1. Learning and Teaching the
Curriculum
Counting
Concepts of Addition and Subtraction

2. Quizz
• What is the biggest number?
• How many numbers are there between 1 and
20?
• Give an example in which the operation of
subtraction leads to a larger number

3. Aims of the session
• To consider how we use numbers in our everyday
lives
• To understand the processes involved in counting
• To consider some basic models of addition and
subtraction
• To become familiar with curriculum documents

4. Activity
• Close your eyes
• Bring to the forefront of your
consciousness an image of the number
three

5. Discussion

6. Key vocabulary
Aspects of number
• Nominal
• Cardinal
• Ordinal

7. The nominal aspect
‘The 3 on a number 3 bus is indeed just a
label, a number being used in what is
called the nominal aspect.’
Haylock and Cockburn, 2008, p.33

8. Sukoku
http://www.jigsawdoku.com/

9. The ordinal aspect
‘...the image of a number line is one that
embodies most strongly the ordinal
aspect of number.’
Haylock and Cockburn, 2008, p.34

10. The cardinal aspect
Cardinal numbers serve ‘as indications of
how many there are in a set of things.’
Haylock and Cockburn, 2008, p.34

11. Connecting cardinal and ordinal
aspects
• The last number you get to when you are
counting a set is the number in the set
– e.g. Seven is one more than six, because it is
the next number after six.
• Teachers to make explicit that the
previous number is alwasy one less

12. Pre-counting experiences
• Sorting objects into sets and
categorisation
– Play sorting games
• Using language such as ‘one more’ and
‘another one’
• Distinguish between sets of different sizes
 understand that sets of different sizes
have different labels

13. Play Alphabetland
• Number names are A, B,
C, D…
• Do not‘translate’ these
number names into the
number names one,
two, three,…

14. Play Alphabetland in pairs
• Can you count backwards from J?
• Answer the following questions:
– C+D
– B+E
– K–B
– G–D
– E+E
– E+F
• Articulate the strategies you use

15. My world in numbers

16. Activity
• Count the number of people in the room
• Make notes about the process
• Consider whether there was
– Recitation
– Coordination (head nodding, pointing, 1-1)
– Keeping track (Where did I start? Who have I
counted/not counted?)

17. Counting exercises
• Count forwards
• Count backwards
• Count from x to y
• What number comes before x?
• What number comes after x?
• Put number mats in order

18. Beautiful numbers

19. International perspectives

20. How many?

21. How many?

22. Resources to support counting
• Cuisenaire
• Numicon
• Bead strings
• Dienes

23. Addition Strategies
• Counting all
• Counting on from the first number
• Counting on from the larger number
• Using a known fact
– (E + E = J)
• Deriving a new fact from a known fact
– (e.g. E + F, using the answer to E + E)

24. Subtraction Strategies
• Counting out
– (e.g. G – D: put up G fingers, fold down D fingers
and count out what’s left).
• Counting back the second number
– (e.g. K – B, saying J, I).
• Counting from one number to the other
– (e.g. counting on from D to G keeping track of
how many have been counted on)

25. What does learning to count entail? Rochel Gelman’s Counting Principles
• Stable order
You need to know the counting words and be able to recite them in the
correct order each time – it is impossible to count up to seven if you know
only the first six counting words.
• One to one
One, and only one, number word has to be matched to each and every
object; lack of co-ordination is a source of potential error.
• Cardinality
When correctly following the first two principles, the number name
allocated to the last object tells you how many objects you have counted.
• Abstraction
You can count anything – visible objects, objects of different shapes and
sizes, things that are too far away to touch, objects that cannot be moved,
moving objects, hidden objects, imaginary objects, sounds, etc.
• Order irrelevance
Objects may be counted in any order provided no other counting principle is
violated.
http://www.teachers.net.qa/Math_CfBT_Workshops/workshop2/Ma2_Session8a.pdf

26. Early Years Foundation Stage

27. Early Years Foundation Stage
Mathematics involves providing children with
opportunities to
•Develop and improve their skills in counting
•Understand and use numbers
•Calculate simple addition and subtraction
problems
•Describe shapes, spaces, and measures

28. EYFS and Counting
Children are able to
•Count reliably with numbers from 1 to 20
•Place numbers in order
•Say which number is one more or one less than
a given number

29. 09/10/12
The Structure and Content of the National
Curriculum for Mathematics
The Structure
• Programmes of study set out what pupils
should be taught at KS1 and KS2.
• Attainment targets- the programmes of
study are sub-divided into attainment targets
• Knowledge, skills and understanding in
the programme of study identify the main
aspects of mathematics to be taught at each
key stage;

30. 09/10/12
The NC Attainment Targets
The knowledge, skills and understanding that pupils of different
abilities and maturities are expected to have by the end of each
key stage.
There are four attainment targets in mathematics
1. Ma 1: Using and applying mathematics
2. Ma 2: Number and algebra
3. Ma 3: Shape space and measures
4. Ma 4: Handling data
Level descriptions – the attainment targets consist of eight level
descriptions of increasing difficulties, ranging from level 1 to
level 5 for KS1 / KS2. The level descriptions provide the basis
for making judgements about pupils’ performance at the end of
a key stage.

31. 09/10/12
The KS1 Programme of Study
By the end of Key Stage 1, pupils should
have developed knowledge skills and
understanding of the following aspects of
mathematics:
Ma 1: Using and applying mathematics
Ma 2: Number
Ma 3: Shape, space and measures

32. 09/10/12
PNS- Primary Framework for Literacy and
Mathematics
Mathematics
There are seven strands:
 Using and applying mathematics
 Counting and understanding number
 Knowing and using number facts
 Calculating
 Understanding shape
 Measuring
 Handling data
Objectives are aligned to the seven strands and these are
subdivided into core learning by year group and core learning by
strand

33. ELPS
Developing Rich Learning Experiences
Experience – give children concrete experience
Language – set up opportunities for children to
talk
Pictures – use visual imagery to develop maths
concepts
Symbols – use symbolic representation
(Pamela Liebeck)

34. Key ideas about addition and subtraction
Developing secure mental arithmetic skills is critical
Subtraction is the inverse of addition – they should be
worked on at the same time as often as possible
Addition is commutative and associative – subtraction
isn’t
There are different ways to calculate. Talk about these,
encourage them. Different strategies suit different sums
Use what you know to work out what you don’t know
Additions to 100 and related subtractions should be done
mentally/informally, not using a formal written method
It’s OK to ‘write down’ mental arithmetic!
Writing down sums horizontally invites
you to try them mentally

35. Activity
• Write a number sentence to
match these objects

36. Suggest a number sentence
1 2 3 4 5 6 7 8 9 10
Start the number track
with 1, not 0

37. Using a number line
0 1 2 3 4 5 6 7 8 9 10
What number sentences can we make?

38. Using a blank number line
+7
12
9 17
What number sentences can we make?

39. Why are these visual models important?
One reason is that many young children get
stuck with questions like:

40. Using visual models
12 + = 19
•In pairs role play the teacher and child
•The teacher must help the child use an
appropriate visual model in order to find the
answer
•Set the child a more challenging problem to
solve

41. Discussion
What is your view about
What is your view about
using ICT to support the
using ICT to support the
learning and teaching of
learning and teaching of
Mathematics in the Early
Mathematics in the Early
Years?
Years?
Refer to
Refer to
examples from
examples from
placement to
placement to
support your
support your
view.
view.
?

42. BREO task 1
Prepare a list of resources for your Early Years
Mathematics box

43. Success Criteria
• I can explain to my friend
– What ELPS stands for and why how it can be used
to support young children to develop their
counting skills
– What is involved in counting
• I can suggest some models and images that
might help a child who is stuck with a question
like 32 -  = 19

44. 09/10/12
Task for Monday 15 October th
• Identify an online resource to share with the
class, e.g. a game, a puzzle, a show, that
supports children’s understanding of simple
addition
• Be prepared to show it to the class
• Be prepared to justify your choice and
suggest some disadvantages