I believe in articulate discussion (in monologue or dialogue) of how one solves problems, of why one goofed that one, of what gaps or deformations exist in one's knowledge and of what could be done about it. I shall defend this belief against two quite distinct objections. One objection says: "it's impossible to verbalize; problems are solved by intuitive acts of insight and these cannot be articulated." The other objection says: "it's bad to verbalize; remember the centipede who was paralyzed when the toad asked which leg came after which."

J.S. Bruner tells us (in his book Towards a Theory of Instruction) that he finds words and diagrams "impotent" in getting a child to ride a bicycle. But while his evidence shows (at best) that some words and diagrams are impotent, he suggests the conclusion that all words and diagrams are impotent. The interesting conjecture is this: the impotence of words and diagrams used by Bruner is explicable by Bruner's cultual origins; the vocabulary and conceptual framework of classical psychology is simply inadequate for the description of such dynamic processes as riding a bicycle. To push the rhetoric further, I suspect that if Bruner tried to write a program to make an IBM 360 drive a radio controlled motorcycle, he would have to conclude (for the sake of consistency) that the order code of the 360 was impotent for this task. Now, in our laboratory we have studied how people balance bicycles and more complicated devices such as unicycles and circus balls. There is nothing complex or mysterious or undescribable about these processes. We can describe them in a non-impotent way provided that a suitable descriptive system has been set up in advance. Key components of the descriptive system rest on concepts like: the idea of a "first order" or "linear" theory in which control variables can be assumed to act independently; or the idea of feedback.

A fundamental problem for the theory of mathematical education is to identify and name the concepts needed to enable the beginner to discuss in mathematical thinking in a clear articulate way.

-- Seymour Papert, distinguished mathematician, educator, computer scientist, and AI researcher, in his 1971 essay "Teaching Children to be Mathematicians vs. Teaching about Mathematics".