tion supporting a conjecture is very often accompanied by arguments showing this structure, so that passing from conjecturing to proving would require transformation from an abductive into a deductive structure. This passage presents difficulties that seem to require specific didactical treatment to overcome."
Balacheff explores the difficulties in moving from argumentation to mathematical proof in more detail: "Teachers organize situations in order to “convince” or “persuade” learners (in the vocabulary of Harel & Sowder).
Argumentation seems the best means to this end. It works both as a tool for teaching and as a tool for doing mathematics for a long while. But then learners suddenly face an unexpected revelation: In mathematics you don’t argue, you prove… It must be remembered that at stake is not truth but the validity of a statement within a well-defined theoretical context.
The arguments in such a discussion involve three types of critical considerations: the search for certainty, the search for understanding and the requirements for a successful communication. The complex nature of proof lies in the fact that any effort to improve a proof on one of these dimensions may change its value on the other two. There is no clear standard to decide on the correct balance.
What is produced first is an 'explanation' of the validity of a statement from the subject’s own perspective. This text can achieve the status of proof if it gets enough support from a community that accepts and values it as such; it can be claimed as mathematical proof if it meets the current standards of mathematical practice.
Students have to learn mathematics as social knowledge, they are not free to choose the meanings they construct. These meanings must not only be efficient in solving problems, but they must also be coherent with those socially recognized
Being the product of a choice, actions are considered in terms of validity and the adequateness of effect. Action is related to its aim, and a contradiction becomes apparent when the aim is not fulfilled.
Language must become a tool for logical deductions and not just a means of communication. The elaboration of this functional language requires in particular:
- a decontextualisation, giving up the actual object for the class of objects, independent of their particular circumstances;
- a depersonalisation, detaching the action from the one who acted and of whom it must be independent;
- a detemporalisation, disengaging the operations from their actual time and duration: this process is fundamental to the passage from the world of actions to that of relations and operations."
Balacheff describes the process of proof as consisting of: “...four main types of 'proof': naive empiricism, the crucial experiment, the generic example and the thought experiment."
He gives many examples of these proof types in his papers, and provides the following analysis: “[we] assert the existence of a break between naive empiricism and the crucial experiment on the one hand, and the generic example and the thought experiment on the other. This divide can be characterised as one of passing from a truth asserted on the basis of a statement of fact to one of an assertion based on reasons. We find here an example of an operational cohabitation between empirical pragmatism and logical rationalism...
Although naive empiricism disappears once conceptual proofs are brought into play, the crucial experiment can continue as an ultimate test to guarantee conviction, most noticeably when the assertion has been founded on a generic example.
The crucial experiment means something different in social interaction, where it becomes a means of resolving completely a conflict over the validity of an assertion, or over the choice between two conjectures. We no longer consider it as a means of proof except when it constitutes the refutation of an assertion.
Another connection is that between the generic example and the thought experiment. Passing from the former type of proof to the latter relies on a linguistic construction which involves