In the early days of engagement research, engagement was measured as happening or not happening in planned activities. Our research, first with Carl Dunst and then with Don Bailey, and much since both of them, has emphasized the fact that engagement is not dichotomous variable. It is instead one that ranges from nonengagement (same as the dichotomous way of looking at it) to sophisticated behavior--a pot into which we put encoded, constructive, persistent, and symbolic). We usually array these codes in a developmental sequence, acknowledging that there is a "cognitive" component to the construct. Our studies have shown that Battelle scores were somewhat correlated with engagement levels.
In a study currently under review, conducted by the brilliant young Portuguese researcher, Cecilia Aguiar, sophisticated-engagement levels in classrooms were associated with sophisticated-engagement levels in homes. Does this mean children are carrying around their engagement trait from one setting to another?
Aguiar found that sophisticated engagement was correlated with age: Older children have more of it. Because children obviously carry around their age from one setting to another, ipso facto, engagement will look like a trait.
Engagement and developmental age are not exactly correlated, however, so we recognize that there is something of a developmental or "cognitive" dimension to engagement. But don't discount the environment. Within children who spend quite a lot of time engaged in a sophisticated manner, those in interesting environments are more likely to spend time in sophisticated engagement. Even more important, perhaps, children who spend much time nonengaged are more likely to be nonengaged in uninteresting settings.