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Practice makes perfect, right? By Suneeta Vyas
Practice makes perfect, right?
At Foxfield, there has been much discussion around mastery in maths and achieving depth. The 5 big ideas of maths mastery highlight explicitly the importance of ‘small steps’ and ‘coherence’ through a sequence of lessons. This allows enough time and opportunity for fluency using a concrete, pictorial and abstract approach. Assessment for learning, teacher and pupil voice all identified the need for more attention to essential skills, including times tables and arithmetic.
We opened up the discussion: ‘how do we ensure that we are not building on sand?’
We want children to be meaning makers, critical thinkers and problem solvers in their own right. We want mathematics to enable our children to be equipped to create their futures, instead of simply ‘entering’ a future that has been created for them. The fundamental principles of kagan cooperative learning states that learning best occurs through positive social interactions. This allowing learners to build learning communities underpinned by mutual respect, shared goals and empathy. Active engagement forms a core pillar in the rudiments of teaching and learning, within the inspire partnership. In ‘Shaking up the schoolhouse,’ (How to support and sustain educational innovation 2000) Phillip Schletchy describes learners who are engaged as:
- Attracted to their work
- Persistent in their work despite challenges and obstacles
- Taking visible delight in accomplishing their work
“Learning… is more a reaching out than a taking in. It is a participation. It is a process of remembering – in the worlds original sense of pulling together the parts of a body into a more complex unity. That is, even though it is often convenient to speak of an agent’s knowledge as though it resided within the agent, this knowledge is what defines the agent’s relationship to the rest of the world. The agent’s activity and identity are inseparable from his, her or its knowledge. Knowing is doing is being.” (Davis, Sumara et al. 2000).
Building in these strategies for participation through carefully crafted learning intentions is the key to ensuring that children have the opportunity to ‘see, hear and do.’
When it comes to teaching mathematics, it is no surprise that in order for all children to really master a concept or achieve depth, they must first be shown why. Specifications from the previous national curriculum framework saw us jumping through concepts, teaching abstract strategies and methods and differentiating in too many ways. However, all we had to do was look a little closer to home. The origins of the concept of ‘mastery’ and ‘learning for mastery’ were first proposed in 1968 by well-known educational psychologist, Benjamin Bloom. He stated that in order for children to move on to another mathematical concept and acquire subsequent information, they must first master the knowledge and skills before. Also during the 1960s, American psychologist Jerome Bruner focused much of his work on scaffolding learning. He, like many others, believed that the abstract nature of maths learning, was a mystery to many children.
As outlined by the NCETM, learning should be structured in such a way, that children are able to build on each stage in order to gain a greater understanding of the concept being learnt. The concrete, pictorial, abstract approach (CPA) allows for just that. The introduction of a new mathematical concept, through the use of effective representations and concrete manipulatives, ensures that new knowledge and information will be internalised to a greater degree. Being able to show a mathematical concept means being able to prove a mathematical concept. Being able to build a mathematical representation requires a deeper understanding of a concept. In order to make connections, children need to be able to see connections.
If we know all of this and use this pedagogy as the basis for our design for learning, why do we see gaps in the knowledge of times tables or arithmetic? It was this question that prompted the analysis of the way in which we teach essential skills.
Let’s look at the teaching of English. It’s everywhere, in everything – whole class reading lessons, grammar lessons, writing lessons, history, geography, the list goes on. The diet of English far outweighs the time dedicated to the teaching of mathematics. This constraint is even more reason to ensure that every single opportunity for maths learning is ‘real.’
At Foxfield, this started with regular and consistent sessions, dedicated to the explicit teaching of times tables. In order to develop multiplicative reasoning, children first needed to have a strong understanding of the concept of zero and be able to unitize. With this in mind, we designed a whole school progression map that incorporated the sequential building of knowledge – the first stage of the journey.
The next stage to consider was the structure and learning progression within a lesson. The goal was to enable practitioners to carefully craft learning to provide pupils with frequent opportunities for intelligent practice, in order for the new knowledge to evolve from simply memorisation, to automaticity. We decided that the use of the array model would be the best structure to represent multiplication, which would be accessible by all children, regardless of their stage in education.
Arrays – or simply put, an arrangement of objects, pictures or numbers in rows and columns – are all around us. They could be the cheeky Crispy Cream donut box that we buy from the supermarket, or the tiles that cover our kitchen floors. Making it real is the key, maths is everywhere. To do just that for the children, we created a multiplication table with them, to introduce the time table of focus and to explore it’s place in the world, by asking: “what comes in groups of…”
Over the course of the half term, children would add objects they find, interact with the different representations and make links to previous tables they have met before.
Following the construction of the table, we felt the logical next step would be to link the real life representation with the mathematical concept and build the times table together. The year 3 teachers at the time, devised a model for what this could look like in practice, to share with the rest of the staff team.
In order to build on learning and for lessons to follow a natural learning progression, teachers applied their usual ‘I do, we do, you do’ dialogic teaching structure to the lessons. ‘I do’ would consist of modelling representing a multiplication fact using resources or the array hundred square. During this modelling, we would verbalise the thinking or metacognition, highlighting key vocabulary such as multiplicand and multiplier and how this visual representation would then translate into a written calculation. The ‘we do’ and ‘you do’ elements of the session provide pupils the opportunity to communicate, collaborate and practice.
‘How does this develop depth of learning though?’, I hear you ask. Well firstly, the children’s knowledge and thinking has already evolved from additive to multiplicative reasoning and secondly, their underlying conceptual knowledge has provided them with the tools to be able to justify, generalise and prove, drawing on a multitude of concrete and pictorial representations. Again, carefully crafting learning to systematically and sequentially build on knowledge and skills was the key.
In his book ‘Unlocking student talent,’ Kern and his co-authors explain how engaging in repeated tasks, following active engagement, with immediate and specific feedback, can build skills within a given area. He goes on to talk about how planning for guided instruction and practice can help all learners to improve their performance and achieve to a higher level. Biologically verified by Daniel Coyle (2009) in ‘The Talent Code, “he more nerves fire, the more myelin wraps around it…the faster the signal travels, increasing velocities up to one hundred times over the signal sent through an uninsulated fibre.”
Intelligent practice makes perfect, right?