The central hypothesis of this paper is that physics students learn to understand equations in terms of a number of conceptual elements that are referred to as "symbolic forms." Each symbolic form associates a simple conceptual schema with a pattern of symbols in an equation. Taken together, the set of symbolic forms constitutes a vocabulary of elements out of which novel expressions can be constructed, and in terms of which expressions can be understood. The work described here is based on an extensive analysis of a corpus of videotapes of moderately advanced university students solving physics problems.

Student engagement in classroom activities is usually described as a function of factors such as human needs, affect, intention, motivation, interests, identity, and others. We take a different approach and develop a framework that models classroom engagement as a function of students' . conceptual competence in the . specific content (e.g., the mathematics of motion) of an activity. The framework uses a spatial metaphor-i.e., the classroom . activity as a territory through which students move-as a way to both capture common engagement-related dynamics and as a communicative device. In this formulation, then, students' engaged participation can be understood in terms of the nature of the " regions" and overall " topography" of the activity territory, and how much student . movement such a territory affords. We offer the framework not in competition with other instructional design approaches, but rather as an additional tool to aid in the analysis and conduct of engaging classroom activities. © 2011 Elsevier Inc..