If you think you’re thankful for computers, just think about how NASA feels about them. Before the days of computers, NASA employees had a much more laborious way of calculating and presenting their data.
Before the days of computers, employees at NASA would have to go about conveying their knowledge in a much more laborious way: chalk, board, and likely tears. The scientists used math and physics to calculate complex spacecraft trajectories, navigation and the orbits or spacecraft, and much more. The calculations were tedious and long.
As for this photograph, probably the photojournalist asked them to fill the board with complicated-looking equations for the shot. These are general equations of motion, so nothing that you would not have in a printed sheet of paper. There are no “calculations” in that board, just reference equations. To be fair it was the middle of the Space Race, they would not publish anything other than generic equations in Life magazine.