I am doing a problem solving activity in a middle school classroom and I came across this video that I made in 2007. I had completely forgotten about it!
In this case, I had two ambitious students who did all the recording for me. I compiled the clips and then wrote a script. They did the narration and I put it together.
I think this would make an excellent classroom project, but the editing was more time consuming than most teachers will be able to accommodate. One solution would be to divide the work between a math class and a video production or ELA class.
My only other advice is to plan ahead of time how much you'll include (how many people, how much of their solutions or attempts). Otherwise editing can take a long time as you figure out the direction based on all the video you've recorded.
I used to use this in class by stopping as soon as the problem is displayed. After students had time to think about a solution I would play the rest. A lot of the fun comes from the fact that they knew the students and teachers, but some have told me this will still be useful in their classes.