The social perspective

The main construct from the social perspective is ‘pattern of interaction’, other related constructs are ‘negotiation of meanings’, and ‘taken as shared’. These constructs will be outlined in this section.

Communication is one important notion accounting for the social or collective perspective. The individual student must actively construct meanings from the experiences in the environment. Language is seen to be the main medium for the individual’s available meanings. However, language cannot be transmitted.

A speaker's utterances can function for the listener like ‘pointing at something’ on- ly, or better as directing the focus of attention, whereas the construction of what might be meant, the construction of references, is with the listener. The speaker's utterances and intentions have no direct access into the listener's system. What the listener’s senses receive, undergoes spontaneous interpretation” (Bauersfeld, 1995, p. 273).

For the interactionists this includes more than language; also gestures, body language etc. (Bauersfeld, 2000). The communication in the class- room is seen as a process in which the participants mutually adapt to each other. This is done through negotiation of meanings (Bauersfeld, 1980, 1994).

Negotiation of meanings

Negotiation of meanings (Bauersfeld, 1995; Voigt, 1994, 1995) concerns the notion of the interaction going on in the classroom. With the view that meanings are not transmitted by language, and that each individual con- structs its own interpretation from what is experienced, it is not possible to know if two people are assigning the same meaning to one utterance. It is evident from classroom situations that teachers and students often differ in what meaning they assign to an object of classroom discourse. To find a way to resolve this problem of ambiguity, the participants in the class- room have to negotiate meanings in order to arrive at a meaning that is taken-to-be-shared. They are continually modifying their interpretations.

This does not mean that this meaning is held by the individuals in the classroom, but that each individual meaning is compatible with the others. Meaning-taken-as-shared

Meanings-taken-as-shared is the notion for the meanings negotiated in the classroom, but does not mean that these meanings are actually shared. The discourse is developing as if the participants hold the same conceptions.

Students in the classroom and the teacher come to the classroom with different background experiences and thus interpret the classroom situa- tions in different ways. The tension between these different interpretations Voigt (1989) claims to be the motor for the negotiation of the meaning in the classroom. In spite of these tensions the classroom discourse mostly happens to go on in a smooth way because of a “provisional willingness to cooperate” (ibid, p. 652). This willingness is based on a tacit and implicit agreement, but there is always a risk that the discourse will break down. However, regularities constituted in the classroom minimises the risk. Routine actions are one type of regularities. One example of such a rou- tine in the mathematics classroom is for students to reduce verbal utter- ances to numbers and key words, which enables the teacher and other stu- dents to identify their own expected meanings in what is said.

The term taken as shared describes the participants' conviction that meanings are shared, or the participants' willingness to neglect doubts in view of inevitable am- biguities, or the presumption that the meanings will be shared if the others will "read between the lines." (Voigt, 1995, pp. 172-173).

As time goes on in the classroom, the participants in the classroom consti- tute a culture through the negotiation of meaning, which leaves the stu- dents holding an impression that they know what mathematics and math- ematics learning is (ibid). Cobb called this the ‘institutionalized

knowledge’ which might differ from one classroom to another (Cobb, 1989).

Patterns of interactions

Patterns of interactions are to be found in all social groups. When observ- ing everyday life, it is possible to reconstruct patterns of interactions, so also in the mathematics classroom. The teacher and the students are most- ly unaware of them (Bauersfeld, 1995, 2000; Voigt, 1989, 1995):

If the observer looks at the classroom life in the way an ethnographer does who investigates a strange culture, the observer might be astonished by what is taken for granted by the members of this classroom culture. However, in the treadmill of everyday life, the participants would say that they know what mathematics and the classroom practice really are. In everyday classroom situations, the teacher and the students often constitute the context routinely without conflicts and without being aware of the ongoing accomplishments. So, in the participants' experiences the context can seem to be pre-given. In everyday classroom practice the teacher and

the students assume that the context is known which is, in fact, taken-to-be-shared, vague, and implicit (Voigt, 1994, p. 181).

In mathematics classrooms the negotiation of meaning is fragile. One rea- son is that the mathematical objects in focus are ambiguous to students. Patterns of interaction minimises the risk of collapse, in that the students take their classroom culture to be given and definite, although it is consti- tuted jointly by the participants in the classroom.

Routines reduce the complexity in the classroom and are necessary el- ements of the classroom. In addition, both teacher and students feel obli- gations of different kinds in ongoing discourses. When a mathematical task, for example, is intended by the teacher to give students the oppor- tunity to construct conceptual knowledge, and the students do not respond as expected, the teacher may feel obliged to make the problem easier, and to lead students step by step, resulting in students focusing on the proce- dures. At the same time students might feel obliged to react to teacher’s question in a way they interpret teacher’s expectation to be (Voigt, 1989). This will result in the kind of pattern which is called the funnel pattern or elicitation pattern, in which the teacher leads the students in small steps to the solution of the problem.

Obligations and routines are prerequisites for patterns of interactions. These patterns are always jointly constituted and offer one reason for classroom discourses to go on, and to develop without too much effort and without being ever changing (Bauersfeld, 2000; Voigt, 1995). They might, however, be an obstacle for learning mathematics if the patterns turn out to be only joint work in order to arrive at a correct solution narrowed down to small steps (Steinbring, 1989; Voigt, 1989, 1995). It is also a risk for observers to equate successful participation in patterns of interactions with the learning of mathematics. The routines and felt obligations make students think that they know what classroom praxis is and should be (Voigt, 1995).

The result is that it is often difficult to change practice, in that students avoid changes from what is understood to be mathematics and mathemat- ics classroom. Often a culture is constituted in the classroom in which mathematical symbols are mostly connected to particular conventions and methodical rules (Steinbring, 1997, p. 50). Mathematics might then be experienced as rules and conventions rather than as relations and struc- tures.