 Hello, and welcome to this very brief introduction to a powerful method of moving from arithmetic to algebra, which I call tracking arithmetic. Based on an idea by Mary Everest Bull, she was married to George Bull, brought up five daughters, and he died when she was only in her thirties, and so she needed income, and she was teaching mathematics and writing books about mathematics, teaching, and other aspects of philosophy. So the basic idea is as follows. Would you please think of a number? Add two, multiply by three, subtract four, multiply by two, add two, divide by six, subtract the number you first thought of, and I think your answer is one. Not only is your answer one, but so is everyone else's. Now, children, the early, secondary, and late primary often find this really intriguing. How on earth can you possibly know when they've been doing all this arithmetic? And so the first thing to do when working with them, what I would do is to say, well, let's try a particular number. I always choose seven, because it's a number that people think is hardest for doing arithmetic on. And since I'm not actually going to do any arithmetic with the seven, it's a really good choice. So think of a number I choose seven. Add two, multiply by three, all I get is three times seven plus six. The idea is to keep the seven isolated and only do arithmetic around it. Subtract the four, multiply by two, you got two times three times seven, six times seven, plus two times two, it's four. Add another two. Now you can see what's going to happen. You divide by six, you get seven, add one. Subtract the number you first thought of, which of course was seven, and the answer is one, and you smile. Now, the whole point about this of tracking arithmetic is to treat the seven as a placeholder. So instead of just having a seven, I'm going to put a cloud around the seven to indicate that's the number I was thinking of. Now, somebody else might have been thinking of a different number. I can use the cloud to represent or to stand for, to denote what they were thinking about. And to do that, I simply take the sevens out of the cloud. Now I've got all the computations that somebody would go through with some number that I don't even know. It's in the cloud. The clouds all disappear, and the answer is always one. The next slide on the PowerPoint gives you another chance of doing something similar with think of a number, showing you some of the variations you can do with that. I want to show you quickly a way of doing this to an ordinary routine task taken from a GCSE exam. In order to find the total area, I think it's sensible to find the area of a brick. So let's see if we can just find the dimensions of a brick. So there's the diagram. Now, the important thing to notice is that the vertical height is a length plus a width. And on the right hand side, I've got a length plus a width, but sideways. So I know that length. That's also seven. Now I can find the length of a brick. It's 11, subtract seven. It's not four. I mean, it is four, but I'm going to denote it as 11, subtract seven. So I can retain how I got it. And now I know that the length of a brick is 11, take away seven. I can find the width of a brick by seven, take away 11, take away seven, which we would often write as two times seven, take away 11. So the brick is 11, take away seven by two times seven, take away 11. Now I want to remind people about a mathematical habit of mine that's really useful, which is whenever you've done a single question, ask yourself, how would you do a similar question in the future? And you do that by asking what could be varied? I can vary the seven. I could vary the 11. I could also vary the configuration, but let's leave that, I'll leave that to you later. I'm going to replace each 11 by a cloud. So it's just some number that someone's thinking about, which in this case happens to be 11, but needn't be. And then replace all the sevens with a brown cloud. And now I do that in the assembly of the answer. So now I know that for whatever number of people put in the green cloud and whatever number of people put in the brown cloud, I know the size of the brick and from that I can find out other things that the question wants me to find. Now here's another configuration, but what's important about it is the task of make up your own like these and writing down instructions as to how to do that task principally by using tracking arithmetic and eventually people can just use letters instead of having to use clouds. The rest of the PowerPoint shows you a number of other contexts in which you can use tracking arithmetic and some of the wrinkles that might arise from that. And I want to end this presentation with tracking arithmetic is seen as an aid to generalization because it's shifting your attention from the particular answer to a general method. If you're trying to practice for an exam, what you want to general methods, not trying to memorize specific questions and how you do them. It's tracking arithmetic can be seen as an introduction to algebra, gradually shifting from the cloud to a letter as a symbol for something that is yet unknown. Tracking arithmetic gives you an opportunity to shift from doing things expressing generality to undoing using algebra to resolve problems. From since the 15th century, most algebra texts have said that algebra is the arithmetic with letters. And I find that really, really unhelpful. And one of the reasons why a lot of people stumble with algebra. Algebra for me is the expressing and manipulating of generalities and tracking arithmetic is one jolly good way to introduce that. And then the algebra arises when, if you know the answer, you want to find what the question was or use the algebra to resolve a problem. So please use the rest of the PowerPoint questions to develop your sense of what tracking arithmetic is or could be about.