Five-year-olds, 5 1/2-year-olds, 6-year-olds, and adults were tested for the presence of two intuitions about thinking hypothesized to be part of the adult theory of mind (a) when engaged in a mentally demanding activity, a person's thinking will be directed toward that activity, and (b) when the activity is not mentally demanding, the person's thinking is likely to wander elsewhere. The adult participants gave clear evidence of possessing these intuitions, both in their responses and in their justifications of their responses. Many of the child participants, especially the older ones, also responded as if they had at least some grasp of them. These intuitions appear to form part of a family of fundamental intuitions concerning the what and when of thinking that are in the process of acquisition during the late preschool and middle-childhood years. (Contains 3 tables.)