Dunno whether these controllers are approach/departure, or en route, but the charts are Los Angeles area, and 1952 looks good a guess as any. Did any en route controllers in the US sit at radar screens then? Probably LA approach/departure controllers didn’t have radar in 1952? So flights would report passing so-and-so place at so-and-so time and the controller would write that on the flight strip? What are all those switches (?) in front of the foreground guy?
I don’t have a direct answer, but for context you might want to read a bit about a midair collision between two airliners over the Grand Canyon in 1956. As I understand it, pilots just reported where they were going and when they expected to cross certain waypoints, and the controllers just took their word for it. The two planes that collided were off of any defined routes that may have been more strictly controlled. That accident was one of the things that prompted tighter control and use of things like radar.
Decent bet that was southern California regional control. I don’t know what the proper period terminology was for that. There was some manner of en route control center back then, but not the same arrangement as the post ~1960 Enroute Centers still used today.
Those switches up front look exactly like standard telephone company equipment as would be installed on a contemporary telephone switchboard. Which leads to this surmise …
Controllers then and now have two independent communication streams going on. They’re talking to airplanes over the radio, and they’re talking to their nearby peer controllers over dedicated telephone circuits. So they’ll get a call from some other controller to inform them of an airplane about to enter their jurisdiction and make similar calls to other controllers to tell them of airplanes leaving their own jurisdiction for somebody else’s area. These are called “hand-offs”. There are lots of other reasons controllers talk to one another, but handoffs are the bulk. Inter-controller comm has a totally different lingo from anything a pilot ever hears.
Decent bet the black switches in front of each controller connect their headset to the phone lines leading to their neighbor controllers. Flip the switch for who you want to talk to, key the mike, and start talking. Nowadays they have a foot pedal to trigger talking on the phone and a button on their headset cord to talk over the radio.