 Hello my name's Dave Hewitt and I'm going to be talking a bit about awareness and about thinking a bit about fractions later on as well. I'm going to start off by sharing a joke that I heard on a radio and it went something like this. A child comes home from school and says to their parent, hey today I learned that two apples plus three apples is five apples. And their parent said, oh wow, that's impressive. Well tell me, what are two bananas plus three bananas? And the child says, no idea, we haven't done bananas yet. Now there's something about this joke from me that sums up what can happen in classrooms. That students can sometimes be waiting for teachers to tell them what they have to do and they try and memorize what they're being told to do. And then they reproduce that in a series of similar questions afterwards. And they get used to the idea of just sort of waiting for the teacher to tell them what to do and then they do it really pretty much something very similar to what they've been told with very similar examples. And I want to work towards sort of encouraging classrooms to be such that children can come through the door with ready to use more of what Gutenio calls powers of the mind. The ability to abstract rules, to be able to apply those rules, to see what's, to consider what's the same and what's different about things. To be creative, these are many of the powers of the mind that we all have as human beings, but sometimes in some classrooms children can get used to the fact that those are not called upon very much. And so they get used to using maybe just memory, which is only one of the powers of the mind that we all have. So, if I think for example about the idea that 2 plus 3 equals 5, then indeed 2 apples plus 3 apples is 5 apples and of course 2 bananas plus 3 bananas is 5 bananas. And 2 rulers plus 3 rulers are 5 rulers and 2 cats plus 3 cats are 5 cats and 2 7s plus 3 7s are 5 7s and 2 glasses plus 3 glasses are 5 glasses and 2 cars plus 3 cars are 5 cars and 2 x plus 3 x is 5 x. And 2 fingers plus 3 fingers are 5 fingers and 2 thirties plus 3 thirties are 5 thirties and 2 spoons plus 3 spoons are 5 spoons. This, we know all of these things. This all comes with the fact that 2 plus 3 equals 5. Now, if we just treat 2 plus 3 equals 5 as just one little thing and we move on to some other addition, then I think we're doing a disservice for what that statement actually means. And the statement is a profound statement and a statement about all of those situations I've mentioned and many, many, many, many more. And if I know that 2 plus 3 equals 5, then I know all of these things that I've just said. And that's partly this bit about getting a lot from the little. There's all these things that come from just 2 plus 3 equals 5. And it's that sort of awareness that all these things come from that one statement. So let's think about something else. I've got here 2 pencils. What have I got here? And I would like you to say it. I'd like you to say 3 pencils. So I've got here and I'd like you to say it. And I've got here and now what have I got? Okay. And now I've got say it. What have I got here? I've got, what have I got here? And now what have I got? So say these things. And then here I've got, what have I got here? These are 3 brushes. So say that. And what have I got here? And what have I got now? And what have I got here? And what have I got here? And what have I got now? And what have I got here? And what have I got here? And what have I got now? Ah, well, I can hear you say. And my guess is, my conjecture is that you ended up saying something like, well I've got, I don't know, I've got 6. I've got 6 things. And what you've done is really significant. What you've done is to realise that these two do not have the same name. And you can't add things together unless they've got the same name. They need the same name. So what you do is you find a name that they have in common, a name that can be applied to both of them. And then adding them together is straightforward. And this is an awareness that we all have. If I do that activity with children, they have the same dilemma. And in the end they grapple around and they find a common name like the name things. And this applies to fractions as much as anything else. But actually to add fractions, once I get a common name, I'm fine. But if I don't have a common name, it's a problem. So this is a really important awareness when I'm thinking about the addition of fractions. I have to have that awareness that in order to add them, I have to find the same name. And that's one of the, what I call key awarenesses that are around within that topic of fractions when I'm considering the addition of fractions. Because it's really a property about addition when I'm adding any two sets of things. If I've got two of something and three of something with the same name, then I've got five of those things. And that awareness is a key awareness. So one of the things that I would like you to think about, what are the other key awarenesses that are involved within the topic of fractions? This is one of them. It's that fact that if I want to add them or subtract them, then I can't do it and I've got to find the same name first. So what I would like you to think about is to think a little about awareness. Things, awarenesses that strike you when I give you a particular activity. And these might be awarenesses that, if you like, here I'm not thinking about things that you've remembered. They're things that are, for me, awareness is slightly different to memory. Memory is something where maybe someone's told me something and I have to memorize it. But that's awareness about needing to find the same name. It came without me saying or you saying, oh my goodness, me, I can remember what somebody told me once. So I have to find the same name. It came because there's an awareness that you've got that came naturally as part of our language. So I want you to be looking at some sort of key awarenesses that are around within that topic of fractions. And one of the things I want you to do now, I'm about to give you an activity to do. And as you do that activity, I want you to try and catch what awarenesses you're calling upon whilst you're trying to do this activity. And also you might be aware of some awarenesses that aren't so available for you. And that you need, you might need to work on a little bit more in order to sort out this task. So there are two levels of awareness I want you to use. One is to just be aware of what you're doing and what you're thinking, what's coming to mind and so on. But then there's that level beneath that which are what are the awarenesses that you're calling upon in order to do this task. And what is it that you're aware that you are not aware of, but you might need to know a bit more about in order to succeed with the task. So the task is I would like you to find a fraction that is in between five sevenths and three quarters. So a fraction that is in between five sevenths and three quarters. That's in a sense, task number one. Task number two is that again I want a fraction between five sevenths and three quarters where the numerator is 11. So a fraction between five sevenths and three quarters where the numerator is 11. And then maybe a third question to consider is that could there be a lower number than 11 for the numerator where it is the fraction is still between five sevenths and three quarters. And maybe the last question I'll ask you is that is there a number like 11, maybe 11 may not be 11. Where by every other whole number after that could also be the numerator of a fraction where it is between five sevenths and three quarters. So if it were 11, I could get 11 over something would be in between. But also there would be a 12 over something else which would be between 13 over something or 14 over something or 15 over something from there onwards. Would always be a fraction that is in between five sevenths and three quarters. A final restriction I'm going to make on all of those tasks is that you have to stay with fractions. No decimals, no calculator. It's always working with fractions. Enjoy the tasks and be aware of your awareness.