Continuity mistake: After Casey has turned down Harvard and is skating on the pond, she falls many times. When she falls for the fourth and final time, her hat falls off. In an overhead shot, the camera even shows it laying on the ground above her head. When she gets back up upon seeing Teddy with the zamboni, her hat has disappeared.
Character mistake: When Casey is using her knowledge of physics to advise a figure skater, she states, "Pull your arms in real tight. That increases your moment of inertia". In fact, this is the exact opposite of how it really works. Reducing her overall radius makes her rotational inertia smaller.