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At Navigation, we are proud of all our curriculum, but we feel maths is area of particular strength. On this page we hope to explain our approach to the teaching of maths. We have broken this page into three broad sections:
Intent: the knowledge and skills that pupils will gain
Implementation: how the curriculum developed or adopted by the school is taught
Impact: the outcomes that pupils achieve as a result of the education they have received
Curriculum Vision Statement:
'By the time children leave Navigation they will be learners who have developed excellence in maths skills in order to equip them to understand the world and have the crucial foundations to access all other areas of learning.'
Our curriculum vision statement outlines our broad goals for maths, whereas below our whole school curriculum progression map gives a detailed breakdown of what each year group is taught. Below this there is also an overview of the different units of work that each year group covers. You can also download these documents at the bottom of the page or click in the top right corner to expand it into a new window. At the bottom of the page, you can also find the national curriculum statements for each year group to download should you wish to view them.
In this section, I will outline how we implement the objectives of our curriculum. It starts with some explanation of some of philosophical underpinnings to maths and then some specific details about classroom practice. We use White Rose Maths as a scheme of work. This is a well-established approach used in over 80% of primary schools and 140 countries.
The central principal that underpins White Rose Maths is a ‘journey to mastery’, which is why we chose this scheme of work to organise the teaching of mathematics at Navigation. Helping pupils along that journey is not always easy, so we use a variety of strategies and approaches to help them, but all are underpinned by some organising principals and themes that allow for consolidation and building on previous learning.
Mastery of anything – playing an instrument, speaking a new language, mathematics – takes a very long time. That is why we talk about the ‘journey to mastery’, rather than ‘having mastered’. For example, children start learning to add in early years and keep developing their skills over many years – from single digit numbers, to multi-digit numbers, then decimals, then fractions, then negatives, addition in different units (such as time calculations '35 minutes after 12:45pm'). It would take several years to master addition, arguably one of the most basic concepts in mathematics. So in order to help learners on that journey, we do is break the journey down into small steps, spending time carefully considering each. Through intelligent practice and building up experience of different contexts, gradually we move towards mastery where pupils are fluent in the unfamiliar and can apply their skills in any new situation.
Time for consolidation
In most schools, the demands of the curriculum mean time is an issue and covering the full content of the curriculum in the depth can pose a challenge. Our approach to maths is designed to give time to think about a topic, develop understanding and also to realise that perfect mastery will not have been achieved by the end of the unit, whatever its length. The small step approach is designed to ensure that students will come back to topics time and time again, both within the study of the same area of mathematics and in other areas so that they will continue to deepen their understanding through this revisiting and interleaving.
Mixed ability grouping
Whilst good teaching can occur in sets and mixed attainment grouping, copious educational research, and our own experience of previously setting children in maths have shown that all children benefit from mixed attainment grouping. It is important to remember that even in sets, there is always a wide range of prior attainment and students have different needs. Equally, to dispel a common myth, teaching for mastery does not mean we do not differentiate (providing different adjustments or work for children based on their current attainment and needs), but we do aim high for all students. By teaching in a more inclusive manner, we ensure that no student is denied access to particular aspects of the curriculum. Similarly, it would not be appropriate for some students to work only on fluency, others on reasoning and a select few on solving problems. We believe all students should have opportunities to develop reasoning and solve problems as well as develop fluency. Differentiation can also be achieved, for example, through varying the degree of support provided, using enabling and extending questions and providing or asking for alternative representations whatever type of grouping a school chooses to adopt. We have found that this has approach has also improve the self-confidence of our pupils and it has provided strong outcomes for all.
We firmly believe in a ‘growth mindset’ approach and that everyone can get better at mathematics. We know from experience that some students can start a year, a key stage or a new school with relatively low prior attainment but over the course of the next term, year etc. rapidly improve and end up as high attainers. It is their attainment that has changed, not their ability and they had the potential to succeed all along. We know what students have attained previously and provide all students with the opportunities to improve and succeed.
Concrete Pictorial Abstract
The concrete, pictorial, abstract approach (CPA) is a highly effective approach to teaching that develops a deep and sustainable understanding of maths in pupils. Often referred to as the concrete, representational, abstract framework, CPA was developed by American psychologist Jerome Bruner. It is an essential technique within the ‘Singapore Method’ of teaching maths for mastery.
Children (and adults!) can find maths difficult because it is abstract. 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. The CPA framework is so established in Singapore maths teaching that the Ministry of Education will not approve any teaching materials that do not use the approach. This is relevant as Singapore is often regarded as an international model for the teaching of maths and many countries have tried to emulate their practices.
Below you can see some examples of how the CPA approach might look.
Concrete is the 'doing' stage. During this stage, students use concrete objects to model problems. Unlike traditional teaching methods, in which teachers demonstrate how to solve a problem, the CPA approach brings concepts to life by allowing children to experience and handle physical (concrete) objects. With the CPA framework, every abstract concept is first introduced using physical, interactive concrete materials. For example, if a problem involves adding pieces of fruit, children can first handle actual fruit. From there, they can progress to handling abstract counters or cubes which represent the fruit.
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. Building or drawing a model makes it easier for children to grasp difficult abstract concepts (for example, fractions). Simply put, it helps students visualise abstract problems and make them more accessible.
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.
CPA in practice
Although here I have presented CPA as three distinct stages, our teachers will go back and forth between each stage to reinforce concepts and build explicit links to previous learning. Our approach encourages teachers to vary the apparatus that children use in class. For example, our pupils might one day use counters, another day they might use a ten frame. Likewise, children are encouraged to represent maths problems in a variety of ways. For example, drawing an array, a number bond diagram or a bar model. By systematically varying the apparatus and methods used to solve a problem, children can craft powerful mental connections between the concrete, pictorial, and abstract phases. When teaching young children numbers, counters and multi-link cubes are more commonly used in the UK. However, concrete materials are frequently shelved by the time children reach KS2 — many teachers believe them to be too childish or distracting. Removing concrete materials exposes children to abstract concepts too early. As a result, they miss out on the opportunity to build a conceptual mathematical understanding that can propel them through their education. This is why even in the older year groups at Navigation we try to make full use of a range of resources to support pupil learning. It is important to recognise that 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. What is important, therefore, is that all learners, however young, can see the connections between each representation.
Each subject is driven forward by a member of staff who monitors the attainment and progress of our children. We do this in a variety of ways including speaking to children, looking at their learning in their books, observing lessons and using data gathered from our school’s assessment system. This analysis can then be used to provide support and resources where needed to maintain high standards across all subjects.
In maths, we also make use of termly testing to see how the children are progressing and as one of many tools to see what their next steps are. Maths in EYFS is often externally moderated for quality assurance. Years 2 and 6 are required to sit national tests in maths, in which our pupils outperform their peers both nationally, within Trafford and compared to similar schools. Year 4 sit a multiplication check test for which the most common score was full marks.
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