1. Maths Presentation for Parents and Carers May Sian Randall-Jones – Deputy Headteacher & Maths Lead

2. - Explanation of: Mastery & Singapore Method Concrete-Pictorial-Abstract Bar modelling - What it looks like at Southfield - Helping at Home - Questions - Your feedback

3. 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.

4. 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.

5. 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.

6. CONCRETE - A range of resources
Discuss the different resources displayed and explain that concrete options are endless. Concrete resources are simply practical items that pupils can hold and manipulate to help them explore abstract mathematical concepts and the relationships between them.
After slide 4, share the White Rose Maths Hub video, The importance of concrete. Give teachers time to discuss what they like and/or dislike about the video and explore the key messages.

7. 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 …
Discuss the different resources displayed and explain that concrete options are endless. Concrete resources are simply practical items that pupils can hold and manipulate to help them explore abstract mathematical concepts and the relationships between them.
After slide 4, share the White Rose Maths Hub video, The importance of concrete. Give teachers time to discuss what they like and/or dislike about the video and explore the key messages.

8. Importance of concrete
Real things and structured images enables children to understand the abstract.
The concrete and the images are a means for children to understand the symbolic so it’s important to move between all modes to allow children to make connections
Morgan, D. (2016)
Jean 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.
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.
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)
Look at and discuss the different quotes. Highlight the fact that Debbie Morgan speaks about three modes: concrete, pictorial and abstract. All pupils (not just KS1 or ‘lower achievers’) should be exposed to all three modes.

9. Pictorial
Pictorial 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.

10. Pictorial
Sian and Jane have £36 to spend
Jane has twice as much as Sian
How 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!

11. Pictorial
Bar model / Bar Method / Singapore Bar Method – pictorial representation of a problem

12. Abstract
Abstract 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.

13. 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
Explain that the concrete-pictorial-abstract approach is based on Jerome Bruner’s important psychological research. Bruner suggested that pupils need to take three vital steps in order to develop their understanding of a concept. This systematic approach is designed to give pupils the time to make connections, notice mathematical patterns and really understand what’s going on.

14. 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).
Explain that the pictorial stage involve the pupils representing the concrete resource or using a bar model to clarify what a maths question actually means.
Give an example for the ‘abstract stage’. For example, when teaching addition, pupils can use or draw base 10 alongside the abstract method (i.e. formal written columns).

15. Concrete or pictorial representations support students to understand
3 + 1 = 4
Concrete or pictorial representations support students to understand
abstract concepts.
Explain that the pictorial stage involve the pupils representing the concrete resource or using a bar model to clarify what a maths question actually means.
Give an example for the ‘abstract stage’. For example, when teaching addition, pupils can use or draw base 10 alongside the abstract method (i.e. formal written columns).

16. Division with lollipop sticks…
Print out the investigation sheet used in the video. Allow time for members of staff to look at the investigation in more detail. Why do we make squares when dividing by 4? What’s the purpose of creating shapes?
Can the pupils represent this resource pictorially? How?

17. Discussion
How else can you solve 13 ÷ 4 using concrete manipulatives and pictorial representations?
In preparation for the next slide, give your staff members time to explore this concept with different resources or pictorial representations. Jot down ideas on poster paper.

18. Multiple representations
13 ÷ 4 = 3 remainder 1
Tens
Ones
Show your staff the different representations.
(i) Counters shared between 4 Can be done physically with bean bags and hoops or counters and circles. However the grid is ideal for bar modelling (sharing).
(ii) Grouping with place value counters Start by making the number 13. This could also be done with base 10. Look at the tens: can I make any groups of 4? No, but can I make an exchange? Yes, I exchange one ten for ten ones. I can make 3 groups of 4 with one left over.
(iii) Rods and rulers for repeated subtraction We want to see how many fours go into 13. Pupils should start at 13 on their rulers and use the Cuisenaire rod that is worth 4. They must always arrive at 0. Three whole fours with 1 left over go into 13.
(iv) Numicon encourage grouping Start by making the number 13. How many fours go into 13? Place the number four alongside the number 13. Three fours with one left over make 13.

19. 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.

20. Cups and counters

21. How can we represent the cups and counters?
What’s in the cup?
How can we represent the cups and counters?
Model
Calculations c c
2c + 2 = c + 5 -2 -2
2c = c + 3 c -c -c
How could this KS3 content be adapted for KS2? (Example: pupils could be asked to solve 2c + 4 = 14)
Could cups and counters be used anywhere in the KS1 curriculum (egg finding the missing digit in a problem)? Pupils could have 10 counters and 2 cups, their partner must hide some counters in the cup and show some. They must write down the calculation 10 = 7 + __ and identify the missing digit.
c = 3

22. However, it should never be a case of concrete 'good', abstract 'bad'
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.

23. 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.

24. 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?

25. Please leave feedback, thank you for coming.
Sian Randall-Jones