Although lectures remain the dominant teaching method in higher education, their effectiveness in improving student understanding is widely questioned. Nevertheless, lectures provide teachers with an opportunity to build on students’ prior knowledge and introduce new concepts by using words, drawings and symbols. Lectures may also contain examples and prepare students for subsequent activities, such as proofs and exercises. As there is little interaction with students, the effectiveness of lectures depends on the discourse used by the teacher.
In this context, the aim of this study is to investigate how university lecturers introduce the formal definition of function limits to first-year students, paying particular attention to the role of teachers’ discourse. More specifically, we examine how teachers use discourse to connect new mathematical knowledge to students’ prior understanding. Our analysis focuses on the concept of ‘discursive proximities’, referring to the explicit links that teachers establish between what students already know and the new concepts being introduced. We distinguish three types of discursive proximities: bottom-up proximities link prior knowledge to new concepts; top-down proximities connect theoretical explanations to examples or applications; horizontal proximities consist of reformulations and links between different semiotic registers, such as natural language, symbolic notation, and graphical representations.
In this ongoing research project, we analysed the discourse of two teachers: one during a lecture and the other during a video lesson. The study of proximities first relied on the construction of a “relief” of the concept of limits, combining epistemological, curricular, and cognitive perspectives. This preliminary analysis made it possible to identify opportunities of proximities for establishing connections between different forms of knowledge and representations.
In the lecture, the teacher begins by eliciting students’ intuitive understanding of limits, using graphical representations to illustrate the idea of a function approaching a value. The discourse involves rephrasing the notion of “closeness” using both natural language and mathematical symbols, particularly through inequalities involving absolute values. These instances constitute opportunities for horizontal proximities, as they aim to link different representations of the same concept. However, when introducing a function designed to highlight the need for a rigorous definition, the teacher does not build on knowledge familiar to students, thereby missing opportunities for bottom-up proximities. Furthermore, discrepancies between oral explanations and written formalism may hinder students’ understanding.
A similar pattern is observed in the video lesson. The teacher presents the formal definition and alternates between a symbolic formulation and intuitive explanations based on the idea of being “arbitrarily close” or “sufficiently close.” Graphical representation and gestures are used to support the explanation. While these elements suggest an attempt to establish horizontal proximities, the links between the different representations are not always made explicit. Consequently, students may find it difficult to understand how the verbal explanations correspond to the formal mathematical expressions.
Overall, this study shows that, although teachers often try to support student understanding through horizontal proximities, these attempts are often implicit and predominantly oral. The lack of explicit connections and written traces may reduce their effectiveness, particularly for students who do not spontaneously make the necessary links between representations. These findings emphasise the importance of establishing more explicit connections in teaching practices and indicate the need for further research to improve our understanding of how different forms of discourse can enhance students’ understanding of abstract mathematical concepts such as limits.

Footnote: This contribution is part of the presentation given at BeNeLux Mathematics Conference (BeNeLux MC), with the title: Evidence-Based Practices in Teaching Mathematics at the University Level, organised by 4TU+.AMI, Annoesjka Cabo