This is the 15962nd most frequent Russian word.
проходящими
"Проходящими" means "passing" or "passing by" (instrumental plural form).
The word 'проходящими' is used as an adjective to describe the people who are passing by.
Я поговорил с проходящими мимо людьми.
I talked with the people passing by.
Here, 'проходящими' characterizes the days as ones that are occurring in sequence, leading to passing moments.
Проходящими днями я успел выполнить много работы.
During the passing days, I managed to complete a lot of work.
In this case, 'проходящими' modifies 'старшеклассниками', referring to high school students who were in the process of passing by.