Mathematics at Airthrie aims to equip pupils with the skills and confidence to deal with mathematical situations and problems encountered in everyday life as well as in a school setting and to develop a lifelong love of maths!
Since 2016, we have used the White Rose scheme of work, which is based around the principles of mastery. The scheme of work organises National Curriculum objectives in blocks within term-by-term overviews and the activities are broken down in fluency, reasoning and problem solving, which are the key aims of the curriculum.
What does it mean to master mathematics?
A mathematical concept or skill has been mastered when a pupil can represent it in multiple ways, has the correct mathematical language to explain their reasoning, and can independently apply the concept to new problems in unfamiliar situations. This is not about just being able to memorise key facts and procedures, which tends to lead to superficial understanding that can easily be forgotten. Mastery is a journey and long-term goal, achieved through exploration, practice and application over time. At each stage, pupils should be able to demonstrate a deep understanding of the topic and when the time comes, be able to select which mathematical approach is most effective in different scenarios and apply it successfully.
Focus on depth
A key principle of White Rose is to deepen understanding before accelerating content coverage. Pupils must be given time to fully understand, explore and apply ideas, rather than accelerate through new topics. This approach enables pupils to truly grasp a concept, and the challenge comes from investigating it in new and alternative ways that are more complex.
A scheme of work based around the principles of mastery is suitable for children of all abilities. While mastery may be challenging for some, the vast majority should be aiming for this standard and at Airthrie, our high staff-pupil ratio allows pupils with SpLD to work in smaller settings. For all pupils, it is important that they fully understand key number concepts, rather than simply memorise a process. Therefore, we extend high-attaining pupils through depth and enhancement, as opposed to acceleration onto new content. This will reap its rewards in the future when mathematics that is more complex is studied.
Concrete, pictorial, abstract representations
Objects, pictures, words, numbers and symbols are everywhere. The mastery approach incorporates all of these to help pupils explore and demonstrate mathematical ideas, enrich their learning experience and deepen understanding. Together, these elements help cement knowledge so pupils truly understand what they have learnt. All pupils, when introduced to a key new concept, should have the opportunity to build competency in this topic by taking this approach. Pupils are encouraged to represent physically mathematical concepts. Objects and pictures are used to demonstrate and visualise abstract ideas, alongside numbers and symbols.
Concrete - is the ‘doing’ stage. To enrich teaching and learning, practical apparatus is used to support pupils' mathematical thinking, reasoning and problem solving. These may include, for example, Numicon shapes, coins, Multilink cubes, Dienes apparatus, counters, bead strings, Cuisenaire rods, sticks divided into 10 equal sections and also those that use numerals such as place value cards, hundred squares, digit cards, dice dominoes and so on.
Pictorial - is the ‘seeing’ stage. Pupils build on the concrete approach by using pictorial representations of the objects to model problems. This stage encourages pupils to make a connection between the physical object and a diagram or picture, which represents the object. Building or drawing a model makes it easier for pupils to grasp concepts, traditionally, they feel more difficult, such as fractions, as it helps them visualise the problem and make it more accessible.
Abstract - is the ‘symbolic’ stage. With the foundations firmly laid, and only once a pupil has demonstrated that they have a solid understanding of the ‘concrete’ and ‘pictorial’ stages, can pupils move to an abstract approach using numbers, notation and mathematical symbols, for example +, -, x, ÷ to indicate the four operations and other key concepts, with confidence.
Fluency, reasoning and problem solving
Teaching the White Rose scheme of work supports the key aims of the National Curriculum of fluency, reasoning and problem solving.
Fluency - Pupils should be able to recall and apply mathematical knowledge both rapidly and accurately. However, it is important to stress that fluency often gets confused for just memorisation – it is far more than this. As well as fluency of facts and procedures, pupils should be able to move confidently between contexts and representations, recognise relationships and make connections in mathematics. This should help pupils develop a deep conceptual understanding of the subject. Frequent practice will help them to achieve a high level of fluency.
Reasoning - The way pupils speak and write about mathematics transforms their learning. Mastery uses a carefully sequenced, structured approach to introduce and reinforce mathematical vocabulary. Pupils explain the mathematics in full sentences. They should be able to say not just what the answer is, but how they know it is right. This is key to building mathematical language and reasoning skills.
Problem solving - Mathematical problem solving is at the heart of the Airthrie approach. Pupils are encouraged to identify, understand and apply relevant mathematical principles and make connections between different ideas. This builds the skills needed to tackle new problems, rather than simply repeating routines without a secure understanding. Mathematical concepts are explored in a variety of representations and problem-solving contexts to give pupils a richer and deeper learning experience. Pupils combine different concepts to solve complex problems, and apply knowledge to real-life situations.
Number at the heart
A large proportion of our teaching time is spent reinforcing number to build competency and fluency. Number is usually at the heart of any primary mastery scheme of learning, with more time devoted to this than other areas of mathematics. It is important that pupils secure these key foundations of mathematics before being introduced to more difficult concepts. At Airthrie, we believe that this increased focus on number will allow our pupils to explore the concepts in more detail and secure a deeper understanding.