Underpinning CPA with research – part 1

So welcome to this video on research on CPA, so the research basis of CPA which stands for, as you hopefully know by now, Concrete-Pictorial-Abstract. This is a particular pedagogical approach made popular in Singapore mathematics. I assume you have watched the previous video by Professor Fan and that you roughly know what the CPA approach entails and also understand what it actually tries to do. Of course you might think that this is actually, sounds quite reasonable and is common sense but you will be glad to hear that contemporary research seems to also indicate that there’s a lot to be admired in this approach.

I will discuss some research articles in this presentation, concisely of course, but to support the statement that it actually works. The first example comes from the raft of literature on Asian mathematics education. Take the overview by Leong Yew Hoong, Ho Weng Kin, and Chang Lu Pien. This article, at least the abstract, is depicted to the right of me and they summarise the Concrete- Pictorial approach. They provide an overview of the origins of the CPA approach. And it actually has some historical roots.

Bruner, Jerome Bruner in his 1966 book ‘Towards a theory of instruction’ talks about enactive, iconic and symbolic modes of representation and one particular adaptation of this is the concrete-representation-abstract sequence which especially has been effective for students with mathematical difficulties. And this of course was already in the 60s. The ‘concrete’ has been a theoretical element in the use of manipulatives; so actually having elements that you can manipulate within mathematics. And like the model method, which we will cover in the next section, CPA has been a key instructional strategy in Singapore since the beginning of the 80s. But if it’s based on existing strategies, what makes the Singapore approach different?

The big achievement, I would say in Singapore is, as the authors describe in that article as well, is how CPA has been adopted by the entire mathematics education community in Singapore, as well as the textbooks. So they have actually integrated CPA in the textbooks, making it a general approach throughout Singaporean mathematics education. Bruner’s enactive-iconic-symbolic distinction was part of a much larger theory of instruction. He says “any set of knowledge can be represented in

three ways”: by a set of actions appropriate for achieving a certain result, enactive representation. By a set of summary images or graphics that stand for a concept without defining it fully, which he calls iconic representation, and the set of symbolic or logical propositions drawn from a symbolic system that is governed by rules or laws forming and transforming propositions, which he called symbolic representation. As part of Bruner’s theory of instruction the choice among multiple modes is dependent on other features of representation; he calls them economy and power.

Economy has to do with the amount of information needed to process the representation, in order to comprehend the underlying knowledge and the latter refers to the potential of the representation in helping the learner to go beyond what is touched upon the surface of the representation and to connect this to deeper or related ideas. What Bruner also finds incredibly important is to look at sequences of mathematical tasks. He describes at length how the enactive stage could be carried out by getting students to work on algebra blocks -and these are three-dimensional forms, for example, of algebra tiles. And you can also think perhaps about Lego bricks that you use to actually make this more concrete.

And then perhaps you gradually guide them to a more iconic representation and then all along the way notation was developed and converted into a properly symbolic system. And again this is in the 1966 book, so he goes from something really concrete towards something more abstract and you can already see the three letter C, P and A emerge from this. Note that these were not described as discrete phases. It was not as if one finishes at one exact point and then a new phase starts. Bruner did emphasise that eventually students would need to get to that symbolic phase.

Also, if a student perhaps struggles with mathematics and he also said that there is a certain risk involved if you bypass perhaps some of these stages. It is always useful to have a fallback that if you’re in the symbolic mode and you don’t know what to do that you’re actually can fall back perhaps on an iconic presentation for example. That doesn’t mean that you have to do them, that’s not what Bruner said, but it can be a useful fallback if you don’t manage to solve for example an equation or in primary school you don’t manage to calculate a certain sum. It should not be forced upon students.

In the enactive-iconic-symbolic that Bruner describes there already is a lot of overlap with concrete, pictorial and abstract, which according to for example Singapore’s syllabus from 2012, has been used in mathematics education for quite some time. Some terms have slightly changed, for example concrete is not restricted to Lego blocks or certain concrete manipulatives but can also be different concrete materials. It can also be concrete experiences which was further explained as comprising activities with suitable manipulatives. This view of concrete is very much in line with Bruner’s enactive, which is also about mathematical knowledge as embodied in actions. Within the same document from the Singapore government relatively little is mentioned about the two other parts, pictorial and abstract.

There is nevertheless a reference to pictorial as representations, so multiple representations, which aligns quite nicely with Bruner’s iconic phase. And the language of ‘guiding through’ is also important and quite in line with the sequencing of different modes, working from first phase to the second phase and the third phase. And then of course abstract is conceptually not far off from the language ‘symbolic’, emphasis of Bruner’s symbolic.