I take it that lecture is rarely the most effective way for a student to learn. Lecture is a case where, I believe, research on learning firmly backs up the common experience that lecture rarely helps students who aren't able to help themselves. And lecture is far from our only pedagogical tool. There are many other ways for students to learn math that are frequently on display in elementary, middle and high school classrooms.
Regardless: when is it appropriate to lecture?
I prefer to think of lectures not as the worst way to teach but rather:
A lecture is the worst way to teach, apart from all the other ways we've tried.
That said, to understand when it is appropriate to lecture we have to understand what lectures do. At its heart, a lecture is a time-efficient way of communicating "stuff" from the lecturer to the students. The problems with this are manifold and mainly focus around the fact that the communication channel is one-way. But that can also highlight when it is appropriate.
When there is no other good channel of information (for example, no standard book or websites).
When changing the system would take more time than is reasonable (at the university level - which I presume this question is aimed at - then teaching is only a part of your job).
When the students resist other means of instruction (sort of similar to 2; it may be possible to change their expectations but it's only going to be a reasonable task if your faculty as a whole take part).
When the students are mature enough to take responsibility for their own learning.
This last one is probably the most important one. If I wanted to learn something then going to an all-singing, all-dancing, interactive event would be the last thing I would choose. I want to know what I need to know and I want it presented efficiently. The point of a lecture, in that circumstance, is to tell me what is important and to give me the benefit of someone else's experience (for example, a book tends to lay out all the options in a fairly neutral way but in a lecture the lecturer can say "This technique is out-moded and basically wrong, that is what you should use.". I can do all the "self-learning" bit on my own and at my own pace. It's a waste of my time and the lecturer's to be doing lots of "interactive activities" in the lecture room.
In other words, if the students already know how to learn and the goal is to tell them what to learn, then a lecture is appropriate.
This is a bit tongue-in-cheek, but it is good to lecture after students think they have mastered something via an active learning experience, provided the lecture is the result of long effort to find an efficient or beautiful path to a result. A lecture should be given with the disclaimer that such efficient and beautiful paths are the result of a long distillation, and should not be demoralizing to those who have struggled hard to find a result for themselves.
The reason for this is reasonably obvious: Discovery learning does not necessarily provide the neatest way through new material, but it is very effective for uncovering difficulties and essential points. After struggling with material deeply, an elegant proof that deals neatly with a horrible difficulty will be highly memorable and will help structure the experience for future use.
Of the common teaching methods used in STEM, evidence from educational research strongly suggests that lecturing is the worst. A 2013 meta-analysis[Freeman] compared the effectiveness of lecturing with that of active engagement techniques. The metrics used were success rates and scores on standardized tests or other assessments. In all STEM subjects and by both types of measures, active engagement gave better results, with the difference being statistically significant at the 95% level in nearly all cases.
The research is not necessarily fine-grained enough to detect whether lecturing or active engagement might be better in one specific situation but not in another. However, there is some evidence that opposes certain hypotheses to the effect that lecturing would sometimes be better. Some of the most detailed evidence comes from physics,[Hake,Mazur] and that's what I'll summarize below.
The evidence does not seem to support the hypothesis that better students are better served by lecturing. In the survey by Hake, students were given a standardized test before and after instruction, and a normalized gain score was calculated for each student by dividing the increase in the student's score with instruction by the maximum increase that would have been possible. Students at highly selective schools often had high pretest scores, but such students still had poor normalized gains when they received instruction by lecturing.
The evidence does not seem to support any belief that lecturing can be a good technique when done by a highly skilled lecturer. The large survey by Hake found no cases in which lecturing resulted in high normalized gain scores.
The evidence does not seem to support the claim that problem-solving and other technical skills suffer due to active learning. Mazur shows evidence that active learning improves over-all performance, not just conceptual performance.
Freeman et al., "Active learning increases student performance in science, engineering, and mathematics," http://www.pnas.org/content/early/2014/05/08/1319030111
Hake, "Interactive Engagement Versus Traditional Methods: a Six-Thousand Student Survey of Mechanics Test Data for Introductory Physics Courses," Am. J. of Phys, 66 (1997) 64
Mazur, Peer Instruction: A User's Manual, 1996, http://www.amazon.com/Peer-Instruction-A-Users-Manual/dp/0135654416/ref
