Reflective processes in a mathematics classroom with a rich learning environment

This article examines reflection in a rich mathematics learning environment characterized by multirepresentational tools and a sequence of problem situations. The study focuses on the development of activities and objects or artifacts during 1 problem situation at the end of a year-long course. Students worked first individually during a preparatory phase. Then, they collaborated to solve a problem in small groups and subsequently wrote group reports. Finally, they engaged in a teacher-led discussion in which all students verbally reported on the processes they underwent, criticized them, and reflected on their learning styles. Activity theory was used to frame the study of reflection within the multiple interactions in the classroom and to cope with the changing context for bestowing meaning to individual actions (the activity), what is constructed (objects or outcomes), and what mediates this construction (artifacts). In an initial problem-solving phase, small groups of students engaged in a reflective discourse (Cobb, Boufi, McClain, & Whitenack, 1997) in which they talked about mathematical objects, such as hypotheses and conflicts among them, and in which actions of individuals were embedded in the social interactions of the group. Producing written reports of the problem-solving phase revealed a process of purification through which students preserved the deep structure of the chain of their actions in the problem-solving phase but suppressed details and regressions. Under the orchestration of the teacher during whole-class discussion, purification progressively yielded judgments about properties of previous actions. Reporting was thus a social practice through which private artifacts became the property of the community. With the help of the teacher, the artifacts were appropriated or rejected by the community on the basis of their properties; these artifacts mediated the construction of high-level mathematical objects. Thus, phenomena such as purification or appropriation stemmed from social interactions through the mediation of the teacher.