Dot card talk 3

Okay mathematicians, what do you notice? Now get your eyeballs ready. That's right, we're about to subitise. Here we go.

So, about how many dots did you see? Yes, there were a lot. Would you like to see it again? Here we go. What do you think?

Yes, we asked the mathematicians, some students in our schools and they when they first saw this a block of or collection of dots like you. They said, well we think it's somewhere between 25 and 78 dots. And we said to them, how certain are you? And they said not at all.

So here's the same collection of dots, just rearranged differently. You might have noticed that some of the dots were big and some were medium size and some were small, so we clumped them together. And now what do you think? How many dots are there altogether?

Yes, so our student mathematicians also said we don't really think that helped much. We still don't really know if it's between any less than 25 or any more than 78. We'd like some more information.

So here come the same collection of dots now just arranged in another different way. Aha and what do you think now, would you revise your estimate? Okay, so the students did too. They were like, well, we think 25 is too small. We think there's more than that, so we think the smallest quantity there might be is 35 and we think 78 is too big. So we think the largest possibility is perhaps 55 and so our range has now changed between 35 and 55.

And we said to them, How certain are you? And they said we feel a bit more confident, but we still don't know. So ready, get ready to look again.

We're going to show you the same collection of dots just organised in a different way. And now what do you think? Aha yes. Most of our mathematician said this too, arrgghh now we know, it's 43.

Would you like to see it again ready? Oh first of all, sorry, we said, How certain are you? And they said we're, almost completely certain. And we said to them, why? Like what's helped you become far more confident in being able to offer pretty good estimate and in fact being able to say, look, we're pretty sure it's 43.

And this is what they said. It is that when you saw it like this, they could see the structure of the ten-frames. Can you see that? Yes, so it was the structure of the ten-frames that help them go oh there's just 4 ten frames and 3 more which you call 43 in rename, with renaming. Yeah.

So, and they said it doesn't matter 'cause here they're wonky, look if we straighten them up a bit, that's what it looks like. Straighter, yes, and we made it intentionally difficult with some of the dots different sizes, but it's the same representation. Here it is there.

And all we did was delete the ten-frames underneath it. Aha, and doesn't that change how you can quantify the collection. Because when we see it like this it doesn't look like we can see anything much of use, but when we have this structure underneath us, it.

Yeah, it really helps us see there's 4 tens and 3 leftover, so that's 43. So here's what our young mathematicians said to us about this when they were reflecting. They said the random dots, the one in the yellow box was the hardest to estimate how many. And their ranges went from 25 to 78. Would you agree with that? That was the hardest one to see initially.

Uh-huh then they said the one that was the next, was helpful, was actually when it was set up or an attempting to set up in an array. So even though it's not array 'cause they weren't the same number of equal rows and columns, that help them revise their thinking a little bit. So they said they thought 25 was too small and 78 was too big.

But that way of arranging the dots helped them see that was somewhere they thought between 35 and 55 and they could feel a bit more confident in their estimate.

Did you, did you feel the same with that? Okay? And then when they saw this one, the green one, they said well, when we saw that structure, we know that we can trust a ten-frame and we could see that it was 4 tens without having to calculate.

And that this then helped us revise our estimate even though the dots were messy in the ten-frame. Then we could just use our knowledge of place value to rename the collection to know how many, 43.

Aha, so what was the mathematics? Well, there's a couple of things in here, but two really big ideas. One is that familiar structures like ten-frames can help you determine how many there are in a collection without having to count them all. And secondly that you can trust a structure even if the dots are a bit wonky. And if we combine this with something else that we also know is that you don't always need a structure to help you know how many either.

Look, ready? Eyeballs ready? How many dots did you see? Okay eyeballs ready again. How many dots did you see?

Yes, so because the collections are small, they are in fact both 5, our brains can sort of see that we can see chunks like 4 and 1 or 2 and 3, and because we're really knowledgeable about the way that numbers are made up, we can say, well, I know 5 is 4 and 1 or when I see 3 and 2 then I just know that's 5, but structure is really useful when we've got big collections, look. Aha. Yes, and so the collection has to be big enough for structure to help us.

Okay mathematicians, I look forward to you investigating ideas of structure. Until next time...