The Role, of Negotiation in Mathematics Learning
Grayson H. Wheatley
This chapter considers the nature of mathematics learning with particular attention to the role of negotiation. This issue rarely arises in direct instruction ( Confrey 1990) since the teacher is viewed as an authority who prescribes how mathematics is to be done. However, when instruction that is informed by constructivism ( von Glasersfeld 1987) is considered, negotiation plays a prominent role.
Negotiation takes place in a social and physical environment that greatly influences the beliefs, intentions, stances, and actions of the participants. Thus, we cannot easily speak of negotiation in general, but only analyze negotiations within particular settings. Political negotiation is characterized by compromises. In negotiating mathematical meaning there is little room for compromise. Unless there are different interpretations of the task, there is usually one answer or one set of answers. This chapter describes a negotiation between two ninth grade boys who had participated in a mathematics program called "Mathematics Achievement through Problem Solving" (MAPS) ( Wheatley, et al. 1988) designed for noncollege-intending students. The central instructional strategy of MAPS is problem-centered learning ( Wheatley 1991). A goal of problem-centered learning is the construction of mathematical knowledge by students. A construction (1) can be explained and justified, (2) has internal consistency, (3) can be reflected upon, and (4) is embedded in other knowledge.
Problem-centered learning (PCL) has three components: tasks, groups, and sharing. Primarily, in preparing for class, a teacher selects tasks that have a high probability of being problematical for students (tasks that may cause students to find a problem). A rich mathematical activity has the following characteristics: