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Presenter

Okay mathematicians, welcome back. I am about to show you a collection of dots and I'd like you to think about, you know, how many dots are there?

But I'm really interested in is, how do you see them?

Okay ready, it's subitising, so you don't get to look at it for too long.

[Screen shows an A4 white piece of paper and coloured markers. The presenter then displays a yellow piece of folded paper with 8 blacks dots, displayed in a formation of 3 horizontal rows, 3 dots in top row, 2 dots in middle row and 3 dots in bottom row with middle row dots spaced in the middle of the first and third row dots, almost like dice dots only there are 8 dots.]

Here we go. How many dots? And how do you see them?

[Presenter now turns yellow paper over, so you cannot see any dots.]

You're right, it was fast 'cause we're getting in on your capabilities to subitise and instantly recognize quantities.

So, there are a lot of dots there.

So, what I'd like you to think about, I'll show you again, but I'd like you to really think about looking for chunks.

So, looking for things that you can see inside all of those dots. Some things that are familiar to you that you could use.

Okay, and then see if you can see a few of those chunks.

Alright, ready to have a look again. Okay, how many dots and how do you see them?

[Presenter turns the yellow paper over to display dots again briefly. Then turns the paper back over and moves it to the top of the white A4 piece of paper.]

Ah, yes, so you've thought about some chunks now. Good.

You're right, it is a little bit like if you've done number talks in your classroom, isn't it? Using a dot card this time?

So, you might be sitting there and thinking, going oh my fist means, I'm just thinking.

[Presenter shows fist and pumps her hand.]

This means, I've got one way of thinking about the chunks and how I see that collection of dots.

[Presenter shows a fist with one thumb up.]

Yeah, and then if you got one way of thinking about it, is there another way you could have seen them or imagined them?

[Presenter shows a fist with one thumb and one index finger up.]

Okay, should we have a look together? Okay, so here is the collection and the first thing we need to work out is, how many dots are there altogether?

[Presenter turns yellow paper over to display 8 dots and turns folded part of the paper down to reveal 8 written as a numeral underneath the 8 dots.]

Yes, there's 8. Okay, so I have 8 in my collection here.

Now what I'm most interested in is, how do you see this collection of 8? What's one way that you could think of?

Ah, okay, some of you are saying you saw a chunk of 5 here, like on a dice pattern. And then three more like an arrow or a triangle?

[Presenter points to dots circling the first 5 dots with her finger and then points to the last 3 dots and traces the pattern of the 3 dots in an arrow formation.]

Okay, let's record that way, as one way of thinking. So, I'm drawing the 4 squares, the 4 dots like a 4 and then one more makes a 5 dice pattern. And then 3 like a triangle. 1, 2, 3 like that. And that's 5 and 3. Better write 'and' in there.

[Presenter uses the green marker to draw 5 dots in a dice formation on the paper and then draws 3 dots in red to the right, in an arrow formation. They then write on the piece of paper to the right of the dots, ‘5 and 3’, writing the numerals in the same colour as the dots.]

Okay, what's another way of thinking about it? Ahhhh, some of you were saying you saw this triangle in this end, but also at this end and then 2 more dots.

[Presenter points with her finger and traces around the 3 red dots indicating triangle and then traces around the 3 on each end of the yellow paper indicating triangles with 2 dots in the middle.]

Okay, so a chunk of 3, a chunk of 2, and another chunk of 3. So, here's one chunk of 3. Here's the chunk of 2. I better write that 3. 2, and the other chunk of 3. Like that. And I need, an and, and an and. So, we see 3, 2 and 3.

[Presenter draws underneath first group of dots, 3 purple dots, 2 red dots and 3 blue dots on white piece of paper in a dice formation and writes the numerals 3 in purple and 2 in red and 3 in blue, then traces with her finger around the dots.]

[Presenter points to the assorted colours.]

Ah, you like my use of color to correspond. Thank you, I do too. Sometimes I don't worry about color, but sometimes it really helps get inside people's brains and understand their thinking and also convey ideas.

And you're right, I could improve this too actually, 'cause if I drew a line underneath there, mmmm, it makes it easier to see one way of thinking compared to another way of thinking.

[Presenter draws a horizontal line in between the 2 groups of dots and points at each group, then draws a horizontal line under the bottom group.]

Okay, I think we have another way of thinking, actually, where someone was saying they read it like a book.

You know across each row. Yeah, and they saw 3 and 2 and 3.

[Presenter points to dots on yellow piece of paper and traces with finger from left to right in rows.]

Okay, so I'm going to do 3 across the top. Oh, I know some of you are going, "That's not my way yet." It's okay, we're coming. 'Cause there's, I know it's amazing, isn't it that you can look at something and go "Well, we know it's 8 dots but there's all these different ways of thinking about it. Yeah, that's why we, well part of the reason why we love mathematics.

[Presenter draws underneath, 3 red dots in the first row. Then they draw 2 green dots in a row underneath centred between each red dot and then 3 purple dots in a row underneath the green dots but lined up with the first row of red dots. They write the numerals 3 in red and 2 in green and 3 in purple and point to demonstrate the 3 rows, then draw a horizontal line underneath the thinking.]

It's 8 dots but look at how many different ways we can imagine those eight dots. Here we have 3 and 2 and 3 more.

Ah yes, I saw this too, and this way, it's going to be tricky for me to draw it, but I'll try.

[Presenter uses her finger to circle the chunk of 5 black dots on the left then the right-side side of the yellow paper and then circles the 2 dots in the middle. Then they spread their hands out and overlap each hand.]

I saw a chunk of 5 and a chunk of 5, and these 2 dots overlap. So, I actually saw 5 and 5 - wheew and then minus 2.

Yes, I know. I'll, I'll draw it and then I will explain it. So, drawing 5 on my dice.

So, in this case, look, I do 5 like 4 and one more. And then in this case I could draw 5 like 3 on a dice.

[Presenter draws underneath on white paper, 5 red dots in a dice pattern and 5 blue dots next to them in a dice pattern.]

And 2 more. I know that's a fun strategy. And then what happened? I need my scissors actually, is, is, this 5 slides across, whoosh, and covers that 5.

[Presenter uses scissors to cut in a line either side of the blue dots. They then cut above the blue dots and slide the 5 blue dots over the 2 red dots on the far right side and then place back into position.]

Um, so it's 5 and 5 minus, yeah, the 2 that I covered, so I might write this as 5 and 5, but then minus, and if I'm going to be careful using color, I might, I might use the 2 to get rid of the 2 here, 'cause they're the ones, yeah, that I sort of see sliding over.

[Presenter writes on the right-hand side, 5 in red and 5 in blue minus 2 in red. They then draw a red cross over the 2 red dots on the right-hand side and slides the cut-out paper with the blue dots across to hide them.]

So, I think that represents my thinking. A 5 and a 5, minus 2 which is also 8.

And now what I'm thinking about actually, is there's lots of different ways to represent 8 and I'm wondering what's another way that we could prove that these are all in fact 8?

Even though there's different chunks of meaning inside of them.

Okay, let's have a look. Okay mathematicians let's use this balance scale to check out some of this thinking.

[Screen shows balance scale, some blue pegs and the white A4 paper with thinking about 8 dots on it at the back.]

So, on this side, I'm going to put a peg on 8 because I knew my representation was 8.

[Presenter places 1 peg on number 8 on the left hand-side of the balance scales and shows the yellow piece of paper with 8 dots and 8 written on it.]

That's how many dots we worked out that we had. And now what we want to do is the first thing we said was eight was 5 and 3 more.

So, let's put a peg on 5. Oh, and nothing's happened, and let's try with 3 and see.

[Presenter place one peg on the scales to the right on number 5 and one on number 3, the scales now balance.]

Yeah, and there I can see that 8 is equivalent to 5 and 3. So the other one we looked at was it it was 3 and 2 and 3. So if I take this off.

[Presenter now points to 8 on the scales and then points to the 5 and 3 on the scales and points to the paper.]

That's right, it goes all the way down, doesn't it?

[Presenter removes the peg from 5 and the scales tip to the left.]

To say to me it is not balanced and if I put the 2 on, it didn't even move it, did it? Okay, and then I guess for the 3, I put that on the other 3 peg. And I just have to balance the white bit so they're the same too.

[Presenter then puts a peg on number 2 and another peg on number 3 and the scales balance.]

Look at that! 8 is 2 and 3 and 3 or 8 is 2 and 2 3’s. Ha!

Nice work mathematicians. Okay mathematicians. What was the maths? Hmm, that's right, we notice that mathematicians can see the same representation and think about it quite differently.

That numbers, but bigger numbers like 8 are made up of smaller numbers like 5 and 3. So we saw 8 is 5 and 3. That 8 is also 3 and 2 and 3. That 8 is 2 and 2 3’s.

Remember how we balance that on the balance scale? Yeah, and that actually 10 is 2 more than 8.

Yeah, and we also really saw today how how a careful or intentional use of colour can help us make and convey meaning as mathematicians.

Now what I wonder for you is.

Did you see that representation in a different way? What about people in your family? How do they see the dots?

Off you go to go and investigate. See you next time.

[End of transcript]