Mathematics is a creative and highly inter-connected discipline that has been developed over centuries, providing the solution to some of history’s most intriguing problems. It is essential to everyday life, critical to science, technology and engineering, and necessary for financial literacy and most forms of employment. A high-quality mathematics education therefore provides a foundation for understanding the world, the ability to reason mathematically, an appreciation of the beauty and power of mathematics, and a sense of enjoyment and curiosity about the subject.
At Bitham Brook our mathematics curriculum aims to ensure all pupils:
- Become fluent in the fundamentals of mathematics, including through varied and frequent practice with increasingly complex problems over time, so that pupils develop conceptual understanding and the ability to recall and apply knowledge rapidly and accurately.
- Reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language
- Can solve problems by applying their mathematics to a variety of routine and non-routine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions.
The curriculum is based on the White Rose scheme of work but has been adapted and designed with knowledge at its centre. The school focuses on the following forms or categories, of knowledge.
- Conceptual knowledge – this equates to the knowledge of mathematical facts, formulas and the relationship between facts. This type of knowledge can be typically described as ‘I know that…’
- Procedural knowledge – this knowledge is the methods that would need to be followed to be successful in answering a particular type of questions. This type of knowledge can be typically described as ‘I know how to…’
- Conditional knowledge – this knowledge is the understanding of strategies which can be used to reason and solve problems. This extends to combinations of conceptual / declarative and procedural knowledge which then become strategies for particular types of problems. This type of knowledge can typically be described as ‘I know when…’
The curriculum is sequenced progressively as children move through the school to build and develop understanding. Each unit of learning is broken down into smaller component steps and these steps are broken down again into smaller component knowledge steps for each lesson. These lessons help the children to develop their conceptual knowledge ensuring they have all the knowledge needed to carry out procedures. Before looking at how they can use this to solve reasoning problems. Misconceptions are identified during lessons and taught explicitly so the children can identify common mistakes.
Mathematical vocabulary is vital to the understanding and the ability to explain reasoning. This is taught during lessons and developed with the use of stem sentences. The vocabulary children encounter builds as children work their way through the curriculum. This allows them to talk confidently about mathematical principles.
Mathematics is an interconnected subject in which pupils need to be able to move fluently between representations of mathematical ideas. The programmes of study are, by necessity, organised into apparently distinct subject areas, but we want pupils to make rich connections across mathematical ideas to develop fluency, mathematical reasoning and competence in solving increasingly sophisticated problems. They should also apply their mathematical knowledge to science and other subjects.
Bitham Brook uses the Concrete, Pictorial, Abstract (CPA) to teaching mathematics. This highly effective approach to teaching that develops a deep and sustainable understanding of maths in pupils.
The CPA approach builds on children’s existing knowledge by introducing abstract concepts in a concrete and tangible way. It involves moving from concrete materials, to pictorial representations, to abstract symbols and problems.
Concrete is the “doing” stage. During this stage, students use concrete objects to model problems. Unlike traditional maths teaching methods where teachers demonstrate how to solve a problem, the CPA approach brings concepts to life by allowing children to experience and handle physical (concrete) objects.
Pictorial is the “seeing” stage. Here, visual representations of concrete objects are used to model problems. This stage encourages children to make a mental connection between the physical object they just handled and the abstract pictures, diagrams or models that represent the objects from the problem.
Abstract is the “symbolic” stage, where children use abstract symbols to model problems. Students will not progress to this stage until they have demonstrated that they have a solid understanding of the concrete and pictorial stages of the problem. The abstract stage involves the teacher introducing abstract concepts (for example, mathematical symbols). Children are introduced to the concept at a symbolic level, using only numbers, notation, and mathematical symbols (for example, +, –, x, /) to indicate addition, multiplication or division.
Teachers will go back and forth between each stage to reinforce concepts. This supports children to craft powerful mental connections between the concrete, pictorial, and abstract phases.
By ensuring that concrete representations aren’t removed to early it allows children to build a conceptual mathematical understanding that can propel them through their education.
The CPA model is a progression. By the end of KS1, children need to be able to go beyond the use of concrete equipment to access learning using either pictorial representations or abstract understanding.