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JANET RANKIN: A common active learning technique that people often use, or maybe the first technique that they use, is personal response systems. So that's an opportunity for students to respond to a question. And that can be anything. It's usually a multiple choice question, but it doesn't have to be. And it can be anything from a show of hands-- who thinks the answer is A, who thinks the answer is B, who thinks the answer is C-- to dedicated devices and applications that allow students to respond to questions. So there's pros and cons for each of the different techniques, but the essence of all the techniques is that students are given an opportunity to think about the answer that they think is correct, and then respond to that. And there is evidence to show, research that shows, that when students make a claim, when they're asked to make a public response, a public claim as to what they think the correct answer is, that makes them more committed to ...

PROFESSOR: The greatest common divisor of two numbers is easy to compute. And that's a factor will play a crucial role in the number theory we're going to develop, and the properties of some of the modern codes that are based on number theory. The efficient way to compute the GCD of two numbers is based on a classical algorithm known as Euclidean algorithm, which is several thousand years old. And let's describe how it works now. So the Euclidean algorithm is based on the following lemma, which we'll call the remainder lemma, and it says that if a and b are two integers, then the greatest common divisor of a and b is the same as the greatest common divisor of b, and the remainder of a divided by b-- providing, of course, b is not 0, because otherwise you can't divide by b. OK how do you make sense out of this? Why is this true? Well, it's actually a very easy proof. Remember that by the so-called division algorithm-- or it's really a theorem-- ...