TL;DR: I have given a lecture once for this course in the past during an emergency, and would like to re-challenge myself once I get adequate prep time, as I feel it is an important experience. Is it a possibility for me to do it in a non-emergency situation? How would you feel if an undergraduate TA asked for your permission to give a lecture?
Some background: I am currently a undergraduate teaching assistant for a junior-level undergraduate research statistics course in the US. There is one other TA besides me who is a third year PhD student (referred to as GTA from here). Both TA's only teach the lab and help grade. We (normally) do not go to the lecture sections.
During the third week of school, there was a big conference that GTA was going to attend, so I was supposed to teach the lab alone. No problem. The Friday before, I have a meeting with GTA to cover the lab content. While there, he informed me that I was also supposed to cover the lecture that third week. Apparently, our professor was ill and would not be returning, and I was the only one available to cover the lecture. The GTA and all three of the faculty members with prior experience teaching the course were all going to this same conference and could not teach it. Apparently, an email was sent to GTA, HR, our advisors, and all the important people except me. In the chaos, our professor forgot to inform me that I was giving the lecture.
The first lecture was the following Monday morning, and I did not have enough time to prepare a PowerPoint with the lab prep also going on at the same time. The professor did not have any material pre-made for the course for me to reference (PPT, etc.), so needless to say, I completely botched that lecture. Fortunately, I was able to prepare a very good PPT for the Wednesday lecture and redeemed myself and managed to get through all the required material. However, the whole experience has left a sour taste in my mouth. I KNOW I can do better if I have more than three days to prepare, and I know that week's performance reflects poorly on me. This experience also made me realize how important it is to practice preparing for a lecture before I become a professor and it have a possibility to poorly reflect on my tenure package. I would really like to have a chance to give another lecture in the future for this course .
Now I know my giving a lecture was a very special case, as I have never heard of an undergraduate student giving a lecture. The professor is currently not back yet, and the graduate TA is teaching the lectures. I was informed that the responsibility should not fall on me again this semester, as there are three experienced faculty members that can cover the lecture before it comes to that.
While I probably will not do it until next semester given that the professor is not back yet, I am just wondering if this is something that would be appropriate for me to even try asking permission for. I am not sure how I should approach him on this topic. How would you feel if an undergraduate TA asked for your permission to give a lecture?
Just to add: The professor is not aware that he did not inform me that I was giving the lecture, so I don't want to bring up the fact that I want a second chance, since I don't plan to make him aware of our miscommunication. I don't want to guilt him into giving me another chance if it would make him uncomfortable.
Thank you for your time.
My first reaction, if a TA or GTA asked to give a lecture, would be to worry about the time. Almost every undergrad math class is in a rush to cover the material. At the end of almost every course, almost every math lecturer is wishing for another day or two. And you want one of my precious, precious days? Even if you do a really swell job, you might leave some loose ends, which I have to spend 5 or 10 minutes the next class polishing up. Then I have to cram that day's 50 minutes into 40 minutes. I say this so you know what you might be asking.
The second thing is that, depending on your location and school, there may be rules about who can teach at what level. In Texas, there are laws (from the state legislature) which say that a master's degree can teach math classes only up to a certain level. And that a grad student can teach classes only up to a certain level. And some schools advertise things like, "All classes are taught by professors." So you might run into a rule or law that simply prevents you from lecturing in a non-emergency situation.
Having said that, I don't see any problem with you approaching the professor and saying that you would like more experience lecturing. You understand there are hurdles, but if possible, you'd like to do another lecture.