The textbook used in the class contained 1000 problems with somewhat cryptic solutions. Each week, Paul picked 6 problems and then each student was responsible for presenting one of the problems on the board. These were typically short proofs. This was in addition to regular problem sets.
Paul notes that while this structure was tricky because it required small sections, it forced students to come to section and was fairly straightforward to implement since it involved using existing problems from the text book.
The description from the syllabus was as follows:
There will be additional small weekly problem sessions (6-8 students in each) led by the course staff on Monday afternoon and evening (1, 2, 4, 5, 7, 8 PM) and on Tuesday (1, 2 PM). These will not be optional "help sessions." Instead, members of the class will be assigned problems to present at the blackboard, drawn from the collection of 1000 solved problems that accompanies the textbook. Participation in these sessions will count for one-third of your homework grade, and graded written problem sets will be shorter than usual.