Student Generalizations of Distance and the Cauchy Property

The Cauchy Property is an important characterization of convergent sequences in complete metric spaces. Students were observed reflecting on the nature of Cauchy Sequences on R, and were then prompted to generalize the definition of a Cauchy sequence into more abstract settings. Their generalizations were implemented into various metrics to eventually be leveraged for the re-invention of the abstract metric definition. This talk will explore some of the students' initial generalizing activity and will describe some of their cognitive schemes developed through interactions with various notions of distance.