Students entering STEM programmes often struggle with mathematics courses, which can lead to high attrition rates [1]. Factors that contribute to this challenge include changes in the nature of the mathematical content and in the amount of support that students receive from instructors. Importantly, university mathematics is much more reliant on conceptual understanding and formal reasoning. This can be viewed as a rift between the mathematical discourse favoured in secondary school and that employed in university [2]. Prior research has indeed identified differences in the discourses of first-year students and their instructors (including their vocabulary and ways of doing), which are thought to complicate learning in first-year lectures [3,4]. However, the process by which students initially imitate, and then internalise mathematical discourse has received less attention. In this study, we focus on how students develop conviction about mathematical statements. In particular, we explore how first-semester students assess the validity of mathematical claims, how they substantiate or refute them, and how these two processes relate to one another. Using video-recorded interviews where student pairs solved a true/false mathematical task, we find two broad trends in how mathematical conviction arises. The first trend involves an intuitive-to-formal approach where students first attempt to develop intuition and only after proceed to formal substantiation. The second trend involves a formal-to-intuitive approach, where students first develop a formal argument to prove or disprove the claim, then seek to acquire intuition that matches their formal conclusion. We argue that the latter approach, though aligned with mathematically literate discourses, can be challenging for students who are simultaneously developing the procedures involved in mathematical substantiation.