The status quo in university lecturing is comfortable. You capture what you know on a set of slides (often using many words - kind of a brain dump, really), and tell students that 'this is the way it is' during class. You normally don't have to open yourself up to potentially embarrassing situations where you realize you don't actually know some of the details you thought you did. This is especially useful the first time you teach a topic and/or when it's not your area of expertise.
But I'm just not into the status quo. Alas, I also felt a little embarrassed during my last class.
I filled in yesterday for the prof I'm TA'ing for. The class is a third year course on 3D computer graphics for the game development students. I decided to make my own slides and build up a really good understanding of how camera viewing worked by going from the canonical view volume to orthographic projection, and from arbitrary view points to transforming a perspective projection into an orthographic projection. This is the approach shown in Fundamentals of Computer Graphics by Peter Shirley, and as an added bonus, the author provides the book's diagrams for free on his website.
(If you're curious, you can check out the slides for my lecture in PowerPoint format - be sure to look at the notes section of the file since that's where the explanations are. I don't like putting lots of words up while I talk.)
The class was going well. I had a few places I wanted the students to try something out for themselves because it's all too easy to look at the numbers on the slide and just accept them as seeming reasonable. I know, because I do that all the time, either in talks or when reading. Even if they couldn't figure out what I was asking them to do, the act of trying would force them to really think about what I just showed them.
At one point, however, I asked them to do something that I hadn't had a chance to do myself. (I didn't find out what I would be lecturing on until fairly last minute and, unfortunately, made my slides the day of the class.) I had done exactly what I was trying to help them avoid: I took for granted what the book was saying and didn't realize that I never tried to understand the details. Not until a specific question came up, that is.
Although I readily admit when I don't know something, I did even more this time: I tried to logically figure it out in front of the class. Dangerous! Especially dangerous because I'm generally not that good at figuring stuff out in front of others. Well, as expected, that didn't go so well. So I said that for some reason my brain appears to be incapable to sorting this out at the moment, but if anyone in the class thought they could see it more clearly, they could try to explain.
A couple of students actually did! I really commend them because as I later confirmed, they were basically right. They are going to understand this topic so much deeper now than they would have if I had just shown them the answer and moved on rather than tried to understand it with them.
Granted, I probably wasted a bit more time on figuring this particular thing out than I should have, leaving some students bored. I suppose finding the right balance comes with experience. This was a small class, making it more reasonable to have spent some time on it, but I don't think it would have made sense to do it in, say, a huge auditorium.
In any case, I promised that I would post a clear explanation on the blog once I had a chance to think about it on my own. I followed through with a post within a couple of hours of class ending since, just as I expected, the answer was clear and obvious once I could think about it away from staring eyes. (Hopefully I got it right - if anyone notices any issues let me know!)
All in all, despite the fact that it was easy to feel embarrassed from my fumbling around, I conclude that it was worth it. Putting yourself out there is uncomfortable, but it generally means that you are going to give students a better learning experience.