Why teach mathematics?
“Mathematics is a creative and highly interconnected 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 all forms of employment. A high-quality education in maths 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 in England, 2014
Fluency, Reasoning and Problem Solving
We follow the National Curriculum aims for mathematics, which are to ensure that all pupils:
- become fluent in the fundamentals of mathematics, including through varied and frequent practise 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.
Fluency development is developing pupil’s understanding of mathematical concepts and skills, how they link and how they ‘work’, in order for pupils to be competent and confident in using a variety of mathematical strategies and approaches independently. It is developed through regular discussion about strategies and methods, a recognition and understanding that there is not one right way of solving a calculation or problem, and regular arithmetic practice (Maths of the Day, arithmetic sessions in Key Stage Two, Times Table practice particularly in lower Key Stage Two).
Reasoning development is developing pupil's language and ability to discuss and explain their strategies, solutions and approaches to mathematical activities. It is developed through regular opportunities to reason verbally and in writing, the use and modelling of high quality mathematical language, the use of diagrams, manipulatives and calculations to support and structure pupil’s reasoning, and the use of a variety of questions, such as Why? Why not? What if? How?
You may see pupils across the school use APE as a strategy to structure their verbal and written reasoning. This strategy is adapted to suit the age of the pupils, and the detail and complexity of the reasoning increases through the year groups.
A= Answer. What is the answer to the question, problem or discussion prompt?
P= Prove. How do you know that the answer is correct? Explain with pictures, diagrams, calculations, manipulatives, sentences or in another way.
E= Explain. Explain their approach, strategies and thinking.
Problem Solving within mathematics refers to activities and experiences where maths is used and applied, where there is more than a simple, straightforward calculation, and where mathematical thinking is required. A variety of skills need to be developed in order for pupils to be successful in solving problems, such as:
- selecting appropriate resources, posing and answering questions,
- representing problems using symbols, words, diagrams or pictures, developing and choosing ways to record,
- making connections,
- concluding and explaining,
- identifying patterns,
- forming generalisations in words, pictures or with resources,
- justifying answers, solutions, methods and conclusions and supporting with examples,
- organising work, working systematically and checking results.
All pupils are given ample opportunity to develop their understanding of mathematics through varied activities that allow pupils to use, enjoy, explore, practice and talk confidently about mathematics. Mathematics teaching and learning in Foundation Stage involves providing pupils with opportunities to develop and improve their skills in counting, understanding and using numbers, calculating simple addition and subtraction problems; and describing shape, spaces and measures.
Key Stage One
The principal focus of mathematics teaching in Key Stage 1 is to ensure that pupils develop confidence and mental fluency with whole numbers, counting and place value. This should involve working with numerals, words and the four operations, including with practical resources (e.g. concrete objects and measuring tools). At this stage, pupils should develop their ability to recognise, describe, draw, compare and sort different shapes and use the related vocabulary. Teaching should also involve using a range of measures to describe and compare different quantities such as length, mass, capacity/volume, time and money. By the end of Year 2, pupils should know the number bonds to 20 and be precise in using and understanding place value. An emphasis on practice at this early stage will aid fluency. Pupils should read and spell mathematical vocabulary, at a level consistent with their increasing word reading and spelling knowledge at Key Stage 1.
National Curriculum in England, 2014
Lower Key Stage Two
The principal focus of mathematics teaching in lower Key Stage 2 is to ensure that pupils become increasingly fluent with whole numbers and the four operations, including number facts and the concept of place value. This should ensure that pupils develop efficient written and mental methods and perform calculations accurately with increasingly large whole numbers. At this stage, pupils should develop their ability to solve a range of problems, including with simple fractions and decimal place value. Teaching should also ensure that pupils draw with increasing accuracy and develop mathematical reasoning so they can analyse shapes and their properties, and confidently describe the relationships between them. It should ensure that they can use measuring instruments with accuracy and make connections between measure and number. By the end of Year 4, pupils should have memorised their multiplication tables up to and including the 12 multiplication table and show precision and fluency in their work. Pupils should read and spell mathematical vocabulary correctly and confidently, using their growing word reading knowledge and their knowledge of spelling.
National Curriculum in England, 2014
Upper Key Stage Two
The principal focus of mathematics teaching in upper Key Stage 2 is to ensure that pupils extend their understanding of the number system and place value to include larger integers. This should develop the connections that pupils make between multiplication and division with fractions, decimals, percentages and ratio. At this stage, pupils should develop their ability to solve a wider range of problems, including increasingly complex properties of numbers and arithmetic, and problems demanding efficient written and mental methods of calculation. With this foundation in arithmetic, pupils are introduced to the language of algebra as a means for solving a variety of problems. Teaching in geometry and measures should consolidate and extend knowledge developed in number. Teaching should also ensure that pupils classify shapes with increasingly complex geometric properties and that they learn the vocabulary they need to describe them. By the end of Year 6, pupils should be fluent in written methods for all four operations, including long multiplication and division, and in working with fractions, decimals and percentages. Pupils should read, spell and pronounce mathematical vocabulary correctly.
National Curriculum in England, 2014
Pupils are encouraged to develop and use a variety of formal and informal methods for recording their mathematical learning, appropriate to their age. Pupils are encouraged to compare and discuss different methods, and are supported in choosing an appropriate and effective strategy for the activity. Mental mathematics and arithmetic are incorporated throughout all lessons and mental calculation strategies are discussed and developed continuously. A secure foundation in mental calculation, mental strategies and recall of number facts is established, before written strategies and methods are introduced.
In order for pupils to understand and reason why concepts, approaches and strategies are true and effective, pupil's conceptual understanding is developed alongside their understanding of how strategies and concepts work. Pupils are taught new strategies using CPA approach (concrete, pictorial, abstract). This means that pupils will first use a strategy or method with concrete resources and manipulatives (such as dienes, counters or cubes), then pictorially (such as diagrams, bar model or pictures of manipulatives), and finally in their head or on paper.
Resources and Manipulatives
A wide range of resources and manipulatives are used throughout the school to develop understanding. All pupils are encouraged to use these to support their learning, and each class has similar resources to ensure progression, consistency and continuity.
Place Value Counters
Dienes or Base Ten
Bead Strings to 20 or 100
Hundred Square or Thousand Square
Place Value Chart
Sorting objects such as dinosaurs, bears, fruit, vehicles.
Pupils are encouraged to develop their understanding of number and calculation by representing them in different ways. The representations that we use in school are:
Part- Whole Model or Cherry Diagram:
These can have as many parts as you like and show ways in which a number can be regrouped. The parts added together make the whole.
This can be one bar, two bars on top of each other (as shown) or more. Bar models can be used in lots of different ways. In this example, the parts add together to make the whole (7 + 7 + 7 + 7 = 28), but it also could show a multiplication (4 x 7 = 28) or division (28 ÷ 4 = 7).
Bar models can also be used to compare fractions, add or subtract fractions or solve fraction problems.
A 5 x 2 rectangle that is used with counters or dots to represent numbers up to 10. Two frames are then used to represent numbers to 20 which is useful to show how to count up to and beyond 10.
These are used to represent multiplications and divisions. This array represents 6 x 3 or 3 x 6 and 18 ÷ 6 and 18 ÷ 3
The use of high quality mathematical language is important in the teaching and learning of mathematics.
Calculation or operation: add, subtract, multiply or divide.
Equivalent: equal or the same.
Number fact: a fact that pupils should be able to recall rapidly.
Representation: a way of drawing, writing or showing a calculation. This could be a number line, with practical resources (dienes, numicon etc.), a diagram or written calculations.
Resource or manipulative: equipment used to support their calculations.
Inverse: calculations that are opposite (addition and subtraction, multiplication and division).
Exchanging or regrouping: when we change one place value to another (e.g. ones to tens, hundreds to tens etc.) This is used when we complete written methods, and is used instead of the terms borrowing and carrying.
Place Value: the value of digits in a number (note that in the new curriculum, we use the terminology ones, rather than units).
Partition or regroup: a number is split into its place values (256 = 200 + 50 + 6).
Regroup: represent the number in a different way. There are many ways in which the pupils could do this which they will explore as they move through the school. e.g. 46 could be 40 + 6 20 + 20 + 6 10 + 10 + 10 + 10 + 5 + 1 etc.
Addition: add, more, and, sum, total, altogether, double, one more, ten more, how many more to make…?, how many more is … than…?, plus, near double, how much more is…?, addition, increase, plus.
Subtraction: take away, less, how many are left?, how many have gone?, one less, ten less, how many fewer is… than…?, difference, how much more is…?, subtract, minus, how much less is…?, subtraction, decrease.
Multiplication: double, near double, lots of, groups of, times, multiply, multiplied by, multiple of, once, twice, times as, repeated addition, array, row, column, equal groups of, multiplication, product, square, square root.
Division: halve, half, share, share equally, one each, equal groups of, divide, divided by, divided into, left, left over, division, remainder, factor, quotient, divisible by… factorise, prime number, prime factor.
Mathematics at Home
Each child in years 2-6 has an individual login to Times Tables Rockstars, a website designed to support the development of multiplication facts.
Other useful websites may be:
Useful documents for ways in which you can help your child at home:
There are a variety of number facts and concepts that pupils need to learn. The aim is for pupils to be able to recall these facts quickly and accurately, and have a secure understanding of these concepts and terms. Regular and continuous practice of these at home will be invaluable.
EYFS- Counting games and activities to 10, then to 20, then beyond.
Showing numbers in different ways and with different objects.
Find one more or one less for numbers up to 20.
Year 1– Addition facts to 10 and subtraction facts from 10.
Addition facts to 20 and subtraction facts from 20.
Practice of simple addition, subtraction, multiplication (as groups of) and division (as sharing). Practical practice of these is ideal.
Counting in steps of 2, 5 and 10 from 0.
Find one more and one less for numbers up to 100.
Counts to and across 100, forwards and backwards from any number.
Regrouping one-digit in different ways (two-digit numbers if confident)
Year 2- Addition facts to 20 and subtraction facts from 20.
Addition facts to 100 and subtraction facts to 100 (multiples of ten).
Multiplication and division facts for the 2, 5 and 10 times tables (and 3, 4, 6 and 8 if confident)
Odd and even numbers.
Counting in steps of 2, 3 and 5 forwards and backwards, from and to 0.
Count in steps of ten from any number.
Regrouping one-digit and two-digit numbers in different ways.
Year 3– Doubling and halving one-digit and two-digit numbers.
Regrouping two-digit and three-digit numbers in different ways.
Multiplication and division facts for the 2, 5, 10, 3, 4 and 8 times tables (and the other times tables if confident)
Counts in steps of 4, 8, 50 and 100 forwards and backwards, from and to 0.
Finds 1,10 or 100 more or less than one, two and three-digit numbers.
Year 4– Multiplication and division facts for all the times tables up to 12 x 12.
Multiples, factors, factor pairs.
Understanding what a decimal number is.
Count in steps of 6, 7, 9, 25 and 1000 forwards and backwards, to and from 0.
Count beyond zero to include negative numbers.
Count in intervals of 10, 100 or 1000 from one, two, three and four-digit numbers.
Regrouping three-digit and four-digit numbers in different ways.
Year 5- Multiples, factors, factor pairs, common factors, prime numbers, prime factors, composite numbers, square numbers, cube numbers.
Multiply and divide whole numbers and decimal numbers by 10, 100 and 1000.
Count forwards and backwards in powers of 10 from any number up to 1,000,000.
Year 6- Revision and practice of previous years.
Positive attitudes towards mathematics are encouraged so that all pupils, regardless of race, gender, cultural background, ability, SEN or disability, including those for whom English is an additional language, develop an enjoyment and confidence within mathematics. All pupils receive high quality inclusive teaching, and high expectations are set for all.
A mastery approach is used within mathematics teaching and learning at St. Michael’s. A mindset that all are able to achieve is adopted by all adults and continually developed within pupils through the use of mixed ability flexible groupings and collaboration; the careful planning of tasks and activities; opportunities for pupils to reflect on their learning and understanding and choose their task, approach or resource; and the use of manipulatives for all pupils and abilities.
“The expectation is that the majority of pupils will move through the programmes of study at broadly the same pace. However, decisions about when to progress should always be based on the security of pupils’ understanding and their readiness to progress to the next stage. Pupils who grasp concepts rapidly should be challenged through being offered rich and sophisticated problems before any acceleration through new content. Those who are not sufficiently fluent with earlier material should consolidate their understanding, including through additional practice, before moving on.”
National Curriculum in England, 2014
Class teachers provide appropriately differentiated activities to ensure all pupils are challenged at an appropriate level for their current learning. Teachers use a variety of differentiation strategies, including different content of activities, different approaches to completing activities, and different use of resources. Pupils are encouraged to be reflective about their own understanding, and are given opportunities to choose the level of activity they complete, through the use of chilli challenges.