In this part of the book, I illustrate the workings of the commognitive approach by applying it to the special case of mathematical thinking. In so doing, my intention is to show what difference commognitive analysis makes in our interpretation of observed phenomena and in our practical decisions about teaching and learning. The discussion will eventually take me back to the dilemmas presented in chapter 1. The hope is that when scrutinized with the commognitive eye, at least some of the puzzles will be solved, whereas some others may disappear.

Being interested in learning, I focus in my analysis on the development of mathematical discourses of individuals, but I also refer to the historical development of mathematics whenever convinced that understanding this latter type of development may help in understanding the former. Considering the fact that communication is inherently collective, the term discourse of an individual or personal discourse may seem to be an oxymoron. Indeed, borrowing Ed Hutchins's words, one can say that those who equate human development with the development of discourses “move the boundaries of the cognitive analysis out beyond the skin of the individual person” and start speaking, instead, about teams of discourse participants as “commognitive systems.” Let me repeat then that thinking has been defined as self-communication.