Few if any of us who are involved in mathematics education will be unfamiliar with the name Jerome Bruner. Born in 1915, his ideas have had a huge influence on the way we have taught mathematics to children in the last 60 years or so.

Bruner’s work on developmental psychology is almost certainly a ‘de-rigeur’ component of all high-quality teacher-training course in universities around the world, yet it is my suspicion that he has been massively (and potentially dangerously) misunderstood.

Bruner’s central and most famous theory centres around the way in which we assimilate new ideas. While Zoltan Dienes talks of six stages of learning, Bruner talks instead of three modes of representation: the enactive, the iconic and the symbolic. (If you like alliteration, think of Doing, Drawing, Discussing.) It has become common to refer to this approach as ‘CPA’, or concrete, pictorial, abstract.

This is all probably familiar to all who have studied or been taught about Bruner, but the crucial misunderstanding of his work is this. While Dienes proposed that his six stages were linear, Bruner never said that. His idea was that these modes of representation are very much connected, and not always linear.

This has not perhaps always been appreciated in many of the hundreds of wonderful schools it has been my privilege to visit over the years. The pattern is often as follows:

Practical Equipment such as Multilink cubes, Base 10 equipment, counters, Cuisenaire Rods, Bead Strings, Place Value cards, Numicon etc. are all used in the Early Years but their use is ‘faded out’ as children move up the school until at some point during Key Stage 2 there is barely more than a pencil and a ruler to be seen on maths tables. The two justifications given for this are “The older children don’t need the stuff to think with any more” and “Well they can’t use the equipment in the tests so we’re not helping them long-term by continuing to make it available.”

Both of these arguments, I believe, are fundamentally flawed. The first implies that thinking is somehow formless and fails to take into account how much the brain relies on images it has created. Thinking is a conversation with myself, and in conversation I will often try to paint mental pictures to explain and explore. The second argument is perhaps more worrying – the children can of course use the equipment in the tests, but not in a literal concrete sense – their use of such materials involves recalling memories of how it helps them see numbers. Thus it is vital that for each new experience, children have the opportunity to build their understanding starting with a concrete experience, then moving towards imagery and finally the bizarre but powerful world of symbols and abstraction. A mastery approach can ill-afford to omit any of these stages, lest learning is superficial and ‘built on sand’.

It has been my experience working with young children that the jump from the first to second may take place in an instant. For example, show them a photo of a relative and they will tend to say simply “That’s grandma!” Rarely if ever do they say “Oh look a very small 2-dimensional representation of grandma!” So, in a mathematical context, this means thinking carefully about new ideas and how they are introduced in a way that gives children shared language, images and experiences to discuss.

The idea of starting with a learning objective is somewhat at odds with a constructivist approach, yet in far too many schools in the United Kingdom, teachers are still required to display just such an objective at the start of every lesson, despite there being no evidence that this achieves very much at all. Just to be clear, a learning objective is a good and important thing, but children are not mere machines – sometimes their thoughts will lead them ‘off script’ and they may make important connections and realisations that fall outside the narrow scope of an objective. Thus it is as pointless as showing them the end of a film before they have had a chance to work through the story and watch the plot develop.

Kinaesthetic investment moulds memory. By this I mean that as almost all educational theorists and practitioners agree, ‘doing’ mathematics helps the brain deepen its understanding. This is one of the keys to mastery. A simple example is the important idea of rounding numbers to the nearest multiple of 10. I have observed teachers draw marks on a numberline and invite children to say ‘which ten’ (we should at least say which ‘multiple of ten’ as there is actually only 1 ten) is closest.

Bruner’s ideas suggest using a different approach. For example, why not take the children outside and chalk a numberline on the ground. Of course number mats work just as well; the important thing is that there is a numberline on the ground along which children can walk, marked 0,10,20,30 etc. Ask children to stand on various multiples of ten, then to stand on, say, 25. Discuss how they decided where 25 is; they will probably talk about ‘halfway’ or ‘in the middle’ or ‘same distance’ etc. Next ask two children to stand on 24 and 26 respectively-and then get them to run to the nearest multiple of ten; this helps them to see 25 as the ‘tipping point’ and the word ‘nearest’ has some meaning that is now real and in the children’s experience. Drawing a picture of the chalked line now gives children a shared image from which to draw – they can imagine themselves on the line walking between two multiples of ten and so can begin to think abstractly.

In no way does using a real line on the ground detract from their ability to think; in fact I would argue that the reverse is true; they are now MORE able to think in the abstract precisely because they started, however briefly, with a concrete experience.

I don’t assume to do justice to Bruner in this short article, but I hope it provides food for thought as you plan your children’s future learning.

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We are saddened by the death of Jerome Bruner on June 5^th 2016, aged 100 years.

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Andrew Jeffrey taught for 20 years, and has been an inspector, lecturer, author and mathemagician! His passion for helping children learn, love and understand the patterns and structures of maths has led him to become an international conference speaker and teacher trainer. 

You can hear more from Andrew on the online home of school improvement, Oxford Owl. Simply register for free to watch this video, from the Professional Development & Best Practice hub, where Andrew discusses strategies for building children’s confidence in maths.