Whichever way you think about Bruner’s three modes of cognitive representation, the starting point is the use of manipulatives that will help a learner make sense of a concept — the foundations on which to anchor and build a solid understanding. Yes, Bruner’s theory underpins our maths teaching, but for some teachers, this crucial first stage can feel like a step too far for some of their pupils.
The potential to demonise children’s behaviour is a real threat. The government’s introduction of ‘behaviour hubs’ and the separation of Ofsted’s previous judgement of “personal development, behaviour and welfare” into two judgements, one for ‘behaviour and attitudes’ and one for ‘personal development’, must surely make some teachers want to play it safe and avoid any distractions in the classroom.
Withdrawing the use of manipulatives can be seen as a reasonable way to minimise the potential for disruption but it should not be seen as the solution. Rather concrete representation offers learners a way into conceptual understanding that is exciting and motivating.
The paradigm shift
When my school adopted the Singapore approach I thought I was going to be fine with it. After all, I already used manipulatives with my struggling learners so I had no worries! It took me a while to make the paradigm shift that all learners can gain a deeper conceptual understanding through the use of manipulatives, not just struggling learners. My thinking changed after I read ‘Concrete-Pictorial-Abstract: Surveying its origins and charting its future’ by LW Hoong, et al., thanks to their interpretation of Bruner: “If a student learns purely in the symbolic mode, in the event that he cannot recall its workings, he can then have something to ‘fall back to in order to recover the meanings of the notations in the symbolic mode. There is no way to ‘fall back’ if there is no history of meaningful learning in other modes of representation”.
However, once I had moved into the realm of ‘resources for all’ there was a second paradigm shift to make — the expectation that all learners use manipulatives. And all learners meant all learners. Not just those who I thought would be ‘sensible’ with them. For some learners who struggle to focus, a pile of toys might well be a temptation they can’t avoid, and I imagined that I would easily lose control of the classroom once they had the colourful plastic bits in their hands. That loss of classroom control is every teacher’s nightmare. But it doesn’t need to be — the use of manipulatives in maths is the crucial first stage of representational understanding.
“Never trust a man, who when left alone with a tea cosy…doesn’t try it on.”
Imagine yourself alone at a table with a pile of multi-link in front of you. Would you play with them? Join some together to make a cube? What if you were in a roomful of adults in a staff meeting or at a conference? Would your fingers, almost of their own accord, reach across the table to make something? Would it mean you couldn’t focus on the person in front of you or beside you?
It’s tempting to think that as adults we wouldn’t, that 100% of our attention would be on the speaker. But if we’re being honest we probably would. Piaget tells us there are important differences between our thinking and children’s, but the temptation of dipping into that pile of cubes demonstrates they’re probably more similar than not.
It’s the thought of what I would do if I had that pile of manipulatives in front of me that helped me with my second paradigm shift. Skemp talks about attitudes and emotions affecting learning, and in this instance, I can think of nothing worse than being left out. So how can we ensure all our learners are in a good place to use manipulatives?
- Spend time exploring the use of manipulatives as mathematical representations but recognise the need to shift some children’s perception of resources as ‘stuff to play with’.
- Allow time at the start of a lesson or concept to enjoy the feel and look of the resource. Can the children make a tower before making 10? Make a face in the mirror before symmetry? Balance the 3D shapes on top of each other before exploring their properties?
- Bag them up — store resources in pre-packed ziplock bags. It’s easy, safer and efficient for distributing. That temptation to touch is still going to be there but it’s going to be harder to satisfy the itch.
- Can some resources remain on desks to take away their novelty value? What needs to be tidied away?
- Give parameters in terms of where resources can be used and stored. Are your learners ‘owners’ of their learning materials? Can some children be given the role of sharing resources? Can they be responsible for maintaining storage areas?
- Model the importance of proper treatment of manipulatives. Do the children’s resources look loved? Are you modelling enthusiasm and respect for resources?
“Have patience. All things are difficult before they become easy.”
After the madness of the last couple of years, the safe bet of withdrawing or not using manipulatives is very tempting. But we know the best way for children to learn maths isn’t by missing out on the concrete stage or by limiting it to only some of our children.
With the latest EEF findings showing the attainment gap between disadvantaged primary school pupils and their classmates has grown since the onset of the pandemic, we could be forgiven for trying to make our maths lessons as efficient as possible. Teach the rule and move on! But for some pupils, the lack of manipulatives available to them during their period of home learning must have surely contributed to their widening gap? As Skemp says, “Merely learning ‘a method which works’ is an impoverished kind of learning which leads to a weak understanding of the nature of mathematics.”
Now that we are back, we might need to make some small changes to our classroom practice, but we can teach efficiently with manipulatives and all our pupils can learn effectively with them. Giving all learners the opportunity to develop their understanding through all of Bruner’s stages is more important than ever.
The Dienes quote “Give me a mathematical structure and I’ll turn it into a game,” always makes me smile. It makes me feel that maths teachers have been given an amazing gift — we get to nurture generations of learners to discover conceptual understanding, learn to problem-solve, use creative thinking and ultimately love maths. How lucky are we!
References, work cited and informative reads:
- Concrete-Pictorial-Abstract: Surveying its origins and charting its future, Leong, Yew Hoong. Ho, Weng Kin. Cheng, Lu Pien. 2015: https://www.academia.edu/29658199/Surveying_its_Origins_and_Charting_its_Future
- Mathematics in the Primary School, Richard R. Skemp, Routledge, 1989