Yeah, Roman slaces mostly lived in households and were pretty well educated. Some would tutor the kids.
Though there were of course some that would work in fields
Evenif the most "Paedagogi" (in the classic period) were "Servi" the translation " tutor-slaves" is weird. It does not reflect the reputation and prestige of a "Paedagogus". Private teacher, private Tutor is a much better tanslation. Also educator and pedagogue could be valid translations.
Yeah, educator slave sounds way better :) (Paedagogi were slaves unlike magistri)