1. COUNTING, EARLY NUMBER AND PLACE VALUE
Day 2

2. DAY 2 We will consider: the role of counting in classrooms;
What is involved in ‘number sense’?
Consider the importance of ‘Place Value’ in the teaching of Number;
Review a range of models and images that can be used to develop Number understanding;
Associated issues for teaching
Review a range of ‘starter’ activities and how to use them to engage learners.
Consider simple resources and the role of practical work in the classroom. (CPA)
What is Mastery in mathematics? How can we identify mastery?
The use of questions & elicitation tasks – how to find out what learners already know and their next steps in learning.

3. Mental and Oral Starters
What are they for?
How can they be used?
What have you seen?
What makes a ‘good’ MOS?
What do you need to consider if planning a mental/ oral starter?

4. Kolb - cycle of continuous development

5. GROUP READING: EYFS readings: Pedagogy and Practice
What does the curriculum look like in EYFS? What is involved?
What does practice look like in an early years classroom? What is the same/ different to primary practice?
GROUP READING:
Ian Thompson articles
‘How much does it count?’
‘Have we got it right?’
‘An alternative counting model’
‘1-100 rules OK?’
‘Issues in Teaching Numeracy in Primary Schools’ [Second Edition]
Chapter 8 Shaking the Foundations EYFS
Chapter 5 Making Connections using resources
Deboys and Pitt ‘Lines of Development in Primary Mathematics’ Section 1 Number N1 to N8.
EYFS & Primary curriculum:
Pedagogy and Curriculum knowledge
What is involved in early number?
What are the key messages?
What is ‘number sense’? Examples?

6. Talk through 4 themes, and 4 overarching principles – flag up colours same throughout card
Next slide shows commitments on reverse of card 6

7. These are the commitment statements.
There is a Principle Into Practice card for each commitment 7

9. There are six broad developmental stages
Mathematics
There are six broad developmental stages
All children develop in different ways & at different rates.
Birth to 11 months
8 to 20 months
16 to 26 months
Notes:
Each area of learning and development is organised in broad phases of development.
The age ranges overlap to create broad developmental phases. This emphasises that each child’s progress is individual to them and that different children develop at different rates. A child does not suddenly move from one phase to another, and they do not make progress in all areas at the same time. However, there are some important ‘steps’ for each child to take along their own developmental pathway. Effective practice means being aware of these and supporting the child in achieving them.
Note 60+ months – not all children will have met the early learning goals by the time they are five or by the end of reception class. Key Stage 1 practitioners will therefore need to be aware of how to use the EYFS and the EYFS Profile data to plan appropriate learning experiences for those children.
The faces are always the same – they represent the ages and stages of development throughout the EYFS They replace previous icons of BTTM and stepping stone colours in CGFS.
22 to 36 months
30 to 50 months
40 to 60 + months 9

10. Number

11. The numbers in alphabetland are
a b c d e f g…

12. The Skills of Counting One to one Stable Order Cardinality Abstraction
There are 5 skills you need to count successfully:
One to one
Stable Order
Cardinality
Abstraction
Order irrelevance

13. Schaffer et al (1974) Stages: ‘Recitation’
(saying the names in conventional order)
‘Enumeration’
(using those numbers to count objects in a set)
‘Cardinality’
(knowing the last number counted tells us how many are in the set.)

14. What does progression look like in ‘counting’?

15. Numerals are arbitrary symbols representing number ideas
These are abstract arbitrary symbols we use to represent number ideas, there is nothing in these to tell us what they stand for -- so which one is the biggest? It is understanding the number idea that enables us to answer – not looking at the numerals.
Children are given abstract ideas and arbitrary symbols to think with when they are very young, which many of them find too difficult
If I say 9 what do you think of? What pictures do you have of 9? 15

16. Giving children pictures of numbers
Create a number track with digit cards from 1 to 9
Draw a ‘picture’ for each number
What if the number track
gets messed up?
(oh no!)

17. Structured number representations
(Numicon structured imagery motivates numerals, providing a picture of the number ideas and the relationships between the number ideas.)
Explain that participants will now engage in some practical experiences in order to give them the opportunity to familiarise themselves with the Numicon structured apparatus. Young children would not be expected to do these activities straight away.
Practical A – learning the patterns of the shapes.
I’m going to ask you to
Put one set of shapes in order on the table.
What patterns do you see? (e.g. odd even) To see this it helps to put the 2 shape horizontally.
Put another set of shapes in the bag. Point to shape on table. Can you feel for this one in the bag?
Lots of the Numicon activities involve feely bags. This is because eventually we want children to internalise the image. We want them to have this picture of the number in their heads and begin to manipulate the numbers mentally. Feely bag activities help children make the step between sing the materials physically and beginning to visualise them and use the image mentally. (could mention here a conversation with a group of numbers count teachers saying that some children could calculate with the apparatus but there hadn’t yet been a noticeable improvement in mental maths – these teachers had not focussed particularly on feely bag work or on encouraging children to visualise and talk about the shapes)
With children Could swap 2 shapes and ask them to say which etc.
What do you notice?

18. Nrich

19. Effie Maclellan [quoted by Thompson 2008]
“once children have the knowledge which underpins counting, not only can they make precise quantitative judgements, they are also in the powerful position of being able to penetrate the tasks of addition and subtraction”.

20. Explore Numbers and Patterns (page numbers are … of 190)
p29 Observing: Aide memoire
p32-33 Numbers and pattern phases
p49-73 Enabling Environments
Role of the Adult p79-162
Mark making extracts p180

21. English 2013
“Visual stimulants are powerful tools to enhance concepts that would otherwise be abstract”.

22. Knowledge Quartet - Transformation
How do we use various resources to represent mathematical ideas in order to make them accessible to learners?
How do teachers choose and use examples in teaching?
“…the use of multiple representations in general is an important part of teachers’ knowledge of mathematics and they can play an important role in the explanation of mathematical ideas.”
Leinhardt et al 1991

23. To the benefit of EAL learners
Jerome Bruner (1974)
Representations as important mediators in developing abstract understandings. ‘Scaffolds’ to support learning.
Hierarchical but complementary modes of representation:
Enactive – physical action
Iconic – make use of ‘pictorial’ images
Symbolic – allow mental manipulation
When using resources you will be using all 3 modes of representation - separately and in combination
To the benefit of EAL learners

25. Count out 12 sweets, remove 4, count what’s left.
Gemma had a bag of 12 sweets and she gave four to her brother. How many sweets did Gemma have left?
Abstract
Real
Real example?
Count out 12 sweets, remove 4, count what’s left.
Represent sweets with cubes/ counters
Drawing sweets randomly
Drawing circles or tallies
Organising drawing into rows
12 on a number track and count back 4
Mark 12 on an empty number line and jumping back 4
Writing 12-4=
Before using symbols, pupils need lots of experience of all these model types
Abstract example?

26. What are the ‘Big Ideas’?
Define…
‘Place Value’
(with examples)
What are the ‘Big Ideas’?
Three important features of our number system:
The positional principle (place value)
The symbol zero
Grouping to the base 10.

27. National Curriculum 2014
What are the end of year expectations for Number & Place Value for your year group? ( [EYFS] / Y1 to Y6) Are there other strands of mathematics that require an understanding of Place Value? (Implicit or explicit)

28. Play time…
What models and images would you use for Place Value to support learners in your classroom?
[Numicon/ straws/ Dienes/ exchange boards / coins/ PV cards/ number lines/ Number square, PV chart]
What have you seen being used?
(Demonstrate for your table)

29. Build the Number… H T U Exchange Boards Ian Thompson (2003)
Quantity Value
Column Value H T U
Place Value moves away from just developing understanding through the physical manipulation of materials and towards a concentration on the relationship between the spoken and written forms of numbers.
Use of Base 10 equipment v Counters/ digits?

30. ‘Typical’ PV resources
Arrow cards
Hundred squares
Number lines
Base 10 – Dienes, coins, PV disks
PV chart [Gattegno chart]
Place Value moves away from just developing understanding through the physical manipulation of materials and towards a concentration on the relationship between the spoken and written forms of numbers.
Others…?

31. TASK 2 - Hand in day 3
Consider a model or image relating to ‘number sense’ and/or ‘place value’.
How does your mentor use models and images in their teaching?
What examples can the learners give you that demonstrate a secure understanding of the image or model?
What connections do they make?
How confident are they in using it?
Are there any misconceptions/ barriers to learning?

32. NCETM – ‘Teaching for Mastery’
The materials are offered to indicate valuable learning activities to be used as an integral part of teaching…
Have you seen them?
Have you used them? How?
Impact?

33. ‘Mastery’
Mastery is something that we want pupils to acquire. All pupils.
So a ‘mastery curriculum’, or ‘mastery approaches’ to teaching maths, or ‘mastery teaching’ in maths lessons all have the same aim—to help pupils, over time, acquire mastery of the subject.
Significantly, some of the implications of adopting mastery approaches to teaching maths are new.
One of these is the move away from labelling pupils as ‘high ability’ or ‘low ability’ and giving them different tasks.

34. Mastery Checklist (p.7)

36. Place Value – The ‘Big Ideas’
The position of a digit in a number determines its value.
PV is based on ‘unitising’ – treating a group of things as one ‘unit’. So 3 ‘units’ + 2 ‘units’ = 5 ‘units’ [where the units are the same size]
The language used to name numbers does not always expose the place value e.g. twelve.
Secure understanding in the value of each digit.
Imagining numbers on a line helps us to order them.
Rounding numbers in context may mean rounding up or down.
Large numbers of six digits are named in a pattern of three: hundreds of thousands, tens of thousands, ones of thousands, mirroring hundreds, tens and ones.
For whole numbers, the more digits a number has the larger it must be but this is not true of decimal numbers. [ 0.5 > 0.35 ]
Ordering decimal numbers uses the same process as ordering whole numbers: starting from the place with the highest value i.e. from the left.
[ < because of the ‘tenths’ value]
PV is ‘additive’ 456 =
PV is ‘multiplicative’ one hundred is ten times as large as ten ….

37. Rationale
Understanding , or mastery if you like, is never complete – there is always a growing edge to understanding.
‘Understanding is more than just acquiring concepts or skills… it is a complex and multi-layered mental activity that is merged and deepened in a continuous and generative process’.
Carpenter and Lehrer(1999)

38. Reading for Day 3 GROUP READING:
‘Issues in Teaching Numeracy in Primary Schools’ [Second Edition]
Chapter 13 - The empty number line
Chapters 12/14 mental and written calculation;
Ian Thompson
‘Mental Calculation’
‘Deconstructing the PNS approach to addition Part 1’
‘…subtraction Part 2’.