A Firewire Patch Bay may be the answer -
http://www.signvideo.com/ieee-1394-firewire-patch-panel-patchbay-dv-video-b...
1 port would be used for the DV deck device. The other 15 ports to the students' computers. Then patch whichever computer to the DV Deck when required.
Be aware though that there may be limit to the Firewire cable run. Read somewhere that it is 15ft, unless you use a repeater every 15ft. Can't verify what I read though.
Thomas Leong