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Didactical contract and responsiveness to didactical contract: a theoretical framework for enquiry into students’ creativity in mathematics

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Abstract

One of the manifestations of learning is the student’s ability to come up with original solutions to new problems. This ability is one of the criteria by which the teacher may assess whether the student has grasped the taught mathematics. Obviously, a teacher can never teach the ability to invent new solutions (at least not directly): he/she can ask for it, expect it, encourage it, but cannot require it. This is one of the fundamental paradoxes of the whole didactical relationship, which Guy Brousseau modelled in one of the best-known concepts in didactics of mathematics: the didactical contract. The teacher cannot be confident that the student will learn exactly what the teacher intended to teach and hence the student must re-create it on the basis of what he/she already knows. In this paper, the importance and the role of situations affording mathematical creativity (in the sense of production of original solutions to unusual situations) are demonstrated. The authors present an experiment (with 9–10-year-old children) that makes it possible to show how certain situations are more favourable (for all children) to express some characteristics of mathematical creativity.

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Notes

  1. Students with a specific failure are those children “who have deficiencies in acquisition, learning difficulties or lack of liking, shown in the domain of mathematics but who do sufficiently well in other disciplines” (Brousseau 1978).

  2. The history of science testifies that these works were often recognized only after long time, sometimes centuries after their production: let us recall for example Kepler or Bolzano (1781–1848), whose numerous works were recognized only after 1920.

  3. The school level in mathematics was evaluated by applying a standardized test (scored 0–10) that allows situating a pupil’s level in relation to the whole of the French population. These scores were then distributed into three classes corresponding to indications created by the test creators: ‘Good’: the mark (x) belongs to <8; 10); ‘Average’: x ∈ <5.5; 8); ‘Weak’: x ∈ <0; 5.5). The Chi-squared test (with 4 degrees of freedom) shows that the teachers’ evaluation was in strong accordance with the results of the test (χ2 = 43.53; p < .001).

  4. The three variables (level in mathematics, level in French, and responsiveness to the didactical contract) are strongly interlinked. It would therefore be useful to calculate partial correlations for eliminating the influence of one of the three by keeping it constant, and to estimate the relationship between the two other variables. Let us remind at this point that partial correlation is what can be observed between two variables if the third value is constant.

  5. For a given variable, if a variation is observed between two lessons, this variable has the value ‘1’ or ‘0’. For example, during the first lesson, students worked together in a group, while during the second they worked individually: thus v5 = 1.

  6. We call a ‘profile’ the quartet consisting of the type of the answer to the problem (previously defined), for each school level of students, for each of the four situations. Statistical similarity (measured by Chi-squared test) of these profiles is provided for the variable ‘type of situation’; within the same situation, students give the same answer independently of their level.

  7. Let us mention that the same results were observed with other types of problems: problems of ‘age of captain’, incomplete ones, etc.

  8. The model of the estimation of progress used here is a complex construct; the procedure used (construction of a theoretical model) allows on one side avoiding classical effects of ceiling or of floor and, on the other side, authorizes testifying that the student made progress or regression, with a 10 % threshold of risk.

  9. Guy Brousseau is the first laureate of the Felix Klein medal for long-life research in the field of mathematics education, awarded by the International Commission on Mathematical Instruction.

References

  • Brousseau, G. (1978). Etude de 1’influence de 1’interprétation des activités didactiques sur les échecs électifs de l’enfant en mathématiques. Enseignement élémentaires des mathématiques. Cahier de l’IREM de Bordeaux I, n° 18.

  • Brousseau, G. (1980). Les échecs électifs dans l’enseignement des mathématiques a l’école élémentaire. Revue de Laryngologie Otologie Rhinologie, 101(3–4), 107–131.

    Google Scholar 

  • Brousseau, G. (1997). Theory of didactical situations in mathematics 1970–1990. Dordrecht: Kluwer.

    Google Scholar 

  • Brousseau, G., & Novotná, J. (2008). La culture scolaire des problèmes de mathématiques. In B. Sarrazy (Ed.), Les didactiques et leurs rapports à l’enseignement et à la formation. Quel statut épistémologique de leurs modèles et de leurs résultats? Bordeaux: AFIRSE, IUFM d’Aquitaine—Université Montesquieu Bordeaux IV, LACES—Université Victor Segalen Bordeaux 2 [CD ROM].

  • Bru, M. (1991). Les variations didactiques dans l’organisation des conditions d’apprentissage. Toulouse: Ed. Universitaires du Sud.

    Google Scholar 

  • Cheeseman, J. (2009). Challenging children to think: teacher behaviours that stimulate children to examine their mathematical thinking. In J. Novotná & H. Moraová (Eds.), Proceedings of Symposium on Elementary Maths Teaching SEMT 09 (pp. 11–23). Prague: Charles University, Faculty of Education.

    Google Scholar 

  • Chopin, M.-P. (2007). Le temps didactique dans l’enseignement des mathématiques. Approche des phénomènes de régulation des hétérogénéités didactiques. Thèse de doctorat de l’université Victor Segalen Bordeaux 2 (s/d du Pr. B. Sarrazy).

  • De Montaigne, M. (1976). The complete works. Essays, travel journal, letters (Donald M. Frame, Trans.). Stanford, CA: Stanford University.

  • Drévillon, J. (1980). Pratiques éducatives et développement de la pensée opératoire. Paris: PUF.

    Google Scholar 

  • Ervynck, G. (1991). Mathematical creativity. In D. Tall (Ed.), Advanced mathematical thinking (pp. 42–53). Dordrecht: Kluwer.

    Google Scholar 

  • Flanders, N. A. (1966). Interaction analysis in the classroom: A manual for observers. Ann Arbor, MI: University of Michigan School of Education.

    Google Scholar 

  • Kant, E. (1991). Qu’est-ce que les Lumières? In E. Kant (Ed.), Vers la paix perpétuelle. Que signifie s’orienter dans la pensée? Qu’est-ce que les Lumières? (pp. 41–51) (Translated from the German by J.-F. Poirier and F. Proust.) Paris: Flammarion. English version: What is enlightenment? http://www.columbia.edu/acis/ets/CCREAD/etscc/kant.html. Accessed 6 Mar 2013.

  • Kattou, M., Kontoyianni, K., Pitta-Pantazi, D., & Christou, C. (2011). Does mathematical creativity differentiate mathematical ability? In M. Pytlak, E. Swoboda, & T. Rowland (Eds.), Proceedings of the Seventh Congress of the European Society for Research in Mathematics Education (pp. 1056–1065). Rzeszów: University of Rzeszów, ERME.

    Google Scholar 

  • Kintsch, W., & Greeno, J. G. (1985). Understanding and solving word arithmetic problems. Psychological Review, 92(1), 109–129.

    Article  Google Scholar 

  • Kuhn, T. S. (1962). The structure of scientific revolutions. Chicago: University of Chicago Press.

    Google Scholar 

  • Levav-Waynberg, A., & Leikin, R. (2009). Multiple solutions for a problem: A tool for evaluation of mathematical thinking in geometry. In V. Durand-Guerrier, S. Soury-Lavergne, & F. Arzarello (Eds.), Proceedings of Sixth Conference of European Research in Mathematics Education (pp. 776–785). Lyon: Institut National de Recherche Pédagogique.

    Google Scholar 

  • Marchive, A. (2007). La pédagogie à l’épreuve de la didactique: Approche historique, recherches empiriques et perspectives théoriques. Rennes: Presses Universitaires de Rennes.

    Google Scholar 

  • Nakakoji, K., Yamamoto, Y., & Ohira, M. (1999). A framework that supports collective creativity in design using visual images. In E. Edmonds, & L. Candy (Eds.), Proceedings of the third conference on creativity and cognition (pp. 166–173). New York: ACM Press. http://www.informatik.uni-trier.de/~ley/db/conf/candc/candc1999.html. Accessed 18 Feb 2013.

  • Novotná, J., & Hošpesová, A. (2009). Effet Topaze et liaisons dans les pratiques des professeurs de mathématiques. In A. Kuzniak, & M. Sokhna (Eds.), Espace mathématique francophone. Dakar, Sénégale. http://fastef.ucad.sn/EMF2009/colloque.htm. Accessed 20 Feb 2013.

  • Novotná, J., & Sarrazy, B. (2009). Mathematical and psychological modelling in mathematics education. In J. Novotná & H. Moraová (Eds.), Proceedings of International Symposium Elementary Mathematics Teaching SEMT 09 (pp. 190–197). Prague: Charles University, Faculty of Education.

    Google Scholar 

  • Sarrazy, B. (1995). Le contrat didactique. Revue Française de Pédagogie, 112, 85–118.

    Article  Google Scholar 

  • Sarrazy, B. (2002). Effects of variability on responsiveness to the didactic contract in problem-solving among pupils of 9–10 years. European Journal of Psychology of Education, XVII(4), 321–341.

    Article  Google Scholar 

  • Sarrazy, B. (2008). Ostension et dévolution dans l’enseignement des maths: Anthropologie wittgensteinienne et théorie des situations didactiques. Education et Didactique, PUR, 1(3), 3–46.

    Google Scholar 

  • Sarrazy, B. (2010). Il contratto didattico: Un contributo teorico alla chiarificazione di alcuni paradossi della relazione insegnante/alievo. In B. D’Amore, M. I. Fandiño Pinilla, I. Marazzani, & B. Sarrazy (Eds.), Didattica della matematica alcuni effetti del “contratto” (pp. 81–92). Bologna: Archetipolibri.

    Google Scholar 

  • Sarrazy, B., & Novotná, J. (2005). Didactical contract: Theoretical frame for the analysis of phenomena of teaching mathematics. In J. Novotná (Ed.), Proceedings SEMT 05 (pp. 33–44). Prague: Charles University, Faculty of Education.

    Google Scholar 

  • Sarrazy, B., & Novotná, J. (2011). Didactical vs. mathematical modelling of the notion of competence in mathematics education. Case of 9–10-year-old pupils’ problem solving. In Proceedings of the Seventh Congress of the European Society for Research in Mathematics Education (pp. 1125–1132). Rzeszów: University of Rzeszów, ERME.

  • Sheets, C., & Cifarelli, V. V. (2011). Developing instructional sequences in pre-service elementary mathematics teacher education. In J. Novotná & H. Moraová (Eds.), Proceedings of Symposium on Elementary Maths Teaching SEMT 11 (pp. 313–320). Prague: Charles University, Faculty of Education.

    Google Scholar 

  • Singh, S. (1998). Le dernier théorème de Fermat. Paris: Editions J. C. Lattès.

    Google Scholar 

  • Vergnaud, G. (1983). L’enfant, la mathématique et la réalité: Problèmes de l’enseignement des mathématiques à l’école élémentaire. Berne: Peter Lang.

    Google Scholar 

  • Wittgenstein, L. (1961). Tractatus logico-philosophicus. Paris: Gallimard (first published in German in 1921).

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Acknowledgments

The research was partially supported by the project GAČR P407/12/1939.

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Authors and Affiliations

  1. Equipe Anthropologie et Diffusion des Savoirs, Laboratoire Culture, Education, Sociétés (LACES EA 4140), Univ. Bordeaux, 33000, Bordeaux, France

    Bernard Sarrazy & Jarmila Novotná

  2. Université Bordeaux Segalen, 3 ter Place de la Victoire, 33076, Bordeaux Cedex, France

    Bernard Sarrazy

  3. Faculty of Education, Charles University in Prague, M.D. Rettigové 4, 116 39, Prague 1, Czech Republic

    Jarmila Novotná

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  1. Bernard Sarrazy
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Correspondence to Jarmila Novotná.

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Sarrazy, B., Novotná, J. Didactical contract and responsiveness to didactical contract: a theoretical framework for enquiry into students’ creativity in mathematics. ZDM Mathematics Education 45, 281–293 (2013). https://doi.org/10.1007/s11858-013-0496-4

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