Didactics of mathematics: an epistemological approach to mathematics education

Abstract

I present some key ideas and examples of (mainly French) work in the didactics of mathematics with the aim of identifying the didactics approach to education in relation to more general approaches. In particular I discuss three crucial features of didactics:

	its epistemological character, i.e. the central focus on specific knowledge in different forms, and the conditions for enabling pupils to acquire it;
	its relations to the background science (in casu, mathematics as a science);
	its actual and potential roles in relation to teaching and teachers (in particular, didactics as a design science and as a part of teachers' education and professional knowledge base).

Keywords:

• Didactical situation
• Didactical transposition
• Didactics
• Mathematics

Notes

1. The equivalent of the term ‘pedagogical research’ is often used in other European languages to designate the general study of educational activities, institutions, processes, etc.

2. No doubt, a note is in place to explain how this word is to be understood in this article (and, increasingly, in the literature). In fact, the noun didactics will be used here to express in English what the same noun is commonly used to mean in German (Didaktik), French (didactique), Danish (didaktik), etc. In short, I mean here the study of the teaching and learning of specific knowledge, usually within a disciplinary domain. A more extensive discussion of the advantages of using the word ‘didactics’ in English too was given by Chevallard (1999).

3. Similar polygons are, informally, ‘of the same shape’. In mathematical terms, two triangles are similar if they have two common angles. More generally, two polygons A and B are similar if they can be subdivided into corresponding patterns of similar triangles.

4. In other words, the curriculum for a teaching context (discipline, level, institution, etc.) refers, in this article, to directives and materials produced outside the school institution and which the teachers must or could use when constructing their teaching within this context. I am, of course, aware that there are several other uses of this term.

5. The biannual summer schools are organized by the ARDM (Association pour la recherche en didactique des mathématiques). Participation is limited to 120 scholars; 40 places are reserved for non-French participants.

6. A decimal number is just a sum of quantities with respect to different units, e.g. 709.03 = 7·100 + 9·1 + 3·10⁻².

7. Once the question has been asked ‘what will happen to a side of 1 cm?’ (even if no such side is present!), one can reason as follows: 4 cm is made up of four units; if these units are enlarged, together they become 7 cm; so each of them will become 7 cm/4, that is 1.75 cm. If 1 cm is enlarged to 1.75 cm, then 2 cm (made up of two units) becomes 2 times 1.75 cm, etc. In general, a side of length x cm (made up of x units) becomes x times 1.75 cm.

8. It is interesting to note that university positions in France within didactics of mathematics are usually classified in the section of applied mathematics, even if few of them are within departments of mathematics.

9. In fact, research on mathematics teacher education is currently an emerging field. It remains to be seen if it will eventually develop into a didactics of didactics, i.e. a systematic study of the conditions for teaching and learning particular pieces of didactical knowledge.

10. The hesitancy of mathematicians to admit the need or worth of didactics could perhaps also be interpreted as an instance of a more general scepticism, among mathematicians, with respect to educational research. This affects also the Anglo-American ‘math ed’ research, sometimes in quite radical ways that have given rise to the notion of ‘math wars’. When didactics is (dis)placed in faculties of humanities and the like, the separation from mathematics is institutionally amplified. Alas, the didactician may not even find a pleasant exile in this situation: ‘Other communities (especially in psychology or pedagogy) have a very false idea of our area of investigation’ (Brousseau, 1999, p. 47).

11. A similar vision for the didactics of science is proposed by Lijnse (2000).

12. The expression is attributed to the famous mathematician Euler (1707–1783) (cf. Martinez, 2006, p. 37). Nowadays imaginary numbers are used, as a basic and uncontroversial construct, in many areas of mathematics.