What was the purpose of the panels of blinking lights on those big mid-century computers? Were they showing calculations in progress?
Excellent question, this is one of my favorite subjects! Blinkenlights serve a number of functions. Hollywood tended to use just the lights to make it look like a computer was busy doing something, but real computers had more than just lights on their front panel. Let's walk through a few examples of use cases with photos of computers I've seen over the years at museums and vintage computer festivals:
Some front panels were built to be used for diagnostics. Computers like these were primitive enough that they required constant care and debugging to do their jobs, especially the early vacuum tube machines (everything pictured here is transistorized). You could tell what peripherals were being used, but also check the status of registers, carry flags, status flags, data, various buses, etc. It was also a way to see if a program had "gone off into the weeds" and started doing things that were irregular, possibly due to a software bug, or a problem with the hardware.
On many of these machines, you can enter programs directly into the main memory using the front panel, but it's an incredibly tedious process -- something to be avoided if possible. Consider it a last fallback.
Other times, it's a starting point, which we call "bootstrapping" (this eventually evolved into the term "booting"). You aren't likely to program everything on such a limited interface, but you are more likely to enter in a small program that can tell the computer how to run a more complex peripheral, like a paper tape or punch card reader, or maybe some type of magnetic storage device. Once you can get a program loading off of a larger permanent storage device, you can load up software to interface with a terminal of some kind which is much easier.
Eventually, the microprocessor made home computers a possibility, but many were only equipped with a front panel out of the box. You would have to add in a serial card, more RAM, possibly some ROMs, and either a teletype or glass terminal in order to get a more sophisticated and intuitive interface from the computer, capable of programming in a higher level language. Some were considered more like trainers, or hobbyist devices, and simply lacked that ability, meaning all you got was a front panel with switches and lights.
I made my own front panel to see what the experience was all about:
Then everything changed in 1977, with the introduction of these three machines: the TRS-80 Model I, the Commodore PET 2001, and the Apple II. They were what you might call "appliance computers" and they had no need for a front panel.
Hopefully that answered your question!