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 a recognition and differentiation between the objects and relations involved in the solution of the problem; in other words a cognitive construction."
The question is then: How can we as teachers create opportunities to enable this cognitive construction?
This will be the basis of my next post.