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Mathematical Working Spaces in schooling: an introduction

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Abstract

The theoretical and methodological model of Mathematical Working Space (MWS) is introduced in this paper. For over 10 years, the model has been the object of collaborative research among various researchers, generally coming from French and Spanish speaking countries. Articulating epistemological and cognitive aspects, the MWS model is aimed at providing a tool for the specific study of mathematical work in which students and teachers are effectively engaged during mathematics sessions. The abstract space thus conceived refers to a structure organized in a way that allows the analysis of the mathematical activity of individuals dealing with mathematical problems. Thus, analyzing mathematical work through the lens of MWS enables tracking down how meaning is progressively constructed, as a process of bridging the epistemological and the cognitive perspectives—these being modelled as two planes at different levels in the diagrammatic structure—in accordance with different specific yet intertwined genetic developments. Each is identified as a genesis related to a specific dimension in the model: semiotic, instrumental and discursive geneses. A general overview of the different papers included in this issue is given, and shows how the model can be used to study different tasks, teaching situations and activities set in specific mathematical fields or domains. Some perspectives are finally drawn, while reflecting on the possibility of ‘networking’ different theoretical frames with the MWS framework. Indeed, the latter is not proposed as a holistic theory, but rather should function as a tool interacting strongly with other approaches.

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Notes

  1. See the Revista Latinoamericana de Investigación en Matemática Educativa (RELIME), 17-4 (2014).

  2. This quotation could also be related to the notions of connaissances opératoires versus connaissances prédicatives, proposed by Vergnaud (2001).

  3. We are here referring to Duval’s ‘déconstruction dimensionnelle des formes’ (2005, p. 20).

  4. In this article and other papers of the special issue, ‘field’ and ‘domain’ will be used more or less as synonyms, fields being somewhat broader than domains. We may for instance speak of the domain of sequences and series inside the field of Analysis.

  5. See Bolema (Mathematics Education Bulletin), April 2016, vol. 30, n° 54.

References

  • Artigue, M. (2002). Learning mathematics in a CAS environment: the genesis of a reflection about instrumentation and the dialectics between technical and conceptual work. International Journal of Computers for Mathematical Learning, 7, 245–274.

    Article  Google Scholar 

  • Arzarello, F. (2006). Semiosis as a multimodal process. Revista Latinoamericana de Investigación en Matemática Educativa (RELIME), 9(1), 267–299.

    Google Scholar 

  • Berthelot, J. M. (2008). L’emprise du vrai. Connaissance scientifique et modernité. Paris: Presses universitaires de France.

    Google Scholar 

  • Boaler, J. (2002). Exploring the nature of mathematical activity: using theory, research and working hypotheses to broaden conception of mathematics knowing. Educational Studies in Mathematics, 51, 3–21.

    Article  Google Scholar 

  • Brousseau, G. (2002a). Cadres, jeux de cadres et théories des situations. Actes de la journée Douady (pp. 73–82). Irem: Université Paris-Diderot.

  • Brousseau, G. (2002b). Theory of didactical situations in mathematics. New York: Springer.

    Google Scholar 

  • Chevallard, Y. (1985). La transposition didactique—Du savoir savant au savoir enseigné. Grenoble: La Pensée sauvage.

    Google Scholar 

  • Cogan, L. S., & Schmidt, W. H. (1999). An examination of instructional practices in six countries. In G. Kaiser, E. Luna, & I. Huntley (Eds.), International Comparisons in Mathematics Education (pp. 68–85). London: Falmer Press.

    Google Scholar 

  • Coutat, S., & Richard, P. R. (2011). Les figures dynamiques dans un espace de travail mathématique pour l’apprentissage des propriétés mathématiques. Annales de Didactique et de Sciences Cognitives, 16, 97–126.

    Google Scholar 

  • Derouet, C., & Parzysz, B. (2016). How can histograms be useful for introducing continuous probability distributions? ZDM Mathematics Education. doi:10.1007/s11858-016-0769-9. (This issue)

    Google Scholar 

  • Desanti, J. T. (1975). Qu’est ce qu’un problème épistémologique ? In J. T. Desanti (Ed.), La philosophie silencieuse (pp. 110–132). Paris: Le Seuil.

    Google Scholar 

  • Drouhard, J.-P. (2006). Prolégomènes « épistémographiques » à l’étude des transitions dans l’enseignement des mathématiques. In N. Bednarz & C. Mary (Eds.), Actes du 3e colloque Espace Mathématique Francophone: L’enseignement des mathématiques face aux défis de l’école et des communautés, CD-Rom. Sherbrooke: Université de Sherbrooke.

    Google Scholar 

  • Duval, R. (1993). Registres de représentation sémiotique et fonctionnement cognitif de la pensée. Annales de Didactique et de Sciences Cognitives, 5, 37–65.

    Google Scholar 

  • Duval, R. (2005). Les conditions cognitives de l’apprentissage de la géométrie: développement de la visualisation, différenciation des raisonnements et coordination de leur fonctionnements. Annales de Didactique et de Sciences Cognitives, 10, 5–53.

    Google Scholar 

  • Eco, U. (1988). Le signe. Trans: J.-M. Klinkenberg. Bruxelles: Éditions Labor.

  • Elia, I., Özel, S., Gagatsis, A., Panaoura, A., & Yetkiner Özel, Z. E. (2016). Students’ mathematical work on absolute value: Focusing on conceptions, errors and obstacles. ZDM Mathematics Education. doi:10.1007/s11858-016-0780-1. (This issue)

    Google Scholar 

  • Ernest, P. (1999). Forms of knowledge in mathematics and mathematics education: philosophical and rhetorical perspectives. Educational Studies in Mathematics, 38(1), 67–83.

    Article  Google Scholar 

  • Fischbein, E. (1993). The theory of figural concepts. Educational Studies in Mathematics, 24, 139–162.

    Article  Google Scholar 

  • Freudenthal, H. (1971). Geometry between the devil and the deep sea. Educational Studies in Mathematics, 3, 413–435.

    Article  Google Scholar 

  • Giaquinto, M. (2005). Mathematical activity. In P. Macosu, K. F. Jorgonsen, & S. A. Pedersen (Eds.), Visualization, explanation and reasoning styles in mathematics (pp. 75–87). New-York: Springer.

    Chapter  Google Scholar 

  • Gillies, D. (Ed.). (1992). Revolutions in mathematics. Oxford: Oxford Science Publications, The Clarendon Press.

    Google Scholar 

  • Gómez-Chacón, I., & Kuzniak, A. (2015). Networking of theories as a research practice in mathematics education. Spaces for geometric work: figural, instrumental, and discursive geneses of reasoning in a technological environment. International Journal of Science and Mathematics Education, 13, 201–226.

    Article  Google Scholar 

  • Gómez-Chacón, I., Romero Albaladejo, I. M., & del Mar García López, M. (2016). Zig-zagging in geometrical reasoning in technological collaborative environments: a mathematical working space-framed study concerning cognition and affect. ZDM Mathematics Education. doi:10.1007/s11858-016-0755-2. (This issue)

    Google Scholar 

  • Gonseth, F. (1945–1952). La géométrie ou le problème de l’espace. Neuchatel: Editions du Griffon.

  • Granger, G. G. (1963). Essai d’une philosophie du style. Paris: Armand Colin, rééd. (Odile Jacob 1987).

    Google Scholar 

  • Guzman, I., & Kuzniak, A. (2006). Paradigmes géométriques et géométrie enseignée au Chili et en France. Paris: Irem Paris-Diderot.

    Google Scholar 

  • Hadamard, J. (2011–1975). Essai sur la psychologie de l’invention dans le domaine mathématique, suivi de Poincaré, H., L’invention mathématique. Paris: Éditions Jacques Gabay.

  • Hanna, G., & de Villiers, M. (Eds.). (2012). ICMI Study 19 Book: Proof and proving in mathematics education. Dordrecht: Springer.

    Google Scholar 

  • Hitt, F., & González-Martín, A. (2015). Covariation between variables in a modelling process: the ACODESA (Collaborative learning, Scientific debate and Self-reflexion) method. Educational Studies in Mathematics, 88(2), 201–219.

    Article  Google Scholar 

  • Hitt, F., Saboya, M., & Cortés, C. (2016). An arithmetic-algebraic work space for the promotion of arithmetic and algebraic thinking: triangular numbers. ZDM Mathematics Education. doi:10.1007/s11858-015-0749-5 (This issue).

    Google Scholar 

  • Houdement, C., & Kuzniak, A. (1999). Un exemple de cadre conceptuel pour l’étude de l’enseignement de la géométrie en formation des maîtres. Educational Studies in Mathematics, 40(3), 283–312.

    Article  Google Scholar 

  • Kidron, I. (2016). Epistemology and networking theories. Educational Studies in Mathematics, 91(2), 149–163.

    Article  Google Scholar 

  • Kuhn, T. S. (1966). The structure of scientific revolutions (2nd ed.). Chicago: University of Chicago Press.

    Google Scholar 

  • Kuzniak, A., Nechache, A., & Drouhard, J.-P. (2016). Understanding the development of mathematical work in the context of the classroom. ZDM Mathematics Education. doi:10.1007/s11858-016-0773-0.

    Google Scholar 

  • Kuzniak, A., & Rauscher, J.-C. (2011). How do teachers’ approaches to geometric work relate to geometry students’ learning difficulties? Educational Studies in Mathematics, 77, 129–147.

    Article  Google Scholar 

  • Lagrange, J.-B., & Hoyles, C. (Eds.). (2009). Mathematical education and digital technologies: rethinking the terrain. New York: Springer.

    Google Scholar 

  • Lakatos, I. (1976). Proofs and refutations. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • Lave, J., & Wenger, E. (1990). Situated learning: Legitimate peripheral participation. Cambridge, UK: Cambridge University Press.

    Google Scholar 

  • Legrand, M. (2001). Scientific debate in mathematics courses. In D. Holton (Ed.), The teaching and learning of mathematics at university level: an ICMI study (pp. 127–135). Dordrecht: Kluwer Academic Publishers.

    Google Scholar 

  • Lerouge, A. (2000). La notion de cadre de rationalité. A propos de la droite au collège. Recherches en Didactique des Mathématiques, 20(2), 171–208.

    Google Scholar 

  • Minh, T. K., & Lagrange, J.-B. (2016). Connected functional working spaces: a framework for the teaching and learning of functions at upper secondary level. ZDM Mathematics Education. doi:10.1007/s11858-016-0774-z.

    Google Scholar 

  • Montoya, E., & Vivier, L. (2014). Les changements de domaine dans le cadre des Espaces de Travail Mathématique. Annales de Didactique et de Sciences Cognitives, 19, 73–101.

    Google Scholar 

  • Montoya, E., & Vivier, L. (2016). Mathematical working space and paradigms as an analysis tool for the teaching and learning of analysis. ZDM Mathematics Education. doi:10.1007/s11858-016-0777-9.

    Google Scholar 

  • Oktaç, A., & Vivier, L. (2016). Conversion, change, transition in research about analysis. In B. R. Hodgson, A. Kuzniak, & J. B. Lagrange (Eds.), The didactics of mathematics: approaches and issues. A homage to Michèle Artigue (pp. 87–122). New York: Springer.

    Chapter  Google Scholar 

  • Peirce, C. S. (1931). Collected Papers, vols. 1–16. Cambridge: Harvard University Press.

    Google Scholar 

  • Pluvinage, F., Carrión Miranda, V., & Adjiage, R. (2016). Facilitating the genesis of functional working spaces in guided explorations. ZDM Mathematics Education. doi:10.1007/s11858-016-0791-y.

    Google Scholar 

  • Rabardel, P. (1995). Les hommes et les technologies. Une approche cognitive des instruments contemporains. Paris: Armand Colin.

    Google Scholar 

  • Radford, L. (2016). Epistemology as a research category in mathematics teaching and learning. In B. R. Hodgson, A. Kuzniak, & J. B. Lagrange (Eds.), The didactics of mathematics: approaches and issues. A homage to Michèle Artigue (pp. 31–36). New York: Springer.

    Google Scholar 

  • Radford, L., Schubring, G., & Seeger, F. (Eds.). (2008). Semiotics in mathematics education. Rotterdam: Sense Publishers.

    Google Scholar 

  • Reichenbach, H. (1938). Experience and prediction. Chicago: University of Chicago Press.

    Google Scholar 

  • Richard, P. R., Oller Marcén, A. M., & Meavilla Seguí, V. (2016). The concept of proof in the light of mathematical work. ZDM Mathematics Education, 48(6). doi:10.1007/s11858-016-0805-9. (This issue)

  • Ruthven, K. (2014). From networked theories to modular tools? In A. Bikner-Ahsbahs & S. Prediger (Eds.), Networking of theories as a research practice in mathematics education (pp. 267–280). Switzerland: Springer.

    Google Scholar 

  • Santos Trigo, M., Moreno Armella, L., & Camacho Machín, M. (2016). Problem solving and the use of digital technologies within the mathematical working space framework. ZDM Mathematics Education, 48(6). doi:10.1007/s11858-016-0757-0. (This issue).

  • Sierpinska, A. (2004). Research in mathematics education through a keyhole: task problematization. For the learning of mathematics, 24(2), 7–15.

    Google Scholar 

  • Tall, D. (1991). Advanced mathematical thinking. Dordrecht: Kluwer Academic Publishers.

    Book  Google Scholar 

  • Tanguay, D. (2015). Circulation et coordination dans les espaces de travail, pour une activité articulant géométrie et arithmétique. In I. Gómez-Chacón, J. Escribano, A. Kuzniak, & P. R. Richard (Eds.), Actes du 4e symposium Espaces de Travail Mathématique (ETM 4) (pp. 69–85). Spain: Universidad Complutense de Madrid.

    Google Scholar 

  • Tanguay, D., Kuzniak, A., & Gagatsis, A. (2015). Synthesis of Topic 1—The mathematical work and mathematical working spaces. In I. Gómez-Chacón, J. Escribano, A. Kuzniak, & P. R. Richard (Eds.), Actes du 4e symposium Espaces de Travail Mathématique (ETM 4) (pp. 20–38). Spain: Universidad Complutense de Madrid.

    Google Scholar 

  • Tanguay, D., & Venant, F. (2016). The semiotic and conceptual genesis of angle. ZDM Mathematics Education. doi:10.1007/s11858-016-0789-5. (This issue)

    Google Scholar 

  • Thom, R. (1974). Mathématiques modernes et mathématiques de toujours, suivi de Les mathématiques « modernes » , une erreur pédagogique et philosophique ? In R. Jaulin (Ed.), Pourquoi la mathématique ? (pp. 10–18). Paris: Éditions.

    Google Scholar 

  • Thurston, W. P. (1994). On proof and progress in mathematics. Bulletin of the American Mathematical Society, 30(2), 161–177.

    Article  Google Scholar 

  • Vandebrouck, F. (Ed.). (2013). Mathematics classrooms students’ activities and teachers’ practices. Rotterdam: Sense Publishers.

    Google Scholar 

  • Veldhuis, M., & van den Heuvel-Panhuizen, M. (2014). Primary school teachers’ assessment profiles in mathematics education. PLoS One, 9(1), e86817.

    Article  Google Scholar 

  • Vergnaud, G. (2001). Forme opératoire et forme prédicative de la connaissance. In J. Portugais (Ed.), Actes du colloque GDM (pp. 1–22). Montreal: Université de Montréal.

    Google Scholar 

  • Verret, M. (1975). Le temps des études. Paris: Librairie Honoré Champion.

    Google Scholar 

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Authors and Affiliations

  1. Université Paris Diderot, Paris, France

    Alain Kuzniak

  2. Université du Québec à Montréal, Montreal, Canada

    Denis Tanguay

  3. University of Cyprus, Nicosia, Cyprus

    Iliada Elia

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  1. Alain Kuzniak
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  2. Denis Tanguay
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  3. Iliada Elia
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Correspondence to Denis Tanguay.

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Kuzniak, A., Tanguay, D. & Elia, I. Mathematical Working Spaces in schooling: an introduction. ZDM Mathematics Education 48, 721–737 (2016). https://doi.org/10.1007/s11858-016-0812-x

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