Maths at Greenfields

A Mastery approach to Maths

We have high expectations for every child. Teaching for mastery in Maths is essentially the expectation that all children will gain a deep understanding of the maths they are learning. For understanding in Maths to be secure, learning needs to be built on solid foundations.

A mastery approach to the curriculum means pupils spend far longer on fewer key mathematical concepts whilst working at greater depth. Long term gaps in learning are prevented through speedy teacher intervention and those children who grasp the concepts more quickly are given opportunities to deepen their knowledge and improve their reasoning skills rather than accelerating on to new curriculum content.

Problem solving is central and opportunities are given for pupils to calculate with confidence, ensuring an understanding of why it works so that pupils understand what they are doing rather than just learning to repeat routines without grasping what is happening.

Structures and Representations

When introducing a concept, children need to experience multiple representations in order to build an understanding of what it is, what it isn’t and how it connects to other areas of maths.

The aim is not that children use the structures to calculate but that they help to understand the concept and are able to do the maths without it.

PART WHOLE MODEL

Being able to partition numbers is essential for children to be able to understand addition and subtraction. Children need to be fluent in recognising that numbers can be made up in different ways. For example, 5 can be made with 3+2, 4+1, 5+0, 3+1+1 etc.

Part whole models are used practically first. The children physically partition and combine objects or counters so that they can see the relationship between the parts and the whole.

Once children are fluent with concrete representations, they can start to represent the part whole models with numbers:

As we use the models, children use stem sentences to help them explain them. Eg 3 is a part, 2 is a part, 5 is the whole. As they practise they start to link numbers together.

Once children are confident with the part whole model, it can become more complex:

BAR MODEL

The bar model is used to illustrate the structure of a problem. They are used to help children decide which calculations they will need to solve a problem. They link very closely with the work children have done on part whole models.

In Reception and KS1, simple calculations are explored practically and when the children are ready they can also be represented pictorially. For example:

Sally had 3 sweets. Armani gave her 2 more. How many does she have now?

Using red and blue cars: Rasheed had 5 red cars and 3 blue. How many more red cars does he have?

Once the children have had lots of experience from concrete – pictorial – abstract they are ready to start generalising.

This becomes a generalisation where a whole will represent 5 and not distinct squares:

TENS FRAMES

Once children have developed a basic sense of the numbers up to ten, they need to develop a strong 'sense of ten' as a foundation for both place value and mental calculations. (This is not to say that young children will not also have an awareness of much larger numbers. Indeed, there is no reason why children should not explore larger numbers while working in depth on 'ten-ness'.)

A sense of ten

Ten-frames are two-by-five rectangular frames into which counters are placed to illustrate numbers less than or equal to ten, and are therefore very useful devices for developing number sense within the context of ten. Various arrangements of counters on the ten-frames can be used to prompt different mental images of numbers and different mental strategies for manipulating these numbers, all in association with the numbers' relationships to ten.

“Bridging 10” can be difficult for children when they are first learning addition and subtraction. The tens frame helps them to see how they can partition smaller numbers to make 10 and a bit more:

NUMBER LINES

• Putting numbers on a line links discrete and continuous quantities.

• Positioning numbers on the number line helps develop understanding of the number system.

• The number line helps children connect different representations of numbers.

• Placing numbers at equal spaces on a number line is a key skill and marker of understanding.

Number lines can also be used to illustrate what we later want children to be able to do mentally. Working with number lines can enable children to visualise what they are doing as they are adding and subtracting.

Children progress from a number track (with a box for each number), onto numbered lines, partially numbered lines and blank lines.

Number lines are used throughout school and include labelling in 1000s and placing fractions between whole numbers.

ARRAYS

Arrays are key models for supporting children with their understanding of multiplication and division.

We start with arrays to illustrate multiplication as repeated addition.

We keep using arrays as we introduce different multiplication facts and we are able to look at the relationships between them.

When we introduce multiplication of larger numbers we can see how those numbers are partitioned and recombined to make multiplication easier. This links with short multiplication.

We can use shorthand when children are familiar with arrays, just labelling the lines to lead them towards long multiplication. It also helps them to understand how to calculate area.