The Nature of Mathematics
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.
(National Curriculum, 2014)
The purpose of mathematics in our school is:
To implement the current legal requirements of the Early Years Foundation Stage (EY) and the National Curriculum (NC).
For our children:
- to 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.
- to reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language.
- to 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.
In the National Curriculum 2014, the emphasis has been to ensure that all children have access to the following strands of mathematics:
This means that children need to be regularly exposed to opportunities involving increasingly complex problem solving which allows them to apply their Maths knowledge. In doing so, they should be encouraged to develop an argument and line of enquiry that they can prove and justify using mathematical vocabulary. This includes the ability to break down problems, both routine and non-routine, into a series of steps.
We believe that our broad and balanced mathematics education is the entitlement of all children, regardless of ethnic origin, gender, class, aptitude or disability.
The programmes of study set out within each domain in the National Curriculum (2014) will be used to ensure children get the learning experiences that are required.
It is important that children can explore Maths and present their findings not only in a written form but also visually and verbally; to that end the school will adopt the CPA approach: concrete, pictorial, abstract. This will allow the children to experience the physical aspects of maths before finding a way to present their findings and understandings in a visual form before relying on the abstract numbers.
The school have been involved with the NCETM and NNW MathsHub through the Maths Lead who completed the Mastery Readiness course in 2020.
Opportunities for Mathematical Thinking allow children to make chains of reasoning connected with the other areas of their mathematics. A focus on Representation and Structure ensures concepts are explored using concrete, pictorial and abstract representations, the children actively look for patterns as well as specialise and generalise whilst problem solving. Coherence is achieved through the planning of small, connected steps to link every question and lesson within a topic. Teachers use both procedural and conceptual Variation within their lessons and there remains an emphasis on Fluency with a relentless focus on number and times tables facts.
The NNW MathsHub and school have provided training for our Maths Lead and teaching assistants around Mastery in Mathematics.
6 Teaching Principles of High-Quality Teaching and Learning in Mathematics
- Teachers believe in the importance of mathematics and that the vast majority of children can succeed in learning mathematics in line with national expectations.
- Whole year groups are taught maths together*. We do not group children by ability. The learning needs of individuals are addressed through careful scaffolding, questioning and appropriate rapid intervention where necessary, to provide the appropriate support and challenge.
- The reasoning behind mathematical processes is emphasised. Teacher/pupil interaction explores how answers were obtained as well as why the method worked and what might be the most efficient strategy.
- Conceptual variation and procedural variation are used extensively throughout teaching. This helps to present the mathematics in ways that promote deep, sustainable learning.
(Conceptual variation is where the concept is varied and there is intelligent practice. Positive variation is showing what the concept is, and negative variation is showing what a concept isn’t. This clears away misconceptions from the very start. Within positive variation, both standard and non-standard representations are shown.)
(Procedural variation is where different procedures and/or representations are used to bring about understanding. For example, teachers may collect several solutions for a problem, before guiding the class towards the most efficient method. It also involves highlighting the essential features of a concept or idea through varying the non-essential features. Variation is not the same as variety – careful attention needs to be paid to what aspects are being varied (and what is not being varied) and for what purpose.)
8 Classroom Norms to Establish in Mathematics
- Everyone can learn mathematics to the highest levels
- If you ‘can’t do it’, you ‘can’t do it yet’
- Mistakes are valuable
- Questions are important
- Mathematics is about creative and problem solving
- Mathematics is about making connections and communicating what we think
- Depth is much more important than speed
- Maths lessons are about learning, not performing
Teaching and Learning –A ‘Mastery’ Approach
The teaching and learning of mathematics at Great Corby School should include aspects of the following Mastery approach strategies in every lesson and/or over a series of lessons.
Concrete, pictorial, abstract (CPA) is a highly effective approach to teaching that develops a deep and sustainable understanding of maths.
Concrete is the ‘doing’ stage, using concrete objects to model problems. Instead of the traditional method of mathematics teaching, where a teacher demonstrates how to solve a problem, the CPA approach brings concepts to life by allowing pupils to experience and handle physical objects themselves. Where appropriate, a new abstract concept is learned first with a ‘concrete’ or physical experience. For example, if a problem is about adding up four baskets of fruit, the pupils might first handle actual fruit before progressing to handling counters or cubes which are used to represent the fruit.
Pictorial is the ‘seeing’ stage, using representations of the objects to model problems. This stage encourages pupils to make a mental connection between the physical object and abstract levels of understanding by drawing or looking at pictures, circles, diagrams or models which represents the objects in the problem. Building or drawing a model makes it easier for pupils to grasp concepts they traditionally find more difficult, such as fractions, as it helps them visualise the problem and make it more accessible.
Abstract is the ‘symbolic’ stage, where pupils are able to use abstract symbols to model problems. Only once a child has demonstrated that they have a solid understanding or the ‘concrete’ and ‘pictorial’ representations of the problem, can the teacher introduce the more ‘abstract’ concept, such as mathematical symbols. Pupils are introduced to the concept at a symbolic level, using only numbers, notation and mathematical symbols, for example +, - , x, ÷ to indicate addition, subtraction, multiplication and division.
What is fluency?
Fluency comes from deep knowledge and practise. This is the first stage of a pupil’s understanding. When assessing pupils, if a child is fluent in a concept they will be assessed as secure in that learning objective.
Fluency includes: conceptual understanding, accuracy, rapid recall, retention and practise.
Accuracy – Pupils carefully completing calculations with no or few careless errors.
Pace – Pupils are able to quickly recall the appropriate strategy to solve the calculation and progress through a number of questions at an age appropriate pace.
Retention – Pupils will be able to retain their knowledge and understanding on a separate occasion to when the concept was first introduced.
The key to fluency is deep knowledge and practise and making connections at the right time for the child.
What is reasoning?
Verbal reasoning demonstrates that pupils understand the mathematics. Talk is an integral part of mastery as it encourages students to reason, justify and explain their thinking. For example, young learners can voice their thought processes where older students could take part in class debates, giving them space to challenge their peers using logical reasoning.
A mathematical mastery classroom should never be a quiet classroom. The way pupils speak and write about mathematics transforms their learning. Mastery approaches use a carefully sequenced, structured approach to introduce and reinforce mathematical vocabulary.
To encourage talk in mathematics, teachers may introduce concepts by including sentence structure (stem sentences). Pupils should be able to say not just what the answer is, but how they know it’s right. This is the key to building mathematical language and reasoning skills. This gives pupils the confidence to communicate their ideas clearly, before writing them down.
Example stem sentences: The denominator is 5 because the whole has been divided into 5 equal parts. The numerator is 3 because 3 equal parts have been shaded/circled.
Teachers then maintain a high expectation upon pupils to repeat and use the correct mathematical vocabulary to explain their understanding verbally. By displaying the vocabulary during the lesson, pupils will be able to use this independently.
What is problem solving?
Mathematical problem solving is at the heart of the Mastery 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. Through problem solving, pupils are required to select their mathematical knowledge and apply this to a new concept.
Planning – Teaching and Learning
All year groups (EYFS – Year 6) use the White Rose Math’s SOL (Scheme of Learning) and Progression Resources to teach sequenced and coherent mathematics. Each block of learning is taught through Small Steps. Teachers use the WRM Blocks to ensure they teach new knowledge and skills that are underpinned by previous learning and building towards future learning. Teachers draw from a variety or resources including WRM and Primary Stars. Teachers have the flexibility to choose resources they feel are most effective to support the needs of all learners (differentiation) and ensure they achieve the aims of fluency, reasoning and problem solving.
All lessons should include an element of fluency – whether that be revisiting a previous taught objective or part of new learning.
Recording of Learning
Pupils Y1-Y6 record their work in their Maths exercise books and on Showbie (online learning platform). Evidence of learning could be photographs, worksheets of mathematical jottings.
The presentation of mathematics books is to be consistent, age appropriate and show that pupils take pride in the appearance of their work.
- The date to be written in either figures or words and underlined
- A title and underlined
- Sheets to be stuck in neatly
Children will be assessed at the end of each block using the White Rose Maths end of block assessments. Results will be recorded on the Maths Tracker. At the end of each full term the children will complete the White Rose Maths Arithmetic and Reasoning papers and results will be recorded on the Maths Tracker.
Feedback and Marking
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