They provide the intellectual contexts for students' mathematical development. Discourse refers to the ways of representing, thinking, talking, and agreeing and disagreeing that teachers and students use to engage in those tasks. The discourse embeds fundamental values about knowledge and authority. Its nature is reflected in what makes an answer right and what counts as legitimate mathematical activity, argue and thinking. Teachers, through the ways in which they orchestrate discourse, convey messages about whose knowledge and ways of thinking and knowing are valued, who is considered able to contribute and who has status in the group.
Environment represents the setting for learning.
It is the unique interplay of intellectual, social, and physical characteristics that shapes the ways of knowing and working that are encouraged and expected the classroom. Analysis is the systematic reflection in which teachers engage. It entails the ongoing monitoring of classroom life-how well the tasks, discourse, and environment foster the development of every student s mathematical literacy and power. Through this process, teachers examine relationships between what they and their students are doing and what students are learning. In deciding how to present and elaborate the ideas underlying each of the six standards, we confronted two basic dilemmas.