At a campus colloquium a number of years ago, a faculty member (no longer at Berkeley) was speaking. A very sincere graduate student asked a question. The professor talked for a few minutes, then turned back to the student and said, "I'm sorry I'm not answering your question, but it's not very interesting." Below are some tips for making engagement with students more productive than this one.
In every class there should be interaction between the faculty and students, and at the core of student engagement is how questions are asked of students and by students, and how questions from students are answered. Few things can encourage or discourage student engagement more than simply how questions are handled in a class.
1. Asking Questions:
At the very least, asking questions of the class is one way to make sure that students are with you, are understanding where you are at a given point.
At their best, questions you ask, even in a large class, are a way of testing how things are going and of involving students.
You can always ask a question of the entire class, then have students discuss it with those sitting near them, and ask for reports from various clumps of students around the room. You can ask a question and have students jot down the answer on a piece of scratch paper. They pass their answers to their neighbors, and you call on someone to read the answer on the sheet they have. That way, students who are afraid of providing their own answers are relieved of the burden.
Remember that questions don't always have to be directed at individuals: you can also poll the class from time to time on issues, e.g., "Who believes that there is a time and a place for the use of sentence fragments in writing?" Then you can follow up by asking individuals to explain their reasoning.
Cold-calling. There is no right or wrong here. Some faculty find it productive to call on students who haven't volunteered while others find that it can create too much tension in the classroom. This is really faculty-class-personality dependent; however, if it's something you're interested in trying, it's best to announce it in advance, and set some guidelines, e.g., students can simply "pass" if they don't feel prepared to answer. And have a backup plan: if you call on someone who doesn't know the answer, you immediately throw out the question to the class.
Sometimes simple questions are fine, such as the final step in a formula, but most often the best questions are those for which the answer isn't a number or a date or a figure. For that reason, think about asking "how" or "why" questions.
One interesting technique: one faculty member at Berkeley would ask a question to her very large class, and students would raise their hands to answer. But the faculty member wouldn't call on anyone until she was satisfied that enough people had raised their hands.
2. Soliciting students' questions and answering them:
The most common-and-very-worst-way to solicit questions is to look at the class and say, "Any questions?" or the truncated "Questions?" or "Ok?" or "Is that clear?" These often have a deadening effect, as if you are really just pausing, or asking just pro forma. Whether or not you intend it, the subtext can be "You shouldn't have questions." Try out a variety of other formulations: "I'm sure at this point you'll have some questions, so let me try to answer them." "This is a complex point, so please ask me questions about it." Or any other formulation that indicates you are actually interested in answering student questions.
When you answer a question, answer it directly first, then go off on any tangents that come to you. Try not to mix tangents in, so that the actual answer is hard to discern. And when you're done, ask if you've answered the question.
Repeat a student's question before you answer it. Be aware of the acoustics and that students sitting behind someone who asks a question might not hear it. In addition, by repeating it, you give the student a chance to indicate whether that was indeed the question he or she intended.
If the question is a good one, say so.
If the question is tangential, develop a nice way to say that. Often something like, "That's interesting, but a little off our point here. However, I'd be happy to talk about that after class/in office hours." If you have a plan in advance for these questions, you'll be happier. And you'll be less tentative to solicit questions from the class.
Consider turning some questions back to the class to answer. Don't feel that you need to be responsible for answering them all. Get all the students involved in this process.
Finally, when you answer questions, don't focus all your attention on the student who asked, but look at the whole class, so that it doesn't become a conversation between you and a single student.