Mathematical routines in transition: Facilitating students' defining and proving of sequence convergence

The transition to tertiary mathematics requires students to use definitions of mathematical objects instead of intuitions. However, routines of defining and of proving with definitions are difficult to engage in, as they are not familiar to students who come from secondary school mathematics. Defining is highly complex because of its underlying rules, which demand that a definition has to be minimal, formal, deductively ordered, operationalized and so forth. While students can learn these rules by the teacher making them explicit during defining, this paper illustrates a trajectory that aims to systematically build on students' familiar secondary school routines of classifying, describing, symbolizing and proving as resources for learning defining and proving with definitions. This trajectory is informed by a five-session teaching-learning intervention in which upper-secondary students in Germany are engaged in defining and proving with definitions and which specifically focuses on systematically building on students' prior experiences in secondary school mathematics.