A central educational goal of higher-level mathematics undergraduate courses is to foster the development of competences that enable students to use mathematical abstraction and formalism and, at the same time, to develop mathematical creativity. Does it mean mathematical maturity? Research in university mathematics education suggests that to grasp the characteristics of maturity, it might be beneficial to introduce students to theoretical objects by engaging them in the autonomous construction of examples, conjecturing, and proving, especially in geometry. This approach is generally fruitful, but students sometimes have difficulty providing counterexamples to false conjectures, formulating conjectures, or proving statements. Starting from the point that there is no way to acquire mathematical maturity except by doing math, we introduce a path that stimulates students to construct knowledge by themselves in the realm of topology. We face the issue of promoting students’ maturity through designing paths that stimulate geometric reasoning. We present the first findings on some students’ development and self-perceptions of their maturity.