# Abstract

A paper written for my Preliminary examination for my doctorate in Curriculum & Instruction at Texas A&M University.

## Introduction

The term “didactic contract” is a translation of the French contrat didactique created by Guy Brousseau (1997).^{1} Actually, the term appears only one time (p. 141) in the book. He places this concept within what he calls the “Theory of Didactical Situations.” Brousseau (1984) is the first publication in English where the expression “didactic contract” appears. The author himself defines the term as

C’est l’emsemble des obligations réciproques et des ≪ sanctions ≫ que chaque partenaire de la situation didactique impose ou croit imposer, explicitement ou implicitement, aux autres, et celles qu’on lui impose ou qu’il croit qu’on lui impose, (2003, p. 5)

I translate the definition as “It is the set of the reciprocal obligations and sanctions that each partner in the didactic situation imposes, of believes to impose, explicitly or implicitly, on others, and those that are imposed on him or her, or he or she believes that they are imposed on him or her,”

In the same document (2003, p. 2) is given a definition for “didactic situation.” I translate it as “where an agent, the teacher for example, organizes an intervention that manifests its intention to modify the knowledge of another agent or causes it arise. The second agent, for example, is a student that is allowed to express him or herself in actions.” The didactic situation is in opposition to the “adidactic situation.” In this situation the student is asked to “construct new knowledge through the confrontation with an antagonistic milieu (Cazes, Gueudet, Hersant, & Vandebrouck, 2006).

A recent book by Mason and Johnston-Wilder (2004, p. 79) described the concepts situation didactique (didactic situation) and contrat didactique (didactic contract)

The situation didactique identified by Guy Brousseau consists of the learners, the teacher, the mathematical content and the classroom ethos, as well as the social and institutional forces acting upon that situation, including government directives such as a National Curriculum statement, inspection and testing regimes, parental and community pressures and so on. Within the situation didactique, Brousseau identified an implicit contract (contrat didactique) between teacher and learners, together with some concomitant forces, pressures and tensions. The didactic contract is that ‘the teacher is obliged to teach and the pupil to learn’ (Brousseau and Otte, 1991, p. 18), or at least to pass the assessment. |

The idea of interaction between teacher and students today has to be extended to ‘virtual’ teachers and on-line students (Cazes et al., 2006). Likewise there are specific contract clauses that are accepted by students and there are others that are transgressed. In the case that computer based instruction is used alongside traditional teacher in classroom instruction, as is most often the case, there is an opportunity for conflicting contracts that bring about a whole new set of problems.

## A. Critique of Literature

It is worth making two initial observations regarding the didactic contract. The first is that almost all research is confined to mathematics education, where the term itself originated. The second is that the majority of pedagogical research that has been utilizing the concept of didactic contract and Theory of Didactical Situations has been performed in France and French-speaking Switzerland and Canada and the United Kingdom. Indeed, it appears that the concept of didactic contract is not popular among researchers in the United States. For example, it is not mentioned anywhere in the Theories of Mathematical Learning (1996). I have been able to only find two references to the term by US researchers (Schoenfeld, 2001; Selden & Selden, 2001) and they were in the same book whose editor was from New Zealand.

An EBSCO database search for papers that mention the term didactic contract anywhere in the text yielded nine references, dated from 1996 to 2007. Out of those nine, six papers where written by French authors (Sensevy, 1996; Amade-Escot, 2000; Sarrazy, 2002; Cazes et al., 2006; Menotti & Ricco, 2007; Verscheure & Amade-Escot, 2007), two by Portuguese-speaking authors (Brito Lima & Da Rocha Falcão, 1997; César, 1998), and one by a British author (Shinkfield, 2007).

Imagine a mathematics classroom setting where the teacher will assign some problems to the students. Usually they are given in the form of some text and illustrations on handouts or as references to problems in the textbook. These texts contain instructions on how to perform the task. However, the explicit instructions that accompany this task have to be interpreted by the students in light of the students’ previous knowledge of the requirements of the teacher. The students have to determine what the question means, what information is given and what the constraints for this assignment are (Menotti & Ricco, 2007). These authors classify these contracts in weaker (didacticité faible)^{2} and stronger (didacticité forte) contracts. Among the stronger contracts is what Menotti & Ricco call “Formal reproduction.” On the surface it appears that the students are learning, but in reality they just “follow orders.” The main didactic technique here is repetition and imitation. This is a typical behaviorist practice. Another contract is the “Socratic maieutic” where all that counts is the right answer and all other attempts are ignored by the teacher. Often this happens when there is not sufficient time for instruction. Again, this contract does not involve true understanding by the students. The teacher may also use “sensory empiricism” when the center of the instructional activity is visually showing the students what the right answer is. Again, there is no guarantee of understanding. Finally, we have “constructivist contracts” where, according to Brousseau “…the teacher organizes a milieu and entrusts to it the responsibility for the [students’] acquisition…” The authors agree with Brousseau that to have true understanding the students need to “participate in all stages of a mathematical activity”

The concept of didactic contract has been used to describe the dynamics in a mathematics classroom where students come in from a traditional instructional method and are there asked to perform mathematical investigations. According to Sensevy (1996) new social norms have to be established between the teacher and the students. The didactic contract thus needs to be redefined.

The practice of creating mathematical problems by the students is understood as a device that allows the renegotiation of certain norms of the didactic contract. Sensevy makes the claim that the didactic contract is not just the context, but is the actual knowledge itself. This appears to me to indicate an strong social constructivist view by the author.

This type of constructivism is also expressed by Brito Lima and Da Rocha Falcão (1997) who connect the concept of didactic contract to the zone of proximal development (ZPD).^{3} It appears that the authors discovered that students who were younger than the usual grade for algebra would be able to solve algebraic problems were it not for the fact that after several years of arithmetic a certain didactic contract was in place that hampered their algebraic explorations.

Another case where an appropriate didactic contract would be able to improve academic achievement was described by César (1998). The author observed that the didactic contract does have a profound impact on the self-esteem of the students and their behavior. Because it is the didactic contract that “legitimizes what both pupils and teachers expect from each other”. Many educational researchers have documented how challenged students improve their achievement. This challenging behavior by the teachers is a component of the didactic contract. However, the interactions between teachers and students, as well as between students, should occur in the ZPD for them to have a positive effect (César, 1998). For positive peer interaction to occur the teacher needed to establish an appropriate didactic contract for classrooms where such activities had not occurred previously. Basically the teacher had to explain the class how to productively collaborate in solving the mathematical problems.

Students behave differently even within the same didactic contract. Sarrazy (2002) calls this the responsiveness to the didactic contract. Another aspect of the fluidity of the didactic contract is the flexibility of the teacher herself. In the case of a “highly ritualized” teacher, the students will quickly figure out what the teacher requires as an answer and thus delude themselves in thinking that they have understood the problems. On the contrary, when a teacher has variable behavior it is difficult, if not impossible, for the students to pick up clues as to the correct answer. Thus, the students will become more involved in the process.^{4}

Why is the concept of didactic contract generally not used in the USA? Concepts in mathematical education are mental constructs, thus they can be expressed in a variety of ways, all valid in their domain. When they are developed by influential researchers, they enter into the pedagogical discourse and are legitimized by their use in publications, conferences and other means of distribution of ideas. Hence, language and cultural barriers can limit the use certain concepts among researchers. I suppose this has happened with the idea of didactic contract. We like to think of science as being universal, common to all, but in reality that is not the case. Steffe and Nesher (1996) does not use the term didactic contract and none of its authors quote Brousseau. Though Learning and Teaching Mathematics: An International Perspective, a more multicultural oriented publication, contains a whole chapter on this concept (Schubauer-Leoni & Perret-Clermont, 1997).

## B. Future Research Investigations

Presently, mathematics education researchers that employ the pedagogical concepts created by Brousseau seem to operate in a separate sphere from those who do not. However, I understand these concepts to fall under the umbrella of “social constructivism” (e.g. Steffe & Nesher, 1996, Part III). I do not foresee that social constructivism will be abandoned anytime soon. I think that those who have so far not used Brousseau’s terminology will continue not to use them because constructivism has already a sufficiently rich vocabulary.

I have not found any research that relates the didactic contract with the work of Michel Foucault on power and knowledge. As far as I can tell, while much has been written on Foucault and education (e.g. Scheurich & McKenzie, 2005; Ball, 1990), no one has investigated this area. In my opinion, a Foucaultian analysis of the power relationships in the classroom, which is another way of looking at the didactic contract, would be a fruitful endeavour.

## References

Alcock, L., & Simpson, A. (2001). The Warwick analysis project: Practice and theory. In D. Holton (Ed.), *The teaching and learning of mathematics at university level: An ICMI study* (pp. 99–112). Dordrecht: Netherlands: Kluwer Academic Publishers.

Amade-Escot, C. (2000). *How students manage the didactic contract? Contribution of **the didactic perspective to research in physical education classroom*. Available from p2048-ezproxy.tamu.edu.lib-ezproxy.tamu.edu:2048/login?url=http://search.ebscohost.com/login.aspx?direct=true&db=eric&AN=ED442786&site=ehost-live (Paper presented at the Annual Meeting of the American Educational Research Association, New Orleans, LA, April 24-28)

Ball, S. J. (Ed.). (1990). *Foucault and education: Disciplines and knowledge*. New York: Routledge.

Brito Lima, A. P., & Da Rocha Falcão, J. (1997). Early development of algebraic representation among 6-13 year-old children: The importance of didactic contract. In E. Pehkonen (Ed.), *Proceedings of the Conference of the International Group for the Psychology of Mathematics Education* (pp. 2.201–2.208). Lahti, Finland: International Group for the Psychology of Mathematics Education. Available from p2048-ezproxy.tamu.edu.lib-ezproxy.tamu.edu:2048/login?url=http://search.ebscohost.com/login.aspx?direct=true&db=eric&AN=ED416083&site=ehost-live

Brousseau, G. (1984). The crucial role of the didactical contract in the analysis and construction of situations in teaching and learning mathematics. In H.-G. Steiner (Ed.), *Theory of mathematics education: ICME 5 – topic area and miniconference: Adelaide, **Australia*. Bielefeld, Germany: Institut fuer Didaktik der Mathematik der Universitaet Bielefeld.

Brousseau, G. (1997). *Theory of didactical situations in mathematics* (N. Balacheff, M. Cooper, R. Sutherland, & V. Warfield, Eds.). Dordrecht, Netherlands: Kluwer Academic Publishers.

Brousseau, G. (2003). *Glossaire de quelques concepts de la théorie des situations didactiques en mathe**matisématiques*. Retrieved 1 October 2009 as pagesperso-orange.fr/daest/guy-brousseau/textes/Glossaire_Brousseau.pdf.

Cazes, C., Gueudet, G., Hersant, M., & Vandebrouck, F. (2006). Using E-Exercise Bases in mathematics: Case studies at university. *International Journal of Computers for Mathematical Learning, 11*(3), 327–350.

César, M. (1998). *Social interactions and mathematics learning*. Available from p2048-ezproxy.tamu.edu.lib-ezproxy.tamu.edu:2048/login?url=http://search.ebscohost.com/login.aspx?direct=true&db=eric&AN=ED463156&site=ehost-live (Paper presented at the Annual Meeting of International Mathematics Education and Society Conference, 1st, Nottingham, United Kingdom, September 6-11)

Dorier, J.-L., & Sierpinska, A. (2001). Research into the teaching and learing of linear algebra. In D. Holton (Ed.), The teaching and learing of mathematics at university level: An ICMI study (pp. 255–274). Dordrecht: Netherlands: Kluwer Academic Publishers.

Duffin, J., & Simpson, A. (2000). Understanding their thinking: the tension between the cognitive and the affective. In D. Coben (Ed.), *Perspectives on adults learning mathematics* (pp. 83–100). Dordrecht: Netherlands: Kluwer Academic Press.

Larochelle, M., & Bednarz, N. (1998). Constructivism and education: Beyond epistemological correctnesss. In M. Larochelle, N. Bednarz, & J. Garrison (Eds.), *Constructivism and education* (pp. 3–22). Cambridge, UK: Cambridge University Press.

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. 127–136). Dordrecht: Netherlands: Kluwer Academic Publishers.

Mason, J. (2001). Mathematical teaching practices at tertiary level: Working group report. In D. Holton (Ed.), *The teaching and learning of mathematics at university level: An ICMI study* (pp. 71–86). Dordrecht: Netherlands: Kluwer Academic Publishers.

Mason, J., & Johnston-Wilder, S. (2004). *Fundamental constructs in mathematics education*. London: RoutledgeFalmer.

Menotti, G., & Ricco, G. (2007). Didactic practice and the construction of the personal relation of six-year-old pupils to an object of knowledge: Numeration. *European Journal of Psychology of Education, 22*(4), 477–495. Available from p2048-ezproxy.tamu.edu.lib-ezproxy.tamu.edu:2048/login?url=http://search.ebscohost.com/login.aspx?direct=true&db=eric&AN=EJ781356&site=ehost-live

Nickson, M. (2000). Teaching and learning mathematics (2nd ed.). London: Continuum.

Sarrazy, B. (2002). Effects of variability of teaching on responsiveness to the didactic contract in arithmetic problem-solving among pupils of 9-10 years. *European Journal of Psychology of Education, 17*(4), 321–341. Available from p2048-ezproxy.tamu.edu.lib-ezproxy.tamu.edu:2048/login?url=http://search.ebscohost.com/login.aspx?direct=true&db=eric&AN=EJ824352&site=ehost-live

Scheurich, J. J., & McKenzie, K. B. (2005). Foucault’s methodologies. In N. K. Denzin & Y. S. Lincoln (Eds.), *The SAGE handbook of qualitative research* (3rd ed., chap. 33). Thousand Oaks, CA: SAGE Publications.

Schoenfeld, A. H. (2001). Purposes and methods of research in mathematics education. In D. Holton (Ed.), *The teaching and learning of mathematics at university level: An ICMI study* (pp. 221–236). Dordrecht: Netherlands: Kluwer Academic Publishers.

Schubauer-Leoni, M.-L., & Perret-Clermont, A.-N. (1997). Social interactions and mathematics learing. In T. Nunes & P. Bryant (Eds.), *Teaching and learning mathematics: An international perspective* (pp. 265–283). Hove, UK: Psychology Press Ltd.

Selden, A., & Selden, J. (2001). Tertiary mathematics education research and its future. In D. Holton (Ed.), *The teaching and learing of mathematics at university level: An ICMI study* (pp. 237–254). Dordrecht: Netherlands: Kluwer Academic Publishers.

Sensevy, G. (1996). Fabrication de problèmes de fraction par des élèves à la fin de l’enseignement élémentaire (Fabrication of fraction problems by students at the end of elementary schooling). *Educational Studies in Mathematics*, 30(3), 261–88. Available from p2048-ezproxy.tamu.edu.lib-ezproxy.tamu.edu:2048/login?url=http://search.ebscohost.com/login.aspx?direct=true&db=eric&AN=EJ525143&site=ehost-live

Shinkfield, D. (2007). *Turning round year 9. **Mathematics **Teaching Incorporating Micromath* (202), 36–39. Available from p2048-ezproxy.tamu.edu.lib-ezproxy.tamu.edu:2048/login?url=http://search.ebscohost.com/login.aspx?direct=true&db=eric&AN=EJ768924&site=ehost-live

Steffe, L. P., & Nesher, P. (1996). *Theories of Mathematical Learning*. Mahwa, NJ: Lawrence Erlbaum Associates.

Verscheure, I., & Amade-Escot, C. (2007). The gendered construction of physical education content as the result of the differetiated didactic contract. *Physical Education and Sport Pedagogy, 12*(3), 245–272.