# Mathematical understanding - Programme of learning

Statutory contentLearning in this area should include an appropriate balance of focused subject teaching and well-planned opportunities to use, apply and develop knowledge and skills across the whole curriculum.

The programme of learning is made up of:

## Curriculum aims

This area of learning contributes to the achievement of the curriculum aims for all young people to become:

- successful learners who enjoy learning, make progress and achieve
- confident individuals who are able to live safe, healthy and fulfilling lives
- responsible citizens who make a positive contribution to society.

## Why this area of learning is important

Mathematics introduces children to concepts, skills and thinking strategies that are essential in everyday life and support learning across the curriculum. It helps children make sense of the numbers, patterns and shapes they see in the world around them, offers ways of handling data in an increasingly digital world and makes a crucial contribution to their development as successful learners.

Children delight in using mathematics to solve a problem, especially when it leads them to an unexpected discovery or new connections. As their confidence grows, they look for patterns, use logical reasoning, suggest solutions and try out different approaches to problems.

Mathematics offers children a powerful way of communicating. They learn to explore and explain their ideas using symbols, diagrams and spoken and written language. They start to discover how mathematics has developed over time and contributes to our economy, society and culture. Studying mathematics stimulates curiosity, fosters creativity and equips children with the skills they need in life beyond school.

## 1. Essential knowledge

Children should build secure knowledge of the following:

- the range of ways mathematics can be used to solve practical problems, model situations, make sense of data and inform decision making
- different types of numbers (i) and what they represent
- how numbers can be used for measurement, quantification and comparison and applied in different contexts
- how to use geometry to explore, understand and represent shape and space
- how likelihood and risk can be understood, quantified and used in everyday life.

### Explanatory text

**Numbers:** This includes natural numbers, integers (positive and negative whole numbers) and rational numbers (fractions and decimals)

## 2. Key skills

These are the skills that children need to learn to make progress:

- generate and explore ideas and strategies, pursue lines of mathematical enquiry and apply logic and reasoning to mathematical problems
- make and test generalisations, identify patterns and appreciate equivalences and relationships (i)
- develop, select and apply a range of mental, written and ICT-based methods and models to estimate, approximate, calculate, classify, quantify, order and compare
- communicate ideas and justify arguments using mathematical symbols, diagrams, images and language
- interpret findings, evaluate methods and check outcomes.

### Explanatory text

**Relationships:** This includes families of equivalent fractions; the inverse relationship between addition and subtraction

## 3. Cross-curricular studies

This area of learning should provide opportunities for:

- Children to develop and apply their literacy, numeracy and ICT skills
- Personal, emotional and social development
- Enhancing children's mathematical understanding through making links to other areas of learning and to wider issues of interest and importance.

## 4. Breadth of learning

a. When experiencing mathematics as a creative activity and being introduced to its role in the world around them children should:

- be taught to work logically and critically as they undertake focused, practical, problem-solving activities (i) in mathematical, cross-curricular and real-world contexts
- visualise quantities, patterns and shapes and develop strategies for working things out in their head as well as on paper and using ICT
- work individually and collaboratively to explore ideas and pursue lines of mathematical enquiry
- articulate their thinking in discussions and make choices about the strategies they use to solve problems, based on what they know about the efficiency and effectiveness of different approaches
- use mathematics to manage money, make sense of information, assess likelihood and risk, predict outcomes and construct reasoned arguments
- meet with people who use mathematics in their work
- use a wide range of practical resources, including ICT
- use mathematical language to explain, refine and evaluate their own and others' work.

### Explanatory text

**Problem-solving activities:** Problem-solving skills should be developed across the primary phase by providing more substantial and increasingly open questions or tasks

## 5. Curriculum progression

The overall breadth of learning should be used when planning curriculum progression. Children should be taught:

### Mathematical understanding - Number and the number system

#### Early

E1. to estimate the number of objects and count them, recognising conservation of number

E2. to read, write and order numbers to 100 and beyond using a range of representations (i)

E3. to explore and explain patterns (i), including number sequences in the counting system

E4. to group, match, sort, partition and recombine numbers, developing an understanding of place value

### Explanatory text

**Range of representations:**For example, number lines, number squares, structural apparatus

**Patterns:**This includes additive number sequences, such as counting in groups of e.g. 2, 5 or 10, odds and evens; and relationships between numbers, e.g. the sum of two odd numbers is always even. Using calculators to explore number patterns and properties is important here

#### Middle

M1. to understand and interpret negative numbers, simple fractions (i), large numbers and tenths, written as decimals, in practical and everyday contexts

M2. to generate and explore a range of number patterns (i), including multiples

M3. to make and test general statements about numbers, sort and classify numbers and explain methods and findings

M4. to approximate numbers, including rounding (i), and understand when that can be useful

M5. about the representation of number in different contemporary cultures (i)

### Explanatory text

**Simple fractions:**Simple fractions include half, third, quarter, fifth, tenth, two-thirds and three-quarters

**Number patterns:**using ICT for changing values and exploring in a spreadsheet model

**Rounding:**For example rounding to the nearest ten, hundred and thousand

**Contemporary cultures:**For example Arabic, Chinese and Indian numerals

#### Later

L1. to use decimals up to three decimal places in measurement contexts

L2. to understand and use the equivalence of families of fractions and their decimal representation when ordering and comparing

L3. to explore number patterns and properties (i), and represent them using graphs, simple formulae and ICT (i)

L4. about the development of the number system (i)

L5. to interpret computer and calculator displays and round to an appropriate level of accuracy

### Explanatory text

**Number patterns and properties:**This includes factors, primes and square numbers

**Graphs, simple formulae and ICT:**Changing variables and rules in spreadsheet models; using graphing software

**Number system:**For example the number system we use today is Hindu-Arabic; the Roman and Egyptian number systems do not use a place value; Babylonian numbers and Mayan numbers use base 60 and base 20 respectively; Greeks explored square and triangle numbers

### Mathematical understanding - Number operations and calculation

#### Early

E5. a range of strategies for combining, partitioning, grouping and sharing (including doubling and halving) and increasing and decreasing numbers, to solve practical problems (i)

E6. to use number bonds to ten to add and subtract mentally (i) whole numbers with one or two significant figures

E7. to represent addition and subtraction as number sentences including finding missing numbers and understanding the equals sign (i)

### Explanatory text

**Solve practical problems:**This lays the foundations for understanding number operations

**Add and subtract mentally:** For example 700+300=1000; 60+n=100; 57+33=90; 57-8=49; this develops their understanding of the inverse relationship between addition and subtraction

**Equals sign:**For example 3+1=1+3; 3+1=n+2; 3+1=5-n

#### Middle

M6. to compare (i) two numbers by finding the difference between them

M7. to use the relationship between addition and subtraction (i) and addition and multiplication to understand and generate equivalent expressions (i)

M8. to use simple fractions to find fractional parts and express proportions

M9. to select from a range of mental strategies for the addition and subtraction of numbers with two significant figures

M10. to understand division as grouping and as sharing and solve division problems using multiplication facts (i)

M11. to visualise and understand multiplication represented as an array, record multiplication as number sentences and solve problems using multiplication facts

M12. to use estimation to find approximate answers to calculations (i), to record calculations and check answers and methods

### Explanatory text

**Compare:**For example finding how much the temperature changed

**Relationship between addition and subtraction**

**:**For example since 54+37=91, 91-37=54 and 91-54=37

**Equivalent expressions:**For example 3×13=3×10+3×3; 5×19=5×20-5×1

**Multiplication facts:**Multiplication facts should include 2, 3, 4, 5 and 10

**Calculations:**For example to estimate the cost of an apple sold in a pack of four or to recognise that 296+735 will be approximately 1000

#### Later

L6. to use proportional reasoning (i) to compare numbers and quantities and solve problems

L7. to extend their knowledge of multiplication facts to 10×10 and use them to solve multiplication and division problems

L8. to understand and use different models of division, including interpreting the outcome of a division calculation, in relation to the context, where the answer is not a whole number

L9. to recognise and use the relationship between fractions and division and represent division as number sentences (i)

L10. to recognise and use the relationships between addition, subtraction, multiplication and division

L11. to develop a range of strategies (i) including mental and written ones, for calculating and checking, including using a calculator or computer efficiently

L12. to solve multi-step problems involving more than one operation

### Explanatory text

**Proportional reasoning:**Including simple ratio and percentages - for example 45 is three times greater than 15, they are in the ratio 3:1

**Represent division as number sentences:**For example 325÷5=(300+25)÷5=300÷5+25÷5 or 325/5=300/5+25/5

**Range of strategies:**This includes mental methods, informal and formal written methods and using technology

### Mathematical understanding - Money

#### Early

E8. to use coins of different values and recognise the equivalence of different combinations of coins (i)

E9. to compare and order costs of different items

### Explanatory text

**Different combinations of coins:**Including in the context of buying and selling involving role play

#### Middle

M13. to record amounts of money using pounds and/or pence, converting between them as appropriate

M14. how to handle amounts of money (i) in the contexts of shopping, saving up and enterprise activities

### Explanatory text

**Handle amounts of money:**For example to find and compare unit costs of items that are sold in multiple unit quantities

#### Later

L13. to solve problems related to borrowing, spending and saving (i)

L14. to understand and convert between different currencies

L15. how to manage money (i) and prepare budgets for events, including using spreadsheets

### Explanatory text

**Solve problems related to borrowing, spending and saving:**This includes using and interpreting information from external sources and making decimal calculations

**Manage money:**This includes using the context of enterprise activities where children need to work out a range of budgetary options, developing awareness of profit and loss

### Mathematical understanding - Measures

#### Early

E10. to compare and order (i) objects and events

E11. to create and use whole number scales (i) to measure

### Explanatory text

**Compare and order:** This includes mass, time and length, for example answering questions such as 'which is heaviest?' 'which takes longer?' or 'which is longest?'

**Number scales:** Number scales include standard and non-standard units

#### Middle

M15. to recognise when length and capacity are conserved

M16. to use standard units to estimate measures and to measure with appropriate accuracy

M17. to recognise and use equivalent representations of time

M18. to measure angles using fractions of turn and right angles

M19. to explore the development of different measuring systems, including metric and imperial measures

#### Later

L16. to recognise when area, volume and mass are conserved

L17. to convert between units within the metric system

L18. to use an angle measurer to measure angles in degrees

L19. to solve problems involving time and time intervals, including time represented by the 24-hour clock

L20. to use decimal calculations to solve problems with measures

### Mathematical understanding - Geometry

#### Early

E12. to identify, group, match, sort and compare common shapes (i) using geometric properties (i)

E13. to identify, reproduce and generate geometric patterns including the use of practical resources and ICT

E14. to generate instructions for straight and turning movement (i)

### Explanatory text

**Common shapes:**Common shapes include triangle, square, rhombus, rectangle, kite, parallelogram, circle, cube, prism, pyramid, cylinder, cone, and sphere

**Geometric properties:**Geometric properties include edges, vertices, faces, right-angles, straight, curved, closed and open

**Turning movement:**For example using a programmable toy or describing a familiar journey including change of direction/angle of turn

#### Middle

M20. to recognise symmetry properties of 2D shapes and patterns

M21. to make simple scalings (i) of objects and drawings

M22. to understand and use angle as the measure of turn

M23. to understand perimeter as a length and to find the perimeter of rectangles and other shapes

M24. to create sequences of instructions using ICT, including generating symmetric and repeating geometric patterns

### Explanatory text

**Simple scalings:**Simple scales include half, twice and ten times

#### Later

L21. to use and make maps, scale models and diagrams for a purpose

L22. to understand area as the space enclosed by a perimeter on a plane, and find areas of rectangles and related shapes (i)

L23. to solve practical problems involving 3D objects (i)

L24. to visualise geometric objects (i) and to recognise and make 2D representations of 3D shapes

L25. to create and refine sequences of instructions, using ICT to construct and explore geometric patterns and problems (i)

L26. to explore aspects of geometry to find out about its origins (i), and its use in different cultures, religions, art and architecture (i)

### Explanatory text

**Shapes:** This includes triangles and shapes that are made up of triangles and rectangles including the surface area of 3D objects

**3D Objects:**This includes developing understanding of the volume of cuboids by solving problems such as 'what is the smallest possible box to hold six smaller boxes?'

**Visualise geometric objects:**This includes imagining what something will look like in different orientations

**Patterns and problems:**This should include use of procedures to improve efficiency

**Origins:**For example Greek architecture and discoveries, stone circles and pyramids

**Different cultures, religions, art and architecture:**For example Islamic patterns, Japanese temple art, Rangoli patterns, modern art and ancient and modern architecture

### Mathematical understanding - Statistics

#### Early

E15. to generate and explore questions that require the collection and analysis of information

E16. to collect, group, match, sort, record and represent information (i) for a purpose and store it using ICT

E17. to interpret and draw conclusions from information they have collected

### Explanatory text

**Represent information:**This includes using Venn and Carroll diagrams, simple frequency diagrams and simple data-handling software to create tables and graphs

#### Middle

M25. to collect and structure information using ICT so that it can be searched and analysed (i), including using appropriate field headings and data types

M26. to use frequency diagrams and bar charts to represent and record information

M27. to interpret their own and others' data

### Explanatory text

**Analysed:**Analysis should include discussion about 'reasonableness' of outcomes

#### Later

L27. how statistics (i) are used in society today

L28. to use different kinds of averages and range to summarise and compare data sets

L29. to use data to assess likelihood and risk and develop an understanding of probability through computer simulations, games and consideration of outcomes of everyday situations

L30. to discuss, sort and order events according to their likelihood of occurring

L31. to answer questions or test hypotheses by using ICT to collect, store, analyse and present data (i)

L32. to use ICT to represent data (i) on a scattergraph, and proportional data (i) in a pie chart in order to explore possible relationships and interpret the findings (i)

### Explanatory text

**Statistics:**For example statistics are used to inform the public about how the local council spend their money, to monitor safety in factories, to inform decisions about whether to install traffic lights, or to decide what stock to order

**Present data:**For example using data types including text, number, date, currency, yes/no and error checking through inspecting outcomes

**Represent data:**For example height and weight for a chart on a child's development

**Proportional data:**Proportional data means data where fractions of the population are represented, such as how a council spends its budget, or how all the children in a class travel to school

**Interpret the findings:** This should include understanding how these diagrams work and choosing an appropriate representation for the data

## What to do now

### Key actions:

- Download the full programme of learning [pdf 194kb]

### Quick links

### See Also

Here are some useful related resources:

- NRICH - Specialists in rich mathematics
- BEAM (Be a mathematician)
- National Strategies
- ATM (The Association of Teachers of Mathematics)
- The Mathematical Association
- NCETM (The National Centre for Excellence in the Teaching of Mathematics)

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