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Skip to 0 minutes and 0 seconds {Blank} Okay so here we can see an example from the max math textbook. This section is about subtraction of more than two numbers and in designing these pages the authors have thought about the Concrete- Pictorial-Abstract approach. So for example -how can we see this?- we see a certain story over here but we also see that there is a concrete element and in this case it’s about marbles. I’ve used marbles when I was playing as a child and perhaps you could even have real marbles in your classroom. These are concrete materials that you would use to depict the ‘concrete phase’. And then you can see that there is a a story; there are 50 marbles.

Skip to 0 minutes and 51 seconds Toby takes seven marbles away, then Jay takes four marbles away. How many marbles are left? And then you can see that by utilizing a representation of real marbles, the idea is that you then perhaps go to a more pictorial phase. Let’s stay with the concrete we can write a subtraction sentence to find the answer because, well already in the story you can see there are initially 15 marbles then take away seven and then take away four. And here you can see how this whole subtraction is actually taking place. Please keep in mind that when you are teaching this, it doesn’t have to be marbles of course.

Skip to 1 minute and 36 seconds Over here -I’ll just see if I can put it on the screen- over here I’ve recorded a short video where I was actually utilizing some Lego bricks, for example. So here again you can see 15 of these Lego bricks. I had made bars out of them initially so here you can see that I am taking them apart. First there was a bar of 7 long and a bar of 8 long. Doesn’t really matter; there are concrete materials here that I’m using. And we have 15 of these Lego bricks here.

Skip to 2 minutes and 14 seconds And well, the initial sum says take away 7: “Toby takes seven marbles away”, well maybe if we use Legos we take seven Legos away as you can see on the left hand side and then we are left with eight and then there are four more marbles that have to be taken away. Or in this case Lego bricks again. And as you can see we are left with four Lego bricks. So this concrete phase is extremely important to make it concrete but it can just also be a picture or in this case even a movie of course. I can’t deliver these Lego bricks in your hands immediately. What you then see is that we can move to the pictorial phase.

Skip to 3 minutes and 0 seconds So here again you can see blocks and I use the word Lego bricks or Lego blocks. Here you can clearly see that the representation becomes more pictorial, especially in these pictures over here you can see almost that there is a brick pictorial that is being used to denote all the quantities. So for example if there are 58 counting blocks Toby takes 12 blocks away and Jake takes 26 blocks away. How many blocks are left? You can depict this and use it in the pictorial phase. This is not to say that you always have to do this.

Skip to 3 minutes and 42 seconds I can understand that it can be quite annoying perhaps at one point when you know how to do subtraction that you would then have to use all kinds of diagrams to actually show that you understand the representation. We need to keep in mind that in the end we want children to be good at arithmetic, be good at mathematics and these are just support tools to actually get to the final product, and to actually understand what you you are doing. We can basically continue with this so let me just continue here 58 minus 12 take away 12 and then there’s also 26 blocks that need to be taken away so we have 26- here as well.

Skip to 4 minutes and 23 seconds Then we can say oh well that’s the same as 58 minus [take away] these 38 blocks and then we are left with 20. So there are 20 counting blocks left. I think it’s also quite important that you have a final sentence in the task as well, so that you really go back to the initial task and make sure that you’ve actually answered the question. Children often seem to forget that they think, well I’ve done this sum, now I’m done, but really, this is a method of also checking your initial answer. And you can see another example that you can address in a similar way.

Skip to 5 minutes and 11 seconds This example; I won’t do that in this one but you can see that it can be addressed in a similar way. Okay, so we’ve already seen ‘Concrete’, which can be really concrete where you’re actually touching stuff, are using actual objects. ‘Concrete’ is in pictures or images or photographs because well that makes it a little bit more realistic of course when you’re dealing with a book. Then we went to the ‘pictorial phase’ here, where you can see diagrams being used. And then basically we end up with the final part which is the ‘Abstract’ which is the sum that you’re actually trying to calculate.

Skip to 5 minutes and 53 seconds We would ask you to do is to have a look at the last page of this and then ask you to actually work this out so for example here we have Hasan is 162 centimeters tall. Zahra is five centimeters shorter than Hassan. Maddie is three centimeters shorter and an analogous as the previous examples we’re asking you to see if you can address this. So Nadia’s height is 162 take away 5 take away 3 so we have 162 take away 8 so we have 154. So Nadia is 154 centimeters tall. And then we can practice a little bit

Skip to 6 minutes and 46 seconds more: subtract the numbers mentally and every time when you do a sum like this. If you are struggling a little bit, keep in mind that you can use a representation to actually aid your thinking. So I can go through all of these, of course, so we have 13 over here we have 42 minus 8 minus 30. Again I can do that perhaps in my head, maybe not, maybe you want to take first say 42-38 and that then gives you the 4, or maybe you can do it immediately like that. So we can go through all of these exercises and then taken together we can make sure that of course the answers are correct.

Skip to 7 minutes and 30 seconds We finish up in the ‘Abstract’ but we’ve started with representations that are Concrete, then went to Pictorial, finally ending up with hopefully a good and secure basic skill with regard to abstract arithmetic

Explanation: a CPA sequence

In this video CPA is described using some pages from a Grade 5 mathematics textbook.

This is an 8-minute video that describes the sequence of tasks, solves the tasks aloud, and in real time while using CPA terminology.

Note: These tasks come from ‘Max Maths Primary - A Singapore Approach’.

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World Class Maths: Asian Teaching Practice

Macmillan Education