Summation notation is a widely-used standard that can represent all kinds of sums. Despite its utility, the literature on this topic points to the notation being difficult for students. Our research project gives insight into how students think about summation notation and why it is so challenging. This report builds off of the first phase of our project, which proved the existence of students’ uncertainties with elements of the notation. This work considered survey data from 285 undergraduates and suggested that uncertainties are common amongst students. We also found that the act of encoding a sum in sigma notation is more cognitively demanding than interpreting a summation notation expression. In this talk we present models of students’ ways of thinking about summation notation.

Contrary to the norm in every-day Anglo-English conversation (and presumably in students' collaborations) that silences longer than a second be avoided, preliminary research and anecdotal evidence suggest that during their collaboration mathematicians spend lengthy periods of time in nearly motionless silence. How can we account for this seeming lack of interaction during mathematicians' collaborations? In what way can what seem to be lengthy pauses without interaction be part of mathematicians' collaborations? This talk will present examples of silence in mathematicians' and students' collaborations, and attempt to make sense of the some of these paradoxes surrounding mathematicians' collaborative silences.