Gudermann then followed in his father's footsteps by becoming a school teacher. He obtained a post as a teacher of mathematics in a secondary school in Kleve in 1823 and taught there until 1832 when he was appointed to the Theological and Philosophical Academy in Münster. His first appointment in the Academy in Münster was as an extraordinary professor, but later he was promoted to ordinary professor of mathematics there.
Gudermann worked almost exclusively on spherical geometry and special functions but he is not remembered for any original mathematical results in these areas. This is not to say that he did not do useful original research but just that he suffered the fate that many mathematicians have suffered, namely that a comprehensive theory was developed later which meant that his contributions fused into the theory. His own contributions tended to be a whole series of special cases (although this could not have been obvious at the time) which were forgotten later when the general results which included them were found. He did write a book on spherical geometry and :-
In the introduction he pointed out that a plane was a special case of a spherical surface, that is a sphere with infinite radius. For this reason and because of its constant curvature there exist many similarities between spherical geometry and plane geometry; yet at the same time Gudermann considered more interesting the study of cases where the similarity no longer holds.In his more extensive work on the theory of special functions Gudermann published several papers beginning in 1830 which extended work which was developed by Euler, Landen, Legendre, Abel and Jacobi. He summarised his contributions in two monographs published in 1833 and 1844. A third monograph, which he promised to write on the topic, was not written because of his early death.
Gudermann is best known, however, as the teacher of Weierstrass between 1839 and 1841 while Weierstrass worked for his secondary school teacher's certificate. Gudermann, at this time, was particularly interested in the theory of elliptic functions and in the expansion of functions by power series. In particular his use of power series in the study of the hyperbolic functions is of importance. This was to influence
Not only did Gudermann undertake research in the theory of elliptic functions but he was one of the first to teach the topic. Weierstrass spent the academic year 1839-1840 taking Gudermann's course on elliptic functions. Much of the article  is devoted to studying Gudermann's work and how it influenced the direction that Weierstrass's research would take, in particular it played an important role in Weierstrass's habilitation thesis. Manning writes in :-
The concepts on which Weierstrass based his theory of functions of a complex variable in later years after 1857 are found explicitly in his unpublished works written in Münster from 1841 through 1842, while still under the influence of Gudermann. The transformation of his conception of an analytic function from a differentiable function to a function expandable into a convergent power series was made during this early period of Weierstrass's mathematical activity.
Article by: J J O'Connor and E F Robertson