Adobe Scratch Disk

If your students open very large Photoshop files that would explain the message.
Photoshop "extracts" the files upon opening and naturally that uses diskspace. Larger files equals more use of diskspace.

Photoshop might by itself also use additional diskspace simply by starting. If that is the issue identifying un-used presets and deleting those might be the way to go.

Besides that simply freeing up diskspace is your only option. Or upgrading the HDD.

@ccampion - see this page for some tips on managing PS scratch disks, etc.

PhotoShop Scratch Disk Misc

Another thing to consider is the fact that the system stores user cache data in /private/var/folders.
Depending on how you tidy up your local user homes, you may find that there are cache folders left over from decommissioned users.

You can list the size and ownership of these cache folders with the following command:

find /private/var/folders -depth 2 -prune | xargs -I{} bash -c 'printf "$(stat -f "%Su" "{}")	$(du -h -d0 {})
"'

If you find there are cache folders that you can delete, you can do it with this command:

find /private/var/folders ( -depth 2 -prune ) -and -user "SOMEUSERNAME" -exec rm -fR "{}" ;

Do not delete folders owned by root or any of the system users - or your Mac may fail to boot up.
If you delete a folder owned by an existing user, the next log-in by that user will be longer than usual as the cache folder is re-created.

As usual, this is not something that the system expects you to do - so be cautious.

Thanks for the responses. A call to Adobe did not offer much other than the possibility that a user(or users) created a very large document (e.g. 700x800 inches thinking it is700x800 pixels). Each save would place a copy on the scratch disk. I was able to delete 2 or 3 users accounts/computer. There is now over 30GB of space on each.