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particular by social psychologists at Geneva (cf. the collective book edited
by Mugny, 1985). According to this theory, the contradiction coming from
two opposite points of view is more readily perceived and cannot be refuted
so easily as the contradiction coming from facts for an individual. The latter
may either not perceive the contradiction or not take it into account when
wavering between two opposite points of view and finally choosing one of
them. In order to master a task, students working jointly are committed to
overcoming conflict. When attempting to solve the contradiction, they may
manage to coordinate the two points of view into a third one overcoming
both initial points of view and corresponding to a higher level of knowl-
edge. This is the starting point for learning. The above-mentioned social
psychologists have tested the theory on the construction of general schemas
studied by Piaget, like the schema concerning the conservation of liquids or
of lengths. When we organized group work situations with students solving
mathematical problems, we could also observe the construction of a new so-
lution of higher conceptual level and the overcoming of the contradiction
between the partners. Let me give two examples:
In a situation in which two students had to describe a geometrical dia-
gram in a written message meant for two other students who did not know
the diagram, the labeling of some elements of the diagram by the producers
of the message often appeared as a solution overcoming the partners' dis-
agreement about their mutual formulations in natural language: Each pro-
posal was judged as erroneous or too complex by each partner and as pos-
sibly leading the receivers to a misunderstanding. Labeling some elements
provided a means that was accepted as an unambiguous and economical
way when describing elements depending on the labeled elements: Instead
of writing "the line joining the point we made to the other point we have
just drawn," they could write: "join Point A to Point B" (Laborde, 1982).
The example of a situation of ordering decimal numbers also illustrates
how students can construct a new correct strategy when they have to decide
between two strategies giving different results (Coulibaly, 1987). Leonard
and Grisvard (1981, 1983) have shown that sorting a sequence of decimal
numbers may pose a problem even for older students, and that with striking
regularity, two erroneous rules often underlie the students' solutions:
1. A rule R1 according to which among two decimal numbers having the
same whole part, the bigger one is the number with the bigger decimal part,
this latter being considered as a whole number; for example:
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WORKING IN SMALL GROUPS
0.514 > 0.6 because 514 > 6 or 0.71 > 0.006 because 71 > 6.
2. A rule R2 according to which among two decimal numbers having the
same whole part, the bigger one is the number with the decimal part having
the smaller number of digits; for example:
0.6 > 0.514 because 0.514 has three digits after the decimal point, while 0.6 has
only one digit after the decimal point, but 0.5 > 0.514 or 0.71 > 0.006.
One may be convinced of the strength of these rules insofar as, in some
cases, they provide correct results. Teachers are very often not aware of
these erroneous rules followed by their students, because they have access
only to their final answers and not to the reasoning leading to them.
Students are thus reinforced in their erroneous strategies. I leave to the
reader the pleasure to check that, when R1 and R2 give the same answer,
they are correct, while, when the results are contradictory, obviously only
one of them is false. But the consequence of this observation is important
from a didactical point of view. It implies that well-chosen numbers may al-
low the teacher or the experimenter to find which rule is followed by the
student in the task of sorting decimal numbers. We must indeed note that it
has very often been observed that a student's answers can be described by
only one rule.
The experiment carried out by Coulibaly determined the rules underlying
8th-grade students' answers to a written test. Four pairs of students were
formed by putting together students following different rules. Each pair then
had to jointly order five sequences of decimal numbers and to elaborate a
written explanation meant for other younger students on how to compare
decimal numbers. The sequences were carefully chosen in order to provoke
contradictions between R1 and R2. The first question gave rise to a conflict
for three pairs, and for two of them, the conflict led to a new rule R'1 over-
coming the contradiction: This rule consists in giving the same length to the
decimal parts by adding the adequate number of zeros to the right of the
shorter decimal part.
So Chrystel thought that 7.5 is less than 7.55, while Cecile argued for the
reversed order; Chrystel convinced Cecile by proposing that she puts the
same number of digits to both decimal parts: 7.5 equals 7.50 and 7.50 was
recognized by Cecile as less than 7.55.
This new rule, which is adapted from R1, avoids the application of R2
and overcomes the conflict. It never occurred in the prior written test. It is
noteworthy that these pairs elaborating the rule R'1 applied it in the next
questions and could formulate it in the explanation meant for younger stu-
dents.
Three consequences can be drawn from this example:
1. A social interaction could lead to a conflict, because of the choice of
the numbers to be compared and of the composition of the pairs (students
operating according to two different rules).
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2. A conflict did not systematically appear in all cases in which it could
have appeared.
3. Conflicts were not necessarily solved by the construction of a new rule.
This brings me to claim that the outcome of such social contradiction de-
pends on several factors, some of which can be more or less controlled,
such as the choice of the task variables of the problem given to the students.
(By task variables, I mean features of the problem whose variations imply
changes in the students' solving strategies; these variables, when used to
promote learning, are also called "variables didactiques," didactical
variables, in France.) The effect of the other ones linked to the individuals
involved in the interaction is more uncertain: A social negotiation between
two individuals is not predetermined, and all the past experience of each