I can't move that boulder thing with Big the Cat. I think it's on the Ice themed stage. (NOTE: I don't have the game and I am unsure of the stage names and contents. I'm doing it for a buddy and I'm sorry if it is inaccurate.)
The Action Stage is Icecap. You don't really need the bolder to break the ice covering the frozen pools, you just need to find the weak spot(it's dark in color) and just jump on it. It'll crack open.

~ SA Tails