Explanation: more CPA

Here we have another example of the Concrete-Pictorial-Abstract approach. Again in the max math book but now for addition of two and three digit numbers. As you can see over here, you can see that there is a concrete aspect which are these colored pencils. There are 14 colored pens and probably every classroom has a lot of colored pencils around so if for example you would take 25 of them and then say Tya has 14 colored pencils. How many colored pencils do they have altogether? Then I think it’s quite clear that you’ve got a concrete example where you’re actually adding two numbers together.

We can make this example more pictorial so, like here you can see that these bars are being used to actually represent the quantities, hence colored pencils, 25 of them. Tya’s colored pencils 14 of them, and they can be brought together. And this of course can be done in several ways but in this case, sort of, the tens have been brought together as you can see over here. Tens have been brought together and then the units together making 25 plus 14 the same as 30 plus 9 and that together makes 39. So here you can really see how it goes from the concrete to pictorial to abstract which is the sum that you wanted to finish with.

And in this section of the book, the tasks continue to be exactly like that. Find, for example, 25 plus 44 using counting blocks and the column method. So here again you can see that there are examples that are very pictorial and the way that the units and the tens. The tens and the ones, have been organized, are used to actually go to a more abstract representation. You could even say that this, because of all the circles around it that this also has some pictorial elements, but hopefully it’s quite clear that this ‘scaffolding’ which also is a term that Bruner coined and perhaps you remember that the Concrete-Pictorial-Abstract approach originates from a lot of principles that Jerome Bruner formulated.

Here you can see it really in action. You go from something concrete to pictorial and then the abstract. The [PDF] file has some more of these examples, so here is the third page, where you can actually see that you can go to hundreds as well and of course at a certain point it will become quite difficult to actually represent these quantities with cubes or anything else, but hopefully it’s quite clear that then students and pupils can actually use the idea behind it and then at a certain point, they might not need the representation anymore.

It is a fallback, you could say, a ‘plan B’ that is always there and that can lead to further secure understanding and skill and an arithmetic skill to make these tasks. And then the chapter continues, the section continues with some more examples and I really advise you to try and make these examples yourself. The solutions will also be posted in the course, as well of course.