Research showed that university students struggle with the mathematical language, particularly when beginning mathematics studies. Therefore, we investigate a group of 15 mathematics students from a German university that were interviewed at seven different times during their first year of study to describe the process of their language development. Focusing on the aspect of definitions as a specific part of mathematical language, our results imply three types of students with respect to their development of mathematical language. Students of the first type seem to have an adequate understanding of mathematical language from the beginning. By contrast, students of the third type have severe problems and show no real development, whereas students of the second type start also with severe problems, but show improvement during their first year of study. The main aim of this paper is to describe and analyze the development of a student belonging to the second type.