In this paper, we investigate how a research mathematician conjectures and proves when conducting her research. To be precise, the aim of this study is to achieve a comprehensive understanding of the way this research mathematician develops these mathematical practices and thus gain insight to improve the teaching and learning of these two practices in an educational context. For this purpose, we consider Rasmussen, Zandieh, King and Teppo's ([2005]. Advancing mathematical activity: A practice-oriented view of advanced mathematical thinking. "Mathematical Thinking and Learning," 7(1), 51-73) theoretical constructs of horizontal and vertical mathematizing. In particular, we have adopted a case study methodological approach with a single research mathematician. Analysis of the data lead to the identification of eleven categories of activities. Each category is linked either to the practice of conjecturing or to the practice of proving, and also to the horizontal or vertical dimension of such practice. Finally, we compare and contrast our results with other related studies and give some educational suggestions that may derive from our work.