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Response to the posting "When Is a Good Day Teaching a Bad Thing?" (by Timothy F. Slater) on Tuesday, September 14, 2004  --  from Prof./Dr. Heinrich Bauersfeld [Germany].
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Now we read of another magic spell, Slater's "Hidden Contract." And when we learn how to break the spell, it seems, everything will turn out better. Yet, it seems that our profession suffers from a growing historical blindness that may lead us to accept as pseudo-wisdom what, years before, was already developed in a sophisticated way.

In comparison with Slater's article, Brousseau's "contract didactique" notion is such a case. Slater's "contract" is a masquerade of bad teaching, while Brousseau's notion emphasizes productive interaction and the mutual obligations:

" ... in all didactical situations, the teacher attempts to tell the the students what she wants them to do. Theoretically the transition from the information and the teacher's instruction to the expected answer must require students to bring the target knowledge into play. We know, the only way to "do" mathematics is to investigate and solve certain specific problems and, on this occasion, to raise new questions. The teacher must therefore arrange not the communication of knowledge, but the devolution of a good problem. If this devolution takes place, the students enter into the game and if they win, learning occurs.

But what if a student refuses or avoids the problem, or doesn't solve it? The teacher then has the social obligation to help her and sometimes has to justify herself for having given a question that is too difficult.

Then a relationship is formed which determines - explicitly to some extent, but mainly implicitly - what each partner, the teacher and the student, will have the responsibility for managing and ... be responsible to the other person for. This system of reciprocal obligation resembles a contract. ... it is in fact the breaking of the contract that is important. Let us examine some immediate consequences.

-The teacher is supposed to create sufficient conditions for the appropriation of knowledge and must "recognize" this appropriation when it occurs.
- The student is supposed to be able to satisfy these conditions.
- The didactical relationship must "continue" at all costs.
- The teacher therefore assumes that earlier learning and the new conditions provide the student with the possibility of new learning.

If this learning does not occur, the student is put on trial for not having fulfilled what was expected of her, but so is the teacher for not having fulfilled what was expected (implicitly) of her. ... this interplay of obligations is not really a contract." (pp. 31-32 -- see below)

Readers interested in Brousseau's further analyses may look into: Brousseau, G.: Theory of Didactical Situations in Mathematics. (Edited and translated by N. Balacheff, M. Cooper, R. Sutherland, and V. Warfield), Dordrecht/Boston/London: Kluwer Academic Publishers 1997.
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SEE THE FOLLOWING URL FOR MORE INFORMATION REGARDING BROUSSEAU, FROM ICMI  --  http://mathforum.org/epigone/math-learn/jaiwumwoi

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-- Jerry P. Becker
Curriculum & Instruction
Southern Illinois University
Carbondale, IL  62901-4610
Phone:  (618) 453-4241  [O]
            (618) 457-8903  [H]
Fax:      (618) 453-4244
E-mail:   jbecker@siu.edu