 Alright, so let's take a look at another method of subtraction known as counting down. And again, just as a reminder, the goal here is not to learn how to subtract, but rather to be able to perform subtraction at whatever level of knowledge you happen to possess. And this again ties back to this notion of adaptive expertise if you know many ways of solving a problem, choosing the best and most efficient method is known as adaptive expertise. And again, educational studies have shown time and again that students who possess adaptive expertise generally outperform those without adaptive expertise, and that acquiring adaptive expertise is mostly a matter of learning more than one way to solve a problem. Or again in the fairly standard way of looking at it, if the only tool you have as a hammer then the only way you have of solving a problem is to bash it until it falls apart. So let's take a look at this. Counting down is another way to subtract, and the main difference from counting up is that to find B minus C, if I'm counting up, I want to count up from C to B and see how far I've gone. If I'm counting down, I want to count down from B down C units and record where I end up. In general, one of these approaches is going to be more efficient. But given identical problems, the algorithm that is more efficient depends on the student, and in fact the same person working the same problem on a different day may decide one method is more efficient. Let's take a look at this. Let's consider the problem 31 minus 7. If I only knew how to count up, I'd have to count up from 7 to 31, and this could take a while. On the other hand, if I try counting down, then what I want to do is I want to find the seventh number before 31, and that's going to be much faster. Now if I know nothing else other than how to count, then I could still do this problem by just counting. I have to count backwards 31 to 30 to 29 to 28, 27, 26, 25, 24, and that's my 1, 2, 3, 4, 5, 6, 7th number before 31, and so at this point I know that 31 minus 7 is equal to 24. Again, the thing that's worth noting here is this is the problem that you can solve as soon as you know how to count and what subtraction is. Now again, a lot of basic arithmetic is bookkeeping, so let's see if we can record the things that we've done using our arrow notation. So again, I started at 31. I went back 1. I'll indicate that using this minus 1 above the arrow. I went back 1 to 30, back 1 to 29, back 1 to 28, back 1 to 27, back 1 to 26, back 1 to 25, back 1 to 24. And so I've gone back 7, and I end at 24. As with counting up, we can use what we know about subtraction facts to improve the efficiency of our algorithm. So if we do learn some subtractions, we can apply this method much more efficiently. So from 31, I might go back 1 to 30, and if I'm at 30, well, maybe I know that 5 before 30 is going to be 25. So instead of counting back down by 1s, I might count back by 5, down to 25. And so far, I've gone back 1, 5 more. I've gone back 6. I want to go back 7, so I'll go back 1 more, and that takes me down to 24. Well, let's do a little more complicated problem, 103 minus 35 by counting down. And again, just to see the variety that is possible here, let's take a look at this in three ways. Again, mainly because this is a problem that can be solved whatever your level of knowledge happens to be. If the only thing you know how to do is count and count backwards, you can solve this problem. If you know more, you can solve the problem more efficiently. So again, we can read this as trying to find the 35th number before 103. And so what I might do is I might just count down by 10s from 103. So I'll count backwards, that's 103, 10 backwards is 93, 10 back is 83, 10 back is 73. That takes me back 10, 20, 30. That takes me back 30. So I need to go 5 more, and I'll just count back by 1s. So 1, 2, 3, 4, 5, and going back 35 takes me to 68. And so 103 minus 35 is 68. As an alternative, I might make use of those benchmark numbers. So my first benchmark number counting backwards is going to be 100. So I count back to 100. Now I want to go back 35, so I can go back again by 10s. So there's 10, 20, 30, 1, 2, 3, I've gone back 33. And so now I want to go back 2 more. And again, if I've gone back 35 from 103, I get back to 68. Now if I actually know some subtraction facts, I can actually count down by larger amounts. So for example, I might know that 103 minus 30 takes me back to 73. And that's really a collapse of my counting back by 10s. So I don't really need to know subtraction facts to find this one because I can do that by repeatedly going back by 10. But then I may have another subtraction fact. If I go back from 73, back 5, or count down to 68. And so again, I've gone back 35, 103, 35 back takes you to 68. And there's our third method of solving the problem.