How do students revisit school mathematics in modular arithmetic?:conditions and affordances of the transition to tertiary mathematics with a focus on learning processes

In the transition from secondary to tertiary mathematics, students try to participate in tertiary mathematics by replicating familiar school mathematical discourses. The objective of this case study is to investigate the conditions and affordances under which students proceed from familiar school mathematical discourses to new, tertiary discourses with a specific perspective on learning processes. The study was located in an upper secondary transition course in a teaching unit on elementary number theory in which highly proficient students in their penultimate year of schooling participated. The main finding is that there was not a linear progression towards the intended tertiary mathematical discourse of modular multiplication but a coexistence of two discourses developing in parallel: the intended tertiary discourse and a continuation of school mathematical discourses. Students see these two discourses as the same, connected by the same perceived aim of searching for patterns. Further hindering discursive development are utterances, in which elements of the intended tertiary discourse (words and representations) are used superfluously, so that elementary/secondary utterances appear to be tertiary. These findings illustrate the need to explicate and address metanarratives. Metanarratives can connect two discourses developing in parallel in ways that, in this case, hinder discursive development towards the intended tertiary discourse.