The films are meant for a class of teachers at the Mathematics Institute, which I suppose was the name of the learning institute at UI in the 60's. So, in portions of the film he is talking to adults (teachers) and in other portions he is actually teaching junior high school aged school children. You can tell the difference when you see the tops of the (adult) heads in the frame.
I was thrown off at first by the youth of the students because the UICSM course (New Math) was a high school course of 166 lessons, but I guess it started in Junior High School. As you watch the films you will see that these kids mature physically pretty fast. You can even see that in the two films above, especially the girls. So these are probably last year junior high students.
This is similar to the class I started with in 8th grade though we were a year ahead. It had exposition and explanation like this, but we got on with doing more frequently. These are just clips so I suppose that part (the getting on with it) is probably left out. The students are exhibiting more and more ability in answering the questions as the topics progresses so they must be getting on with it somewhere.
In any event, these films, with the teacher exposition and the mix of practical and quirky examples, capture for me (approximately) what I mean by "the golden age of high school mathematics". Throw in a little more "getting on with it" and you're there.