The phrases ‘certainty is an illusion’ and ‘think probabilistically about the world’, both introduced in the first class, are the first two examples … - David Franklin

Now – those who got it right. On the count of three, I’d like you to sit down if you’d seen the question before. 1, 2, 3!” Everyone who was standing sits down. This gets a huge laugh. The students who got the answer wrong suddenly feel much better about their mistake: in a Harvard class, not a single person got it right unless they had seen it before.

What is the probability that at least two people in a group of 20 people will have the same birthday?

With these issues in mind, Dan has a unique way of tying exams into the learning process. After attending a workshop run by the Nobel Prize winner Carl Wieman, he was converted to the idea of two-stage exams. The first stage of the exam is the same as any other. The second stage brings the students together in groups of four, and gives them all a subset of the questions they just answered in the first stage. The students are encouraged to discuss their answers in their groups, and they all submit a second version. Groups who manage to improve upon the students’ individual scores get a small boost in the final outcome.