Title

Didactics of Mathematics: What is It? Examples of the French Contribution 

Abstract

Research in didactics is the study of learning and teaching processes. Because all countries are faced with the problem of the massive failure of students in mathematics, didactics of Mathematics has developed more rapidly than didactics for other disciplines. The main characteristics of didactics, comparatively with pedagogy, is the fact that it studies processes that are more specific of the contents of knowledge. Our research methods:

1. Experiment in the class-room with situations likely to favor and encourage the students’ activity and the identification of relationships;
2. Analyse the contents of students’ activity from a conceptual point of view: identify the implicit mathematical concepts and theorems structuring the students’ activity. What is it that is held to be relevant? What is it that is held to be right? Or wrong?
3. Describe the development of the students’ activity and the conceptualizing process; identify different reasonings, different errors, and different steps. What is it that can be made more explicit? Which obstacles are individual students faced with? "some individuals more than others"? How do they overcome these obstacles, with the help of the teacher, or without?

Observation is essential; but it is also essential to interpret theoretically the observed facts and processes. Didactics makes epistemology crucial. Progress takes place in the short term of activity in the classroom, but also in the long term development over several years (sometimes many years). Some theoretical frameworks, in France, are the theory of situations (see Brousseau), the theory of conceptual fields (see Vergnaud), the pragmatic and anthropological approach (see Chevallard).

Selected References 

• Vergnaud G (1982) A classification of cognitive tasks and operations of thought involved in addition and subtraction problems. In: Carpenter TP, Moser JM, Romberg TA (Eds). Addition and Subtraction: a cognitive perspective. Hillsdale-NJ. Lawrence Erlbaum, pp. 39-59
• Vergnaud G (1983) Multiplicative Structures. In: Lesh R, Landau M (Eds). Acquisition of mathematics concepts and processes. Academic Press, pp. 127-174
• Vergnaud G et al. (l990) Epistemology and psychology of mathematics education. In: Kilpatrick J, Nesher P (Eds). Mathematics and cognition. Cambridge, Cambridge University Press, pp. 2-17
• Vergnaud G (1996a) The theory of conceptual fields. In: Steffe LP, Nesher P, Cobb P, Goldin GA, Greer B (Eds). Theories of Mathematical Learning. Mahwah, Lawrence Erlbaum Ass
• Vergnaud G (1996b) Some of Piaget’s fundamental ideas concerning didactics. Prospects 26/1:183-194
• Vergnaud G (1999) A comprehensive Theory of Representation for Mathematics Education. Journal of Mathematical Behavior. Special issue on représentation 17(2):167-181

In French and other languages

• Vergnaud G (1981) L’enfant, la mathématique et la réalité, Berne, Peter Lang, Translation in Spanish (1991) El Niño las Matemáticas y la Realidad. México, Trillas; in Italian (1994); in Russian (1998); in Portuguese (2010) A criança, a matemática e a realidade. Curitiba, UFPR

Other Referenes in French language

•  Vergnaud G, Bregeon JL, Dossat L, Huguet F, Myx A, Peault H (1997) Le Moniteur de Mathématiques. Cycle 3. Paris, Nathan
• Vergnaud G (2000) Lev Vygotski pédagogue et penseur de notre temps. Paris Hachette Education
• Vergnaud G (Ed) (1983) Didactique et Acquisition du Concept de Volume. N° spécial de Recherches en Didactique des Mathématiques. vol 4  
• Vergnaud G, Brousseau G, Hulin M (Eds) (1988) Didactique et Acquisition des Connaissances Scientifiques. Actes du Colloque de Sèvres, May 1987, Grenoble, La Pensée Sauvage

Contact

vergnaud99@gmail.com