The public parts of my notebook.
[But there are cases where] the conflict [between reality and our expectations] remains obstinately in place however much we ponder the problem. These are the cases where we are tempted to conclude that "intuition cannot be trusted”. In these situations we need to improve our intuition, to debug it, but the pressure on us is to to abandon intuition and rely on equations instead. Usually when a student in this plight goes to the physics teacher saying, “I think the gyroscope should fall instead of standing upright,” the teacher responds by writing an equation to prove that that the thing stands upright. But this is not what the student needed. He already knew that it would stay upright, and this knowledge hurt by conflicting with intuition. By proving that it will stand upright the teacher rubs salt in the wound but does nothing to heal it. What the student needs is something quite different: better understanding of himself, not of the gyroscope. He wants to know why his intuition gave him a wrong expectation. He needs to know how to work on his intuitions in order to change them.