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.
Continuity mistake: When Casey is skating on the pond after she leaves Harvard, she falls multiple times. The second time she falls, she lifts her right leg causing her foot to come into frame. On her feet you can clearly see that she's wearing black and green snow boots, even though seconds before that she was wearing white figure skates.
Factual error: In the science lab when Mr Bast is turning on the Bunsen burner, it doesn't seem as though there is a flame to light it, and it is just lit somehow almost magically. He also toasts the bread beside CHEMICALS. No smart teacher ever does that and food is forbidden in science/physics labs.