Book · January 994 citations 110 reads 2,264 authors

All together, what is it?
Child: It is paper.
Teacher: What is it? not what is it made of? What's its name?
Child: It's written?
Teacher: No.
Child: . . . (silence)
Teacher: You have said that it has eyes, a nose, and so on. What is it?
Child: . . . (silence)

---

MARIA G. BARTOLINI BUSSI
This episode is taken from the observation protocol of a one-to-one interac-
tion between an elementary school teacher (Bondesan, personal
communication) and a low achiever (1st grade): The child already knows
the teacher and the climate is very relaxed. This special interaction (a
remedial workshop) was designed for low achievers in order to foster the
development of planning and designing strategies by means of verbal
language as a prerequisite for mathematical problem-solving (Boero, 1992).
The goal of this session is to build a copy of the puppet while verbalizing
the process. The child is a 1st grader with learning disabilities; she is not
handicapped, but she has lacked family experiences of joint activity in
which action is systematically accompanied by speech. As the protocol
shows, she can name the different parts of the object, but cannot name the
whole. The teacher feels responsible for unblocking the child, because of
institutional needs (the very purpose of that remedial workshop) and for
personal needs (the "revolutionary" will to offer equal opportunities to
every child). What has theory to offer her? Two radical competing positions
are offered by Piagetian versus Vygotskyan researchers: act as a clinical
interviewer, encourage the child to express herself and to build her own
knowledge; act as a guide, help the child, lend her the right gestures and
words. Actually, the teacher behaved as a Vygotskyan and successfully
offered the child actions and utterances to be imitated; maybe, being
Piagetian, in this radical sense, could have resulted in abandoning the child
to her destiny.
4.2 When Mathematical Behavior is Against Everyday Behavior
The problem of mathematical proof seems to be one of the crucial issues of
didactics where advanced thinking is concerned. Balacheff (1990b) studied
the students' treatment of a refutation by means of social interaction. His
work confirmed the usefulness of social interaction, but enlightened its lim-
its too, because of the major role played by argumentation. In a specific
study on deductive thinking, Duval (1991) showed that the rules of deduc-
tive reasoning are very different from the rules of argumentative reasoning.
The strategy that the same author experimented successfully to make the
students (aged 13-14) distinguish between argumentative and deductive rea-
soning is supposed to be more Vygotskyan than Piagetian (actually, in the
paper, disagreement with Piaget is explicitly stated even if Vygotsky is not
referred to): They were given the rules for building an oriented proposi-
tional graph, to connect hypotheses to conclusions (a good example of
semiotic mediation). We could even be critical about such an introduction of
rules to be followed if they are perceived by students as rules of classroom
contract only. Yet what seems to me unquestionable is that deductive rea-
soning depends on social factors: When students are approaching
Teacher: Is it a child?
Child: . . . (silence)
129

---

130
mathematical proof, they are entering a flow of thought that was (and still
is) developed outside school by mathematicians, together with a related
system of values as well as of acceptable behaviors. To cope with this
problem, it is not sufficient to consider mathematics as an individual
subjective construction, it is necessary to consider mathematics as a