • Bautista, A., & Roth, W.-M. (2012). Conceptualizing sound as a form of incarnate mathematical consciousness. Educational Studies in Mathematics, 79(1), 41–59.

    Article  Google Scholar 

  • Bell, A., & Janvier, C. (1981). The interpretation of graphs representing situations. For the Learning of Mathematics, 2(1), 34–42.

    Google Scholar 

  • Brown, L., & Coles, A. (2012). Developing “deliberate analysis” for learning mathematics and for mathematics teacher education: How the enactive approach to cognition frames reflection. Educational Studies in Mathematics, 80(1–2), 217–231.

    Article  Google Scholar 

  • Brown, S. I., & Walter, M. I. (1983). The art of problem posing. Philadelphia: Franklin Institute Press.

    Google Scholar 

  • Butlen, D., & Pézard, M. (1990). Calcul mental, calcul rapide [Mental calculations, rapid calculations]. Grand N, 47, 35–59.

    Google Scholar 

  • Butlen, D., & Pézard, M. (1992). Calcul mental et résolution de problèmes multiplicatifs, une expérimentation du CP en CM2 [Mental calculation and multiplicative problem solving, an experiment of CP in CM2]. Recherches en didactique des mathématiques, 12(2–3), 319–368.

    Google Scholar 

  • Capra, F. (1996). The web of life: A new understanding of living systems. New York: Anchor Books.

    Google Scholar 

  • Darwin, C. R. (1867). On the origin of species by means of natural selection, or the preservation of favoured races in the struggle for life. New York: D. Appleton & Co.

    Google Scholar 

  • Davis, B. (1995). Why teach mathematics? Mathematics education and enactivist theory. For the Learning of Mathematics, 15(2), 2–9.

    Google Scholar 

  • Davis, B. (2004a). Inventions of teaching: A genealogy. Mahwah: Erlbaum.

    Google Scholar 

  • Davis, B. (2004b). Teaching with difference: Dissipating structures. Panel presentation at the 2004 Complexity Science and Educational Research conference. Ontario: Chaffey’s Locks.

    Google Scholar 

  • Davis, B., Sumara, D., & Kieren, T. (1996). Cognition, co-emergence, curriculum. Journal of Curriculum Studies, 28(2), 151–169.

    Article  Google Scholar 

  • Even, R. (1998). Factors involved in linking representations of functions. The Journal of Mathematical Behavior, 17(1), 105–121.

    Article  Google Scholar 

  • Filloy, E., & Rojano, T. (1989). Solving equations: The transition from arithmetic to algebra. For the Learning of Mathematics, 9(2), 19–25.

    Google Scholar 

  • Hazelkemp, D. W. (1986). Components of mental multiplying. In H. L. Schoen & M. J. Zweng (Eds.), Estimation and mental computation. 1986 NCTM Yearbook (pp. 116–126). Reston: NCTM.

    Google Scholar 

  • Heirdsfield, A. M., & Cooper, T. J. (2004). Factors affecting the process of proficient mental addition and subtraction: Case studies of flexible and inflexible computers. The Journal of Mathematical Behavior, 23(4), 443–463.

    Article  Google Scholar 

  • Hewitt, D. (1996). Mathematical fluency: The nature of practice and the role of subordination. For the Learning of Mathematics, 16(2), 28–35.

    Google Scholar 

  • Kieren, T. (1995). Teaching mathematics (in-the-middle): Enactivist view on teaching and learning mathematics. Invited plenary lecture at the Queen’s/Gage Canadian Mathematics National Leadership conference for teachers, Queens University.

  • Kieren, T., & Simmt, E. (2009). Brought forth in bringing forth: The interactions and products of a collective learning system. Complicity: An International Journal of Complexity and Education, 6(2), 20–28.

    Google Scholar 

  • Kieren, T., Simmt, E., & Mgombelo, J. (1997). Occasioning understanding: Understanding occasioning. In J. Dossey, J. Swafford, M. Parmantie & A. Dossey (Eds.), Proceedings of 19th conference of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 627-633). Columbus, OH: ERIC.

  • Maheux, J.-F., Proulx, J., & Simmt, E. (2012). Review of “Scientific and mathematical bodies: The interface of culture and mind”. Educational Studies in Mathematics, 81(2), 279–285.

    Article  Google Scholar 

  • Maheux, J.-F., & Roth, W.-M. (2011). Relationality and mathematical knowing. For the Learning of Mathematics, 31(3), 36–41.

    Google Scholar 

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

    Google Scholar 

  • Maturana, H. R. (1988). Reality: The search for objectivity of the quest for a compelling argument. Irish Journal of Psychology, 9(1), 25–82.

    Article  Google Scholar 

  • Maturana, H.R., & Mpodozis, J. (1999). De l’origine des espèces par voie de la dérive naturelle (L. Vasquez & P. Castella, Trans.) [The origin of species by means of natural drift]. Lyon: Presses universitaires de Lyon. (Original work published 1992)

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

  • Morss, J. R. (1990). The biologising of childhood: Developmental psychology and the Darwinian myth. Hillsdale: Lawrence Erlbaum Associates.

    Google Scholar 

  • Murphy, C. (2004). How do children come to use a taught mental calculation strategy? Educational Studies in Mathematics, 56(1), 3–18.

    Article  Google Scholar 

  • Nathan, M. J., & Koedinger, K. R. (2000). Teachers’ and researchers’ beliefs about the development of algebraic reasoning. Journal for Research in Mathematics Education, 31(2), 168–190.

    Article  Google Scholar 

  • Nùñez, R. E., Edwards, L. D., & Matos, J. F. (1999). Embodied cognition as grounding for situatedness and context in mathematics education. Educational Studies in Mathematics, 39(1–3), 45–65.

    Article  Google Scholar 

  • Pirie, S., & Kieren, T. (1994). Growth in mathematical understanding: How can we characterize it and how can we represent it? Educational Studies in Mathematics, 26(2–3), 165–190.

    Article  Google Scholar 

  • Plunkett, S. (1979). Decomposition and all that rot. Mathematics in Schools, 8(3), 2–5.

    Google Scholar 

  • Proulx, J. (2012a). Le calcul mental au-delà des nombres : conceptualisations et illustrations avec la résolution d’équations algébriques. [Mental calculations beyond numbers: conceptualizations and illustrations with algebra equation solving]. Manuscript submitted for publication.

  • Proulx, J. (2012b). Mental calculations with/in a graph: Operating on functions. Manuscript submitted for publication.

  • Reid, D. (1996). Enactivism as a methodology. In L. Puig & A. Gutiérrez (Eds.), Proceedings of the Twentieth Annual Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 203–210). Valencia, Spain: PME.

  • René de Cotret, S. (1999). Perspective bio-cognitive pour l’étude des relations didactiques. In G. Lemoyne & F. Conne (Eds.), Le cognitif en didactique des mathématiques [Cognition in mathematics education] (pp. 103–120). Montréal: Presses de l’Université de Montréal.

    Google Scholar 

  • Rezat, S. (2011). Mental calculation strategies for addition and subtraction in the set of rational numbers. In M. Pytlak, E. Swoboda & T. Rowland (Eds.), Proceedings of the 7 th Congress of the European Society for Research in Mathematics Education (pp. 396–405). Rzeszow, Pologne: CERME.

  • Siegler, R. S. (1996). Emerging minds: The process of change in children’s thinking. New York: Oxford University Press.

    Google Scholar 

  • Silver, E. A. (1994). On mathematical problem posing. For the Learning of Mathematics, 14(1), 19–28.

    Google Scholar 

  • Thom, J.S., Namukasa, I.K., Ibrahim-Didi, K., & McGarvey, L.M. (2009). Perceptually guided action: Invoking knowing as enaction. In M. Tzekaki, M. Kaldrimidou & H. Sakonidis (Eds.), Proceedings of Proceedings of the 33rd conference of the International Group for the Psychology of Mathematics Education (vol. 1, pp. 249–278). Thessaloniki, Greece.

  • Thompson, I. (2009). Mental calculation. Mathematics Teaching, 213, 40–42.

    Google Scholar 

  • Threlfall, J. (2002). Flexible mental calculations. Educational Studies in Mathematics, 50(1), 29–47.

    Article  Google Scholar 

  • Threlfall, J. (2009). Strategies and flexibility in mental calculation. Zentralblatt für Didaktik der Mathematik, 41(5), 541–555.

    Article  Google Scholar 

  • Towers, J., & Davis, B. (2002). Structuring occasions. Educational Studies in Mathematics, 49(3), 313–340.

    Article  Google Scholar 

  • Varela, F. J. (1987). Laying down a path in walking. In W. I. Thompson (Ed.), Gaia: A way of knowing. Political implications of the new biology (pp. 48–64). New York: Lindisfarne Press.

    Google Scholar 

  • Varela, F.J. (1996). Invitation aux sciences cognitives (P. Lavoie, Trans.) [An invitation to cognitive sciences]. Paris: Éditions du Seuil. (Original work published 1988)

  • 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 

  • von Glasersfeld, E. (1995). Radical constructivism: A way of knowing and learning. London and Washington: Falmer Press.

    Book  Google Scholar 

  • Zack, V., & Reid, D. A. (2003). Good-enough understanding: Theorising about the learning of complex ideas (part 1). For the Learning of Mathematics, 23(3), 43–50.

    Google Scholar 

  • Zack, V., & Reid, D. A. (2004). Good-enough understanding: Theorising about the learning of complex ideas (part 2). For the Learning of Mathematics, 24(1), 24–28.

    Google Scholar 

Download references

Acknowledgments

I wish to thank Nadine Bednarz, Tom Kieren, Jean-François Maheux, and Elaine Simmt for their insightful suggestions and helpful comments on this article.

Author information

Authors and Affiliations

  1. Département de mathématiques, Université du Québec à Montréal, C.P. 8888, Succ. Centre-Ville, Montreal, Qc, H3C 3P8, Canada

    Jérôme Proulx

Authors
  1. Jérôme Proulx
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jérôme Proulx.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Proulx, J. Mental mathematics, emergence of strategies, and the enactivist theory of cognition. Educ Stud Math 84, 309–328 (2013). https://doi.org/10.1007/s10649-013-9480-8

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10649-013-9480-8

Keywords

Search

Navigation