Momentarily put aside the notion of style. Many riders have difficulty on hard snow because of their tactics. Consistency and fluidity of movement are critical, and when movement flow is constantly interrupted, the board is apt to lose its grip. Encourage students to refine their sense of touch and increase their awareness of movement flow from turn to turn. Some people may not be aware that their board's skidding is at the root of their problems on ice.
Snowboards work best on ice when they move along their long axis all the way through a turn.
Unless there is no other way to stop or drastically reduce speed, the edge should never be set across the board's direction of travel. In other words, slice the ice. When a snowboard slices lengthwise across the snow, its construction allows it to dampen vibration that would otherwise cause the edge to chatter and lose its grip. A continuously carving edge is stable and predictable, whereas braking and skidding are not. Imagine that you're driving a car and it begins to skid.