 So the question that I'm asked all the time is why should we learn more than one way to do this problem? And this centers around something called adaptive expertise. And so just for example, I've shown three different ways of performing 37 plus 55 using a method called counting on. And so we had our one, two, three different ways of doing this addition. None of these are probably the familiar way you have of doing 37 plus 55. And so the natural question is why bother? But even then, within that, I have three different ways of doing this, which is the best way to perform the addition. And the answer to that question is the best way to perform this addition is whichever method suits your current level of skill. Here's the problem that you want to think about. The problems stay the same. 37 plus 55. The question is how much do you know at the point where you can solve this problem? Now if we use the standard algorithm for doing this addition, you have to know quite a bit before you can add 37 plus 55. On the other hand, if you know how to count, you can do 37 plus 55 well before you know all the other things you need to learn to do the standard algorithm. So if you know all of your addition facts, well this third method here is actually pretty efficient. On the other hand, if you're still learning the addition facts, the first or second way is better. And the first way actually has the easiest additions to do. This first method here, the additions are the easiest to do, although it does require keeping track of what you have yet to add. So our second method, the additions are a little bit more complicated, but on the other hand it makes it easier to track that I've added 50 and now I've added 5. Here we have to do a little bit more bookkeeping. So which method is best? Well it's whichever method that you have. And this notion of adaptive expertise is the skill of being able to change from one approach to another depending on the problem and on your current level of skill. And again, more generally this has been known in the past as if you only have a hammer. If the only tool that you have is a hammer, the only thing you could do is bash things with it. In real life we have to be able to switch flexibly between a variety of tools. If I want to cut a tree down, I'm going to use a chainsaw. If I want to butter a dinner roll, I'm going to use a butter knife. I'm not going to use a chainsaw to butter a dinner roll. And I'm not going to try to cut down the highest tree in the forest with a butter knife or a herring. So why would I add or perform any arithmetic computation the same way every time? And just as an example, add 998 plus 3. So here's the familiar way that we've all been taught. So the first thing I have to do is while I can't add 998 plus 3, well what I have to do is I have to rewrite the problem in column format. And then I do this complicated thing. 8 plus 3 is 1, carry the 1, 9 plus 1 is 10, write down 0, carry the 1, 1 plus 9 is 10, write down the 0, carry the 1 over, and there's my sum after about 12 steps. So this is the familiar standard method we have of doing this addition. If I try counting on, well let's see, 998 plus 3, that's the third number after 998, that's 999, that's 1000, 1001, and there's my answer. Now here's the thing to keep in mind. If I do this, I can't do this addition until I learn a lot of things. I have to learn my addition facts. 8 plus 3 is 11. I have to learn this standard algorithm. If I get a two-digit number, write down the 1, carry the 1 over the next column, and all this other stuff, I have to learn the standard algorithm. I have to learn my addition facts. I have to rewrite the problem because it's horizontally, but I can only do this if I set the problem down vertically. There's a lot I have to do to add this using the standard algorithm. On the other hand, as soon as I know how to count, and as soon as I know what the names of the numbers are, I can do this addition almost immediately.