Typical Purdue T.A. jobs involve doing recitation, designing quizzes, and grading exams. I also had the experience of doing lectures for MA158 and moderating forums for MA528.
Purdue University
Reports from various students in the Purdue course evaluation reports
Lecturer for MA158
Duties involve weekly lectures, designing and grading quizzes, proctoring exams, and working in the math resource room.
Forum manager for MA528
Duties involve moderating the Piazza system, answering questions, and encouraging discussion among students.
T.A. for MA162 and MA262
Duties involve weekly recitations, designing and grading quizzes, proctoring exams, and holding office hours.
Grader for MA527 and MA528
Duties involve grading weekly handwritten assignments and proctoring (and grading) exams.
National Taiwan University
T.A. in calculus for mathematics majors
Duties involve weekly recitation, providing and grading assignments and quizzes, proctoring exams, and holding weekly office hours.