Content and form in mathematics

The present article is based on the premise that mathematical activity exhibits an important duality. On the one hand pre-school children do mathematics spontaneously; the mathematics they do is often called informal arithmetic. On the other hand, the mathematics we teach is a cultural product developed by generations of mathematicians (Byers and Herscovics, 1977). Whatever premium school mathematics puts on individuality and original thought, it seems obvious that the mathematics done in school is neither spontaneous nor informal; the bulk of it is formal mathematics. We expect our pupils to do mathematics in a conventionally acceptable manner. In short, mathematics is both an individual and a social activity. We shall endeavour to show that duality in mathematical activity reflects a duality of mathematics as a discipline. The latter is a duality between content and form; in fact, it is the importance of form in this discipline that resulted in mathematics being classified as a formal science.