Maths Presentationfor Parents and Carers

March 2017

Jane Cartlidge – HeadteacherSian Randall-Jones – Acting Deputy

Headteacher and LKS2 Phase Leader

- Explanation of:Mastery & Singapore MethodConcrete-Pictorial-Abstract

Bar modelling

- What it looks like at Southfield- Helping at Home

- Questions-Your feedback

The essential idea behind the mastery teaching approach is that all pupils gain a deep understanding

of the mathematics. This ensures that:future mathematical learning is built on solid

foundations which do not need to be re-taught (less breadth but greater depth)

Increasingly, there will be less need for separate catch-up programmes due to some children falling

behind;pupils who, under other teaching approaches, can

often fall a long way behind, are better able to keep up with their peers, so that gaps in attainment are

narrowed whilst the attainment of all is raised.

Concrete Pictorial Abstract

Concrete, pictorial, abstract (CPA) is a highly effective approach to teaching that develops a deep and sustainable understanding of maths. Developed by American psychologist, Jerome Bruner, the CPA approach is the mainstay of

maths teaching in Singapore.

Background To Concrete, Pictorial and Abstract (CPA)Children and adults can find maths difficult

because it is abstract. The CPA approach helps children learn new ideas and build on their existing knowledge by introducing abstract concepts in a more

familiar and tangible way.

CONCRETE- A range of resources

CONCRETE- A range of resources

Challenge:

In your tables can you think how the Concrete Resource you have been given might be used for Maths Learning …

Importance of concrete

Ofsted’s 2012 report ‘Made to Measure’suggests that although manipulatives are

used in some primary schools to support

teaching and learning they are not used as

effectively or as widely as they might be.

Importance of concrete

Used well, manipulatives can enable

pupils to inquire themselves- becoming

independent learners and thinkers. They

can also provide a common language with

which to communicate cognitive models

for abstract ideas.’Drury, H. (2015)

Importance of concreteJean Piaget's (1951) work suggests that

children aged seven to ten years old work in

primarily concrete ways and that the

abstract notions of mathematics may only

be accessible to them through embodiment

in practical resources.

PictorialPictorial is the “seeing” stage, using representations of the objects to model problems. This stage encourages children 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 represent the objects in the problem.Building or drawing a model makes it easier for children to grasp concepts they traditionally find more difficult, such as fractions, as it helps them visualise the problem and make it more accessible.

Pictorial

Sian and Jane have £36 to spendJane has twice as much as SianHow much does Sian have?

Many of you can jump straight to the Abstract, but some pupils who have missed the Concrete / Pictoral stage can’t!

Pictorial

Bar model / Bar Method / Singapore Bar Method –pictorial representation of a problem

AbstractAbstract is the “symbolic” stage, where children are able to use abstract symbols to model problems.Only once a child has demonstrated that they have a solid understanding of the “concrete” and “pictorial” representations of the problem, can the teacher introduce the more “abstract” concept, such as 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.Although we’ve presented CPA as three distinct stages, a skilled teacher will go back and forth between each representation to reinforce concepts.Our approach encourages teachers to vary the apparatus the children use in class, for example, one day they might use counters, another day they might use a ten frame. Likewise, children are encouraged to represent the day’s maths problem 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 they use to solve a problem, we help children to make quicker mental connections between the concrete, pictorial and abstract phases.

Why is CPA so important? (1)

• It gives children a deep understanding of maths

• Concrete resources give time pupils to investigate a concept first - and then make connections when formal methods are introduced

Why is CPA so important? (2)

• The pictorial stage allows pupils to demonstrate and sustain their understanding of mathematical concepts and processes

• The abstract stage should run alongside the concrete - pictorial stage (enables pupils to read mathematical statements and show their understanding using concrete resources or pictorial representations).

3 + 1 = 4

Concrete or pictorial representations support students to understand

abstract concepts.

Division with lollipop sticks…

Discussion

How else can you solve 13 ÷ 4 using

concrete manipulatives and pictorial

representations?

Multiple representations

13 ÷ 4 = 3 remainder 1

Multiple representations

13 ÷ 4 = 3 remainder 1

Multiple representations

13 ÷ 4 = 3 remainder 1

The Concrete and

and Pictoral

Stages are NOT

just for our

younger or less

able pupils – older

and more able

pupils benefit too.

See this pictoral

representation of

division in UKS2.

Cups and counters

CalculationsModel

What’s in the cup?

How can we represent the cups and counters?

c

c2c + 2 = c + 5

-2 -2

2c = c + 3

c = 3

-c -c

c

However, it should never be a case of concrete 'good', abstract 'bad'. 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 but children in KS2 may still use either or both alongside the developing abstract. What is important, therefore, is that all learners, however young (or old!), can see the connections between each representation.

How we have incorporated the Singapore method into our school:

-You can understand why we favour plain paper for Maths - so that we do not inhibit pictoral representation, older pupils working in abstract can lay out calculations on plain paper (squared paper is always available for graphs etc.)- Concrete and Pictoral were already a strong feature in EYFS and

KS1- In KS2, we have reviewed our teaching to strengthen the C – P

– A transition- In KS2, we have re-organised our day to maximise Maths

Learning.

What can you do at home to support your child’s learning in Maths?

- Share a range of Concrete and Pictoral methods- There are lots of Singapore and CPA resources online – including

YouTube examples- Moving on to the Abstract Methods – follow the school’s

calculation Policy – to be on the Website from Term 5 - My Maths- A word about Times Tables

Any questions please?

Please leave feedback, thank you for coming.

Jane CartlidgeSian Randall-Jones