Math videos containing real time rough thinking I find this vlog experiment of Gowers very brave, and I think his idea of having examples of real-time mathematical thinking by experts can be very encouraging for young mathematicians, who imagine the pace of the greats is impossibly far and away above that of mere mortals when they meet new problems. (Online comments by Borcherds, as well as this wonderfully courageous experiment by Gowers suggest otherwise.) A funny anecdote along these lines: Once a graduate student said to me: "I have an image of Terry Tao eating his breakfast one morning, eating his cereal, and as he lifts his spoon to his mouth he pauses, looks off into the distance, and in that moment does more productive mathematics than I will in my entire lifetime."
Have others participated in this experiment? Are there other online video examples of "real time raw thinking" by leading mathematicians, following up on Gowers's experiment above?
Whether this question should be on MO is undoubtedly going to be controversial but I think it has to do with mathematical practice, and particularly the working styles of top mathematicians who frequent this forum.
 A: I feel that after what I said I should offer something constructive here as well. Part of the problem with such movies is that the very act of verbalizing the thought process interferes with it quite severely. At the very least it significantly slows you down (take any computer program you ever wrote for computing something interesting, add the command to print the result after every operation, and watch it running). But that is not all. Interrupting a human thought is worse than interrupting a computer program because most of us are incapable of storing a SSW at the moment of interruption and retrieving it without error once the interruption is over. Also, when talking, we feel like we need to exhibit some clear logic in our transitions, while in fact many of them occur merely because one chain of thoughts suddenly outpaces the other, so we just switch to a more promising approach, and so on, and so forth.
So, simultaneous verbalization is a killer. What to do then? The only way out I see is to think it all through in the usual way and then post all that we wrote on paper in the order of writing with some short comments about the places that would look totally unclear to an outsider. That is like having a fossil record of your thought that tells about it as much as fragments of dinosaur bones about the terrible lizards themselves, which is not everything, but still quite a lot. Here is mine for the same problem. What I had in front of my eyes at the moment of writing was the screen with 4 first determinants, out of which only the last one really mattered, i.e., I was looking at
$$
\begin{matrix}
1&1&1&1
\\
1&2&3&4
\\
1&3&6&10
\\
1&4&10&20
\end{matrix}
$$
What I wrote is here:

(the top scribbled line reads "row - part. sums of prev. rows?").
The rest should be clear from my comments.
I feel like I have to add in the end that it doesn't mean that my thinking is better than that of Gowers. In fact, IMHO, compared to him, I am a hopeless imbecile sliding into senility. It merely means that I believe that the movie he posted does not (and, as conceived, could possibly not) reflect his true thinking. Also, to speak frankly, the Terry Tao's blog post titled "Does one have to be a genius to do maths?" reminds me of the phrase "Superman just helps now and then" in the famous cartoon. It is all fine, but if you watch that cartoon with a timer in hand, you'll realize that this "now and then" occupies about 90% of the shown events and the corresponding percentage in Terry's published papers is fairly close. As to his "generals", his account, if you translate it into layman language, reads to me "On my test I had to jump over the Empire State building and I almost knocked the spire off plus made a couple of unnecessary extra steps when landing, which showed to me that I need some more training before I attempt an unaided flight to the stratosphere". Disagree? Then try to pass his test yourself right now. You have enough information about what the questions were in his post, so just sit down and honestly write the answers to the best of your abilities. If you get happy with the result, let me know. I'm willing to take your word for it. But I give you mine that I failed miserably :-)
I am as curious about how great minds work as the OP, so if somebody has a good idea of how to observe that thought process, I will be happy to hear it. I also admire Tim Gowers for his attempt despite the fact that I still firmly believe that the way he approached it was hopelessly doomed from the beginning unless the true goal was very different from the proclaimed one.
A: When I taught a beginners course in analysis a few years ago I wanted to give the students an impression of how one can work on exam problems. I asked somebody to prepare a test exam (I had my exam prepared already and asked for an exam in a similar style). I got the exam in a closed envelope and walked into an empty lecture hall with a blackboard and document camera. There I recorded myself solving the exam problems thinking out loud all the time. I first went through the problems, commented on then, then went to the blackboard and started solving them and once I was seeing how I should do it, I went to the document camera and wrote down the solution as I would write them to get full marks. I posted the video uncut for the students. It was actually a nice experience for me and I got a lot of feedback saying that this was very helpful for the preparation of the exam (and also it helped some people with "test anxiety" - if that is the correct word in English…).
