What does the professor want me to do now?

I think he mentioned something about cleaning the emblem on the floor of
the rotunda.  That's the room just to the outside of the ring where the
classroom is.

Of course, you'll need cleaning supplies too.  Fortunately, they're within
easy reach.  Look at the rooms along the inner ring of the Academy.
Get both the floor scrubber and the orange safety cones.

Now you can take the turbolift down to the bottom floor.

I would advise dropping the three cones to make sure people don't get hurt.

You're ready to unfold the scrubber and then hop in it.  Unfortunately, you
need to get every last spot -- the sequence will stop by itself when you've
finished.  EGA users may not be able to tell when they're finished, but
Sierra has a patch that will let you skip this scene.