The teaching responsibilities are the most visible: learning the material (this can take years, especially for some upper level courses), keeping up to date with current advances in both research and pedagogy, choosing an appropriate textbook, preparing lectures, supervising teaching assistants, grading, etc.
Research activities involve solving problems that were previously unsolved. These activities are more difficult to describe. Most professors have one or more areas in which they are experts. Part of the job is to keep up with current research activities in their areas of expertise. This includes reading journal articles, attending conferences, and staying in touch with other researchers in their field. It is sometimes as hard - and as important - to find the right questions to ask as it is to answer them. Problems that have been solved are written up in the form of an article and submitted to an academic journal for publication. The editors of the journal send the submission to experts (referees) for their opinion about whether the article deserves to be published. If published, the authors get a pat on the back.
Service activities involve serving on various committees at the Department (e.g., the undergraduate program committee which evaluates the courses we require for majors and minors), College, Campus (e.g., the Liberal Education Committee, which evaluates the courses students must take in order to gain sufficient "breadth" in their education), and the University of Minnesota system. They also perform service activities to the "academic community," such as organizing conferences, editing and refereeing for journals,
Graduate Teaching Assistants (GTA's) at UMD typically spend roughly half of their time teaching, and the other half taking courses and working on their research projects.
GTA's in the Mathematics and Statistics Department are almost always enrolled in the UMD Master's program in the Applied and Computational Mathematics Program. They take between three and four upper level courses each semester of the two year program. They also work on their own research projects or theses under the supervision of a faculty member in the Department of Mathematics and Statistics.
GTA's typically teach discussion sections for courses like Calculus or Precalculus. Their responsibilies include helping students understand material presented in the Lecture part of the course, helping students solve homework problems, holding office hours, and grading student work. Occasionally they may teach a course such as College Algebra (Math 1003) as the sole instructor.