1. Concrete Pictorial Abstract Key Stage 1 Mathematics Session 1
Lyne & Longcross
Concrete Pictorial Abstract Key Stage 1 Mathematics Session 1

2. Use the resources on your table to represent the number 25.
How many different representations can you show?
Use the resources on your table to represent the number 25.
As they come in. Have prepared on the table.
Cuisenaire, numicon, tens and ones, place value counters, counting stick, real life objects, arrow cards, beadstring, whiteboards and pens, tens frame
Walk round to look at each others representations

3. Concrete Pictorial Abstract Key Stage 1 Mathematics Session 1
Lyne & Longcross
Concrete Pictorial Abstract Key Stage 1 Mathematics Session 1
Add dates of your session to your slides
How the project came about and who has been working on it
The project is a trial
Suggestions and recommendations are welcome
Please add questions to the paper if you have any questions etc

4. Aims of the session Theory and research behind CPA.
Applying CPA in Year 1 and 2 classrooms.
Practical ideas for classroom resources.
Time to talk … sharing good practice.

5. There are 7 jelly beans and 5 fruit pastels, how many sweets altogether?
“Show me”
“Prove it!”
Have a go
Share ideas and representations
Actual sweets, counters, beadstrings, numicon, cuisenaire, numberlines, white boards and pens, tens frame, bar model, part, part whole model, tally
Represent your answer using concrete objects (C), pictures (P) and number sentences(A).

6. This is how children in Year 1 and 2 approached the problem …
Demonstration of 10s frame and part/part whole
This is how children in Year 1 and 2 approached the problem …

8. Questionnaire
Please take a few moments to answer this questionnaire about resources you use in your classroom

9. "learning begins with an action - touching, feeling, and manipulating“
So what is CPA?
Children need to experience a mix of three different modes of learning:
Enactive, Iconic and Symbolic.
Jerome Bruner
Not new information. He completed his research in 1966
Research and theory based.
It is what we have always done just with a different name.
"learning begins with an action - touching, feeling, and manipulating“

10. So what about CPA?
Children need to experience a mix of three different modes of learning:
Enactive, Iconic and Symbolic.
Jerome Bruner
New maths curriculum does give teachers the opportunity and time to do this
Steps to mastery
An approach from everyone!
Not something additional – it is best practice
real objects
pictures
symbols
5 = 3 + 2

11. CPA in the Curriculum
The final version of National Curriculum for mathematics aims to ensure all pupils:
become fluent in the fundamentals of mathematics, including through varied and frequent practice with increasingly complex problems over time, so that pupils develop conceptual understanding and are able to recall and apply their knowledge rapidly and accurately to problems
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.
These three aims should not be lost in the detail of the programmes of study. The new draft seeks to strengthen these aims.
These aims come from research carried out by the DfE into high performing jurisdictions
New maths curriculum does give teachers the opportunity and time to do this
Steps to mastery
An approach from everyone!
Not something additional – it is best practice

12. Derek Haylock and Anne Cockburn 2008
A mathematical concept can be thought of as a network of connections between symbols, language, concrete experiences and pictures
Derek Haylock and Anne Cockburn 2008
Linking the whole ideas of
concrete materials –
symbols – arranging cards with numbers on them, pressing buttons on calculators, etc
language – discussing choices, reading, explaining, interpreting
and pictures – drawing number lines, jottings, set diagrams etc
It is this act of manipulation that allows for connections to be made through the different experiences.
Moyer (2001: 176) supports this by stating that it is the active manipulation of materials that ‘allows learners to develop a repertoire of images that can be used in the mental manipulation of abstract concepts’.

13. The Connections Model Linking the whole ideas of concrete materials –
symbols – arranging cards with numbers on them, pressing buttons on calculators, etc
language – discussing choices, reading, explaining, interpreting
and pictures – drawing number lines, jottings, set diagrams etc

14. A resource to share with parents etc

15. Build it Draw it Write it CPA – Child approach
By having various representations of the same thing helps the brain makes connections

16. Every experience builds on the picture.
Piaget’s work on Schemas underlined this need for variation to establish facts.
Every experience builds on the picture.
Leads to depth of learning and mastery.
Piaget (or Benny Hill?)
Piaget’s theory – Well known (1936)
Set stages of development
Schema – You learn something by what you experience or what you have been told then something happens and then it changes your understanding of this. Then with this new information you experience something new or add new information and again your understanding is shaped.

17. My experience is the limit of my world.
If you have provide one view, that is the only way the students will see it.
My experience is the limit of my world.
My language is the limit of my world.
If you are not approaching teaching and learning by providing different representations the children may only understand it in that context
Provide lots of different representations

18. Piaget’s work on Schemas underlined this need for variation to establish facts.
Piaget (or Benny Hill?)

19. Mike Askew: Embodiment
Concrete & Abstract using our bodies
“Getting inside the maths”
Thinking about using hands and fingers for more in-depth number concepts
bunny ears double counting multiplication other ideas?

20. In a study published last year, researchers Ilaria Berteletti and James R. Booth analyzed a specific region of our brain that is dedicated to the perception and representation of fingers known as the somatosensory finger area. Remarkably, brain researchers know that we “see” a representation of our fingers in our brains, even when we do not use fingers in a calculation.
The researchers found that when 8-to-13-year-olds were given complex subtraction problems, the somatosensory finger area lit up, even though the students did not use their fingers. This finger-representation area was, according to their study, also engaged to a greater extent with more complex problems that involved higher numbers and more manipulation. Other researchers have found that the better students’ knowledge of their fingers was in the first grade, the higher they scored on number comparison and estimation in the second grade. Even university students’ finger perception predicted their calculation scores. (Researchers assess whether children have a good awareness of their fingers by touching the finger of a student—without the student seeing which finger is touched—and asking them to identify which finger it is.)

21. Vygotsky – Theory of learning through play 1920-30s
Social aspect of learning and impact on cognitive development
Desire to learn
Zone of proximal development (Area in which they can learn with support)
When a child wants to play they choose toys. They choose concrete items.
Role playing – props and toys they can access things beyond their age.

22. When children play they choose to use concrete things.
When a child wants to play they choose toys. They choose concrete items.
When children play they choose to use concrete things.

23. Zoltan Dienes said that children learn mathematics by means of direct interaction with their environment. He was also clear on the value of play. … and cautioned on the need for “variability”
“Once we have got children to play a number of mathematical games, there comes a moment when these games can be discussed, compared with each other. It is good to teach several games with very similar rule structures, but using different materials, so that it should become apparent that there is a common core to a number of different looking games,”
DIENES (DATE??) – variability principles (showing the same number in lots of different ways)
Multiple embodiments through manipulatives, games, stories and dance.
Agreed with Vygotsky
Perceptual Variability principle – a variety of embodiments eg. Dienes, money but the same concept
Eg my representations of numbers in different ways place value
The Perceptual Variability Principle. This principle suggests that conceptual learning is maximized when children are exposed to a concept through a variety of physical contexts or embodiments. The experiences provided should differ in outward appearance while retaining the same basic conceptual structure. The provision of multiple experiences (not the same experience many times), using a variety of materials, is designed to promote abstraction of the mathematical concept. When a child is given opportunities to see a concept in different ways and under different conditions, he or she is more likely to perceive that concept irrespective of its concrete embodiment.
All theorists are supporting that children need play, variation and physical objects.

24. Here are some of our maths toys …

25. “Every child in my Y2 class knows that 3064 is worth more than 402.”

26. Because they know that this Pokemon is worth more than this Pokemon!
“Every child in my Y2 class knows that 3064 is worth more than 402.”
Because they know that this Pokemon is worth more than this Pokemon!

27. “student’s disposition to learn” is a key influencer on the effect of learning.
Dr John Hattie
If you want to learn it you will learn it.
You may not be able to read a four digit number but in the context of pokemon more confident.

28. Match the different representations
Find your partners with the other calculations eg all the ways of representing 3 x10
When you’ve found all the ways can you put the pictures in order of progression

29. Freddie’s Video – place value
Watch the video of Freddie What do we know about Freddie’s understanding of place value?
Is Freddie ‘stuck in the concrete’
Need to use different resources to represent the number to really deepen his thinking
Freddie – Year 2 (Spring term)
Is now in Year 3 on the autism spectrum
Did not achieve age related expectations

30. Build it, draw it, write it!
How many mints are there in Jar B? There are the same number of sweets in each jar. 25% of the sweets in Jar A are mint. Two fifths of the sweets in Jar B are mint. There are 10 mint sweets in Jar A.
Mrs Brown has two jars of sweets. The jars contain the same number of sweets in total.
25% of the sweets in Jar A are mint.
Two fifths of the sweets in Jar B are mint.
There are 10 mint sweets in Jar A, how many mint sweets are there in Jar B?
Write it:-
2. What information do I need?
The most important parts of this question are 25% of the sweets in Jar A are mint and two fifths of Jar B are mint. It is clear this question is going to involve using fractions and percentages.
The question asks how many mints are in jar B. The answer needs to be a number rather than a percentage or fraction. The number has to be whole as you cannot have a fraction of a sweet.
Two fifths is greater than 25% so there are more sweets in Jar B.
Step A
Look at the information given.
There are 10 mint sweets in Jar A.
25% of the sweets in Jar A are mint ones.
Therefore 25% of Jar A must be 10 mint sweets.
Step B
Now find the number of sweets in Jar A.
The total number of sweets in Jar A is 100%.
If 25% is 10 sweets, 100% is four times 25% therefore there must be sweets in Jar A. 10 x 4 = 40
Step C
Now find the number of mint sweets in Jar B.
There is the same number of sweets in each jar.
Therefore Jar B must also contain 40 sweets.
Two fifths of Jar B are mint sweets therefore 2/5  of 40 needs to be found.
Firstly find 1/5  of 40 by dividing by ÷ 5 = 8
Now find two fifths by multiplying your answer by 2. 8 x 2 = 16
There are therefore 16 mint sweets in Jar B.

31. Build it, draw it, write it!
Rosie has two bags of sweets. Each bag contains only lime and strawberry sweets. There are 20 sweets in each bag.
In the first bag there is 1 lime sweet for every 3 strawberry
In the second bag there are 2 lime sweets for every 3 strawberry.
How many more lime sweets are there in the second bag?

32. Next session
Bring planning objectives to next weeks meeting of your maths objectives for week beginning ….
There will be time for planning a lesson in the following session around the maths topic you will be covering in class. Please have a think about what topic you will be covering.