The Collaborative International Dictionary of English v.0.48:
Tutorship \Tu"tor*ship\, n. The office, duty, or care of a tutor; guardianship; tutelage. --Hooker. [1913 Webster]WordNet (r) 3.0 (2006):
tutorship n 1: teaching pupils individually (usually by a tutor hired privately) [syn: tutelage, tuition, tutorship]