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Skip to 0 minutes and 0 secondsWe can also look at textbooks, for example, in Singapore. This was a government document but we can also look at the textbooks. The CPA approach can also be found in these textbooks. However, they were again slightly changed along the way because of course you can't really put concrete materials within a textbook. What you can do, and this was a significant change and shift from the emphasis from teaching and learning in the 1990s, is that you can actually offer a wide range of learning manipulatives, but actually within the textbook. Rich contexts that are very concrete. Concrete experiences to facilitate learning.

Skip to 0 minutes and 46 secondsConcrete experiences can take the form of activities, for example, and what also changed is that there were, that first was an emphasis on the primary, primary education, but now they also took these principles more to secondary education, So these textbooks in Singapore have added something more to Bruner's original ideas but certainly the core ideas from Bruner are still very much apparent. Another difference perhaps to denote is that in Bruner's original Bruner talked about stages in terms of instructional sequences.

Skip to 1 minute and 26 secondsAnd some people have taken this as levels, so they think that perhaps one stage or one level is higher than the other; you need to go from a lower stage to a higher stage and in our view that is not what Bruner really meant. He really emphasised the instructional sequence, working from one towards a level of competence but using the different modes to actually get there. Problem solving situations are quite complex and they need to work across different modes of representation and not merely mental operations at one single level. So they work together, you try to offer

Skip to 2 minutes and 13 secondsall three of the modes: concrete, pictorial and abstract. An example of this can be seen to the right of me. You can see that one particular mathematical concept is presented in three different ways. This corresponds a lot with what Professor Fan showed you in the last video as well. you've got the concrete, which again now not necessarily is a concrete, touchable object but certainly is more concrete, realistic - it has a certain context. Then you have a pictorial representation, the P, pictorial and then you have the abstract presentation which corresponds with Bruner's original 'symbolic phase'. It also corresponds with research from cognitive psychology, for example research on multiple representations or for example multimedia learning; Mayer is quite a

Skip to 3 minutes and 15 secondswell-known name here: you offer things in different types of representations and that will aid and facilitate learning. Of course these sources mainly emphasise the usefulness of using visualisation and imagery to cater for multiple representations. Is there also some evidence perhaps that actually the direction going from more concrete to

Skip to 3 minutes and 41 secondsmore abstract: concrete-pictorial-abstract, that this sequence also is the best direction? Yes there is. A recent article in the journal Educational Psychology Review by Fyfe, McNeil, Son and Goldstone reviews literature on so-called 'concreteness fading' and that article recommends that materials begin with concrete materials and then explicitly and gradually fade to more abstract situations.

Skip to 4 minutes and 12 secondsTheoretical benefits of this 'concreteness fading' technique for mathematics and science instruction include things like (i) that they help learners interpret ambiguous or opaque abstract symbols in terms of well- understood concrete objects, (ii) they provide embodied perceptual and physical experiences that can ground the abstract, so that means that they can actually reinforce the abstract ideas by actually being able to understand the concrete elements at first, but (iii) it also enables learners to build up a store of memorable images that can be used when abstract symbols lose meaning. So this is what Bruner mentioned when he said there's something to fall back on.

Skip to 5 minutes and 0 secondsIt's great if you can solve perhaps a mathematical task in an abstract way but what if you get a new task and you don't know how to solve it; perhaps you can fall back on more concrete or pictorial imagery and then a last one is (iv) that they guide learners to strip away so-called extraneous concrete properties. Some things that are part of the objects and distil so-called generic, generalisable properties. Later on we will look at for example Variation Theory and look at triangles and there are certain properties that visually immediately become apparent. But does that also mean that they become apparent when you talk about them in an abstract way.

Skip to 5 minutes and 54 secondsOne note of caution: the fact that using different forms of representations from more concrete to abstract is useful does not mean that a teacher should use only one of them. In fact using only one, for example by appealing to a preferred learning style, 'they like pictures more than abstract', that might even backfire. Offering a diverse learning environment with all three elements, concrete, pictorial and abstract in a nicely sequenced way is the way to go. I hope these examples have further emphasised what the concrete-pictorial-abstract approach actually is and also what research underpins it.

Skip to 6 minutes and 41 seconds It is important to realise that although we know the CPA approach now mainly from Singapore, the underpinning ideas have a long history, for example as we have seen, from Jerome Bruner

Underpinning CPA with research - part 2

In this second part, we highlight the Singapore context for CPA and cover so-called ‘concreteness fading’. This technique begins with concrete materials and then explicitly and gradually fades to the more abstract (see Fyfe et al., 2014, for a review).

This technique specifically:

  1. Helps learners to interpret ambiguous or opaque abstract symbols in terms of well-understood concrete objects.
  2. Provides embodied perceptual and physical experiences that can ground abstract thinking.
  3. Enables learners to build up a store of memorable images that can be used when abstract symbols lose meaning.
  4. Guides learners to strip away extraneous concrete properties and distil the generic, generalisable properties.


Fyfe, E., McNeil, N.M., Son, J.Y., & Goldstone, R.L. (2014). Concreteness fading in mathematics and science instruction: A systematic review. Educational Psychology Review , 26, 9–25.

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World Class Maths: Asian Teaching Methods

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