Keeping the 'Rapid Graspers' Engaged
The term ‘rapid graspers’ here refers to children who pick up the concept being taught, more quickly than their peers. This is not a re-wording of ‘high attainers’ or ‘gifted and talented’ because, with a teaching for mastery approach, no attempt is made to identify which particular children will be rapid graspers on any particular day or topic, recognising that children’s aptitudes vary between topics and over time.
Thousands of teachers in England have now had the opportunity to see a Maths Hub ‘Shanghai Showcase’ lesson – that is, to see a primary maths teacher, from Shanghai, teach a lesson to a class of English school children, at a local school. Mostly observers are impressed by the intricacy of the lesson design, the depth with which concepts are broken down and the carefully crafted questions that characterise the Chinese ‘teaching for mastery’ approach. But sometimes, they are troubled by noticing children finish questions quickly and apparently being expected to sit and wait for their peers to finish before the lesson moves on.
Expectations and history differ dramatically between England and China. Clare Christie, maths lead for Ashley Down Schools Federation in Bristol, notices that “…although the standards are undoubtedly extremely high [in Shanghai], they don't seem as 'fussed' about making sure there is a little extra challenge for the highest attainers/children that pick the concept up quickly in that lesson”.
Because of the cultural gulf between the two systems, it is imperative to identify how teaching for mastery can work in English classrooms. This is the challenge that has been taken on by the growing cadre of Mastery Specialists, and one of the big questions they face is how to extend children who complete work set quickly and are ready for more.
A headteacher, concerned about how teaching for mastery was being implemented in his own school, commented:
"15 shared into equal groups of three. Quick graspers do it in five seconds. Others take longer. Quick graspers do nothing while they wait so everyone can move on at broadly the same pace."
Kate Mole (Teaching for Mastery Lead for London South West Maths Hub, and Maths Lead for a federation of schools) identifies that teachers adopting a teaching for mastery approach have rapidly understood the need for a single learning point for each lesson, with the whole class working on the same content. She says:
“As practitioners leading on teaching for mastery, we deeply understand the approach and also know the damage that was, and can still be, caused by children not deepening their learning enough. We know the benefits of children working on the same content and can justify not 'pushing them on'. However, I can totally empathise that there are those times when you feel that there is an additional challenge needed for some children in our classes.”
A Maths Hubs national project is researching this, and due to report at the end of the school year. Teachers are working together to develop understanding of how to provide opportunities in all lessons, for children to work more deeply than the age-related expectations. The project aims to share activities that encourage children to demonstrate depth in different areas of the curriculum, and also to report on how journals could contribute to building to a judgement of ‘working at greater depth’ at the end of the school year.
In the meantime, Mastery Specialists are devising solutions in their own classrooms. Nicola Ballantine (Y6 Teacher and Assistant. Head at Nonsuch Primary School, and a Primary Mastery Specialist for London South West Maths Hub) has written a useful blog, Dive deeper with the ‘high attainers’, that addresses the challenge in a very practical way, suggesting generic pointers that children can be encouraged to use to deepen their learning when they have ‘finished’:
If the class was engaged in a fluency activity, such as multiplying unit fractions, a student who completed the task early could then write a maths story which could cement a connection between the abstract algorithm and the real world, i.e. three friends each had of the of a cake left over from a party, . Next, if the reasoning task asked for the student to explain why the product of two fractions is smaller than the factors, they could extend their learning by drawing a pictorial representation, proving why this is true. Lastly, after completing a problem solving task requiring them to apply their understanding in a new context, they could then deepen their learning by demonstrating what they deem to be a common mistake, such as adding the denominators rather than multiplying them, and they could explain why this error sometimes occurs. By engaging in these deepening tasks, students are able to sit with their learning just a little bit longer, devoting more thinking time to understanding the structure of the concept as well as unpicking any of their own misconceptions.
Read more of Nicola’s blog
Kate Mole points out that using this approach relies on ‘winning over’ those children that consider themselves high-attainers, to help them understand that they are being extended but through depth rather than by being given different work. A powerful way to do this, she says, is to give pupils the answer:
“Look, here’s the answer – but the answer isn’t the most important part, show me how we got there…now show me another way…now prove it. The answer is only the beginning.”
She also says that winning over parents is critical. She recounts a parents’ meeting where she began by asking who could divide fractions. Only a handful of parents put up their hands – which, when she assured them that they would all have been taught it at school, made her point: you can learn a trick (turn the second fraction upside down and multiply), but if you don’t understand it conceptually, you won’t remember it. She then went on to teach the parents, using manipulatives and images, to divide by 2 and then by 2, emphasising an understanding of what they were doing.
As a final point, Kate is keen to clear up a common misinterpretation of the National Curriculum requirement to teach fluency, reasoning and problem solving. These three elements should not be regarded as a hierarchy of difficulty, she argues, with some children only engaging in fluency work. Reasoning and problem solving should not be the ‘extension work’. All children need to be involved in reasoning: it is important for understanding. Extension comes from reasoning and problem solving at greater depth.
And her response to the headteacher observing children divide 15 into three equal groups…? She points out that this difficulty is nothing new:
"This difficulty is not a result of teaching for mastery! It's actually just not great teaching! As teachers, we have always had situations where children answer quickly or finish sooner than other children. It is our job to have a question up our sleeve to enable them to continue to engage with the learning."
Instead of sitting waiting, these rapid graspers might have been asked:
"Can you make any other equal groups from 15? Can you use 15 cubes to make groups that aren't equal? Why are they equal? What are they not equal? What’s the same, what's different about the equal and non-equal groups? How many different ways can you make equal groups from 15 - are there more than two? Can you convince me? How do you know? Can you write me a definition of what equal means?”
Further reading around this issue:
- The 2012 report Raising the Bar, from the Advisory Committee on Mathematics Education (ACME), explains the importance of depth over acceleration for able mathematicians
- In his 2012 paper, Nurturing able young mathematicians, Tony Gardiner explains the importance of providing a rich curriculum for all, of not attempting to identify particularly able mathematicians early, and of encouraging rich, deep study in order to master core curriculum topics, rather than accelerating apparently able students through the curriculum at a superficial level.