Principles of mastery
For many years now, the teaching of mathematics has been surrounded by the myth of a “maths brain” – this idea that some people are naturally good at maths, whilst others are not. A mastery approach rejects this idea, instead advocating the belief that with “hard work, all can achieve.”
The mastery classroom would see all children moving through the learning at broadly the same pace, this will ensure that deeper learning is achieved for all and it will help to reduce gaps forming in children’s learning. Moving through learning in small, coherent steps will help children develop their mathematical fluency and enable children to see patterns and relationships within mathematical concepts.
Depth rather than breadth is a key element of a mastery approach. Longer periods of time are spent looking at the key ideas which are needed to underpin mathematical understanding and put into place the building blocks needed so children can develop a sustained understanding of maths.
The importance of mathematical language and talk within the classroom is vital.
Moving children through the CPA approach in order to master learning enables all children to be successful mathematicians.
The key elements of teaching mathematics through a mastery approach have been identified by the NCETM:
Click below to find out more about the 5 big ideas:
Mathematical talk and language
Being able to talk about mathematics can be challenging for many children. The language around mathematics can be abstract, it can often have more than one meaning and it isn’t necessarily language we hear or use every day. As teachers, we need to expose children to mathematical language and use it every day in the classroom. We should have high expectations that all children are able to use correct mathematical terminology (When completing a subtraction question, rather than “the top number take away the bottom number”, we should expect “the minuend subtract the subtrahend to equal the difference”) The use of sentence stems in the classroom can support this. Sentence stems offer repetition of ideas, a structure from which to develop one’s own thinking and a scaffold for less confident children.
The idea of children learning through talk is not something new. Lev Vygotsky (1896-1934) described how humans are able to learn from each other through social interaction. Through social interaction with peers, children are able to clarify and internalise their own thoughts, whilst learning from each other. Vygotsky went on to describe how a child’s thinking skills are based on the relationship between social interaction and individual thinking time.
The CPA approach is based on the work of Jerome Bruner (1915 – 2016). He identified three stages of learning children move through: enactive, iconic and symbolic. We now know this is the CPA approach (Concrete Pictorial Abstract).
The first stage in children’s learning of a new skill or concept begins at the concrete. Children must first explore a real – life experience to help them develop their initial understanding. Whilst this may not always be possible, we would instead use a manipulative to represent the real life. Once children have gained an understanding at this level, they can then be moved towards a pictorial representation of the concrete. This may be an image or a diagram of the mathematical concept. The pictorial stage should act as a bridge between the concrete and abstract. It is also worth noting that children may need the concrete alongside the image. The final stage is the abstract. Ultimately, the aim is to have all children at this point in their learning; however it will take some children longer to reach the abstract than others.
1. Representation and structure
The fundamentals of mathematics lie heavily in the ability to be able to spot patterns and relationships within mathematical concepts. Mathematics should be seen as a web of interlinking structures all of which build on each other. It is crucial to children’s thinking skills that they are able to identify this, rather than seeing different mathematical concepts in isolation of each other. Microscopic planning will help to draw out these relationships, allowing children to link together their understanding to help form the bigger picture. The activities and tasks we give to children must place emphasis on relationships, patterns and structures within maths.
2. Mathematical Thinking
Seldom is there ever one method to solve a mathematical question – this is the nature of maths. We encourage multiple methods, questions such as “is it possible to solve this another way?” develop children’s abilities to think mathematically. We should encourage children to offer proof and reasoning with their solution, rather than taking their immediate answer at face value. Ideas should be discussed and shared with others to help develop understanding across the whole classroom.
Depth of understanding will only be achieved if children develop mathematical coherence. The links between coherence and representation and structure are evident – identifying relationships and patterns. Whilst representation and structure is focused on children being exposed to such ideas through carefully chosen questions, coherence focuses on building on previously learned knowledge. The work of Jean Piaget (1896 – 1990) places emphasis on the importance of coherence. He talked about assimilation and accommodation. When children learn a new skill or concept, they will build on existing knowledge in order to develop understanding (assimilation). This theory of learning fits with the idea of coherence and a mastery approach. Piaget identified that when this was not possible, children had to accommodate new learning – meaning children have to modify their original understanding in order to make way for new knowledge.
Carefully constructed questions should be used to highlight the essential features of the mathematical concepts being taught. There are two types of variation: conceptual variation and procedural variation.
Conceptual variation focuses on identifying the essential features of a key concept as well as the non-essential features. The key question or problem offered to the children will remain the same; however the representations of the problem will differ:
In this example, the child would need to use the different representations of a triangle to identify what it is that makes a triangle, a triangle (what do they all have in common). It is also an opportunity to discuss what is not important (colour, orientation, size).
Procedural variation focuses on developing children’s mathematical thinking skills through careful choice of questions to exemplify patterns and relationships:
Questions such as: “What is the same/different?” “What happens to the solution each time…?” “Can you describe how the addends change in each question…?” These questions will enable discussion within the classroom and conversation about the patterns observed.
This should not be misunderstood as rote learning. Mathematical fluency is based on: accuracy, efficiency and flexibility. The ability to manipulate numbers and the flexibility to be able to move between different mathematical concepts is key to fluency. Being efficient – “is my method the most efficient?” is a question that children should ask themselves. The learning of number facts is crucial in developing mathematical skills and knowledge, but the most important aspect is being able to apply this knowledge to routine and non-routine problems.