What are some good ways to lead a discussion session for a TA in a graduate-level STEM class? I'm a TA for a class on graduate-level Digital Signal Processing, and one of my responsibilities is to lead a discussion session. It's a 50-minute session that happens once a week, and the professor teaches the class for 3 hours in the week. 

The professor had suggested that I discuss the solutions to the homework problems in this session, so today for my first session, that's what I did. 

However, I felt this was quite a waste of time, since everyone who had had problems in the homework had come to my office hours and got them cleared, and everyone else in class had managed to do the problems. While I went through solving the problems on the board, I wanted to actively engage the students by throwing in questions about details, what-ifs, some important tricks to remember, etc. But it was a mostly unresponsive, bored-looking class with frown faces.

Should I not go through the homework solutions, and instead focus on important topics covered in lectures? 

Also any ideas on how to lead a discussion session in a way that the students who actually managed to do the homework (which is the vast majority of the class) don't get bored?
Clear up common points of confusion you've encountered while grading.

If you are also grading homework for the class (or you know the grader...), you can identify common mistakes or points that students seem confused about. Even if you explain these on the individual homework papers on which students made those mistakes, (a) many students don't bother to follow up if they don't understand the grader's comments, and (b) it's possible that other students would have made those mistakes, and didn't only because they made a different mistake instead.

Discussion sessions are a great time to point out these potential pitfalls, warn students away from them, and discuss why they represent a wrong understanding.

Ask a question as a confused person.

I like to challenge a class (in the context of the discussion - not necessarily on an exam!) by asking a question that is ill-formed, confusing, or based on a mistaken premise. (Not to trick them - I warn them that this is what I'm doing.)

For example, I like to find questions asked by Real People on Stack Exchange sites (like Signal Processing Stack Exchange!) related to what I'm teaching, and challenge students to formulate a good answer. Then we discuss the answers as a group.

Answering a "real" question posed in this manner often requires much greater mastery of the material than solving the "nice" problems we usually give students.

Pose a design problem.

On a similar note, the "nice" problems we ask questions often have boring, "nice" solutions that require straightforward application of what they have learned, but not much original thought and critical thinking.

Raising a design problem - i.e. "How can we design a DSP pipeline to do X?" or "Here are two designs that solve a particular problem. What are the benefits and disadvantages of each?". These, too, can come from mailing lists and forums. Here's an example of the type of design problem I'm thinking of.

(Of course, these should be problems your students can solve, based on what they have learned.)

Have students nominate topics for discussion at the
beginning of a section.

At the beginning of the session, ask the students to suggest a few topics from the lecture or homework that they would like you to review in greater depth. Then have the class as a whole vote on the nominated topics to determine what to do with the session.

Source of this suggestion: TA handbook for UCSC

Ask the professor to provide you with past exam problems for use in the discussion sections.

Students seem to love working on past exam problems :) 

Specifically, you may want to do it as follows:


  Hand out a printed copy of the questions (double-sided, stapled). For each question:
  
  
  Read the question aloud, or ask a student to read the question aloud.
  Ask if the question is clear. 
  Have them discuss the answer in small
  groups. 
  Ask for votes, or ask someone to explain their answer. 
  If the
  class doesn't all agree on the correct answer, then you can have them
  discuss further, or you can try to clarify a difficult point.
  


Source of this suggestion: Jason Eisner

Give mini-lectures about things you think deserve to be retaught.

Especially if you have the time to also attend the professor's lectures, you can identify topics that he may have had to rush through. 

Similarly, you may give a "refresher" on a prerequisite topic that students don't seem adequately prepared in.

Or, you can give a mini-lecture on a topic that, based on homework grades, students just aren't getting.

Source of this suggestion: Jason Eisner