Abstract
In this article, we present episodes from two qualitative research studies. The studies focus on students of different ages and populations and their work on different mathematical tasks. We examine the commonalities in environment, tools, and teacher-student interactions that are key influences on the positive dispositions engendered in the students and their interest and engagement in mathematics. In addition, we hypothesize that these positive dispositions in mathematics lead to student reasoning and, thus, mathematical understanding. The resulting framework is supported by other educational research and suggests ways that the standards can be implemented in diverse classrooms in order to achieve optimal student engagement and learning.
