It's absolutely standard way of working in the corporate network.
In mind to raise HELL and that on each computer the users had worked on their (unique) name. Possible without AD, but users still need to have unique on each computer. Also in this case, you will need all the users computers students duplicate the teacher's computer with the same password.
1.In shared directory create a separate folder for each user.
2.For each user folder:
2.1.Break inheritance rights.
2.2.Give full rights for the teacher, the administrator and the user of this folder.
2.3.Other users and groups and remove (you can leave the System).
In the end, each student will be able to work only with his folder, the teacher will see all the folders.
PS: for teachers, if there are several and they will work under their own uchetku better to have a group and assign rights for the group. Group for students will also be useful, for example, that would give her read permissions on the shared directory (which are user subdirectories), but not necessarily.