 So let's do some arithmetic. So when we add, we are putting two collections together, and we're looking at the cardinality of the joint collection. Now, that's what we're doing. An algorithm is a systematic way of solving the problem. So here's the actual process of addition. But how we do that addition, how we determine the cardinality, falls into the category of an algorithm. So one of the important things is that research in mathematics education consistently shows the following. First of all, students can and will develop their own strategies for solving problems. You just have to let them find these strategies. More importantly, the students who do develop, who are given this opportunity to develop their own strategies, consistently do better in mathematics. And here's an important thing. When you give students an algorithm, when you say here is the systematic way of solving this problem, you're not letting them develop their own strategies. And what that's happening is they are getting a worse education than if you help them develop these strategies. So what am I about to do? Well, I'm about to give you a strategy for solving a problem. So take that for what it's worth. But I won't give it out completely. So let's take a look at this. Let's try and add five plus three. So addition occurs when we join collections. So this corresponds to joining collection, looking something like this. There's our five. There's our three. We've run them together. Addition corresponds to the cardinality of the joint set. And the simplest way we can do that is just determine how many of these squares are present in the combined set. Well, the easiest way to do that is to count them all. Count every single one of them. And we call this a count all strategy. And I count 1, 2, 3, 4, 5, 6, 7, 8. And the sum 5 plus 3. Here's 5. Here's 3. Run them together. I have eight things all together. And I can write that as the arithmetic fact. 5 plus 3 equals 8. Now, we might ask ourselves the next question is, can we do this more efficiently? And this is the heart of all algorithms. How can we improve our process? Well, the process, the obvious thing to do is count everything here because that is what the sum means. On the other hand, can we find a better way? Well, again, research shows that students, if they're allowed to develop their own ideas and they're allowed to work with it, they will find ways of improving the algorithm. They will hit upon important insights that allow them to find 5 plus 3. Rather than counting every single one of these, they'll find ways of improving the algorithm. So, again, students who actually do this tend to learn more and do better. They are better mathematicians. So again, it's important to allow students the opportunity to find their own ways of improving efficiency because they will eventually do more, learn better. Well, one important way to improve the algorithm is to recognize that I don't actually have to count this set 1, 2, 3, 4, 5 because when I put it down, I put down a set of 5. And so in some sense, I already know how many things are in this set. I don't have to count them. What I can do is I can start at 5 and go from there. And this is known as count on. And so the idea is I can start my count at 5. Here I am. And then count on from that point, 6, 7, 8. And so that gives me, again, the same sum. 5 plus 3 is equal to 8. But using a different strategy, count on. Now, let's talk about reducing efficiency. Counting on is a mental strategy for arithmetic. And it works best when we do it in our heads. Or maybe we might count on our fingers or do something like that. But the important thing is that it's a mental strategy. We can try to write it down. And the obvious way to do that is to write down what we're actually counting. But keep in mind that you're trying to record a mental strategy by writing it down. And so the effect is something like recording a symphony by taking a picture of the orchestra. You might not get a good idea of what's really going on. And it's certainly not the best way to hear the music. If you look at a picture of an orchestra as it's playing, you won't really hear much of the real music. Society, unfortunately, likes written work. And it is actually hard to grade what's in somebody's head. So we might represent counting on in some manner like the following. If I want to find 5 plus 3 by counting on, remember I'm starting at 5. And then I'm counting the third number after. And I'll write down what the actual count is. So maybe I'll write down something like this. From 5, I count on. The next number is 6, then 7, then 8. And that's 1, 2, 3. That's my third number after. And there's my 5 plus 3 is equal to 8. Now, later on, it'll be convenient to record the fact that we're actually adding 1 to get the next number in the sequence. So I might expand my written form of counting on this way. 5, add 1 gets you to 6. Add 1 gets you to 7. Add 1 gets you to 8.