Connect with Maths Early Years Learning in Mathematics is an online community to support the teaching and learning of mathetmatics Birth to 8 years old. This presentation by Louise Hodgson, a mathematics specialist addresses counting principles in early years learning.

1. Early mathematical thinking
Louise Hodgson 2013

2. How many cups would you
need to make a triangle
building with four levels?

3. Allow free exploration with the cups
for two weeks before posing the
problem

4. Instead of limiting instruction to counting
skills or writing numerals throughout the
early years, allow youngsters multiple ways
to represent quantity.

5. Overview
1. Sources and observations
2. Early mathematical ideas and processes
3. The role of the early childhood educator
1. Learning opportunities for numeracy -
play spaces

6. 2. Sources and observations

7. Belonging,
Being &
Becoming
Early Years
Learning
Framework
Mathematics
development
Recent
research
findings
Perspectives
from early
childhood
educators -
beginning and
experienced

8. Young children are not ready for mathematics
education.
Mathematics is for some bright kids with
mathematics genes.
Simple numbers and shapes are enough.
Language and literacy are more important than
mathematics.
Teachers should provide an enriched physical
environment, step back, and let the children play.
Common misconceptions

9. Mathematics should not be taught as stand-alone
subject matter.
Assessment in mathematics is irrelevant when it
comes to young children.
Children learn mathematics only by interacting
with concrete objects.
Computers are inappropriate for the teaching and
learning of mathematics.
http://www.earlychildhoodaustralia.org.au/australian_journal_of_early_childhood/ajec_index_abstracts/early_childhood_teachers
_misconceptions_about_mathematics_education_for_young_children_in_the_united_states.html
Common misconceptions

10. About ‘intentional teaching’
•Intentional teaching is one of the 8 key pedagogical
practices described in the Early Years Learning
Framework (EYLF).
•The EYLF defines intentional teaching as ‘educators
being deliberate purposeful and thoughtful in their decisions
and actions’.
Intentional teaching is thoughtful, informed and
deliberate.
Intentional teaching and the Early Years
Learning Framework

11. Intentional educators:
•create a learning environment that is rich in materials
and interactions
•create opportunities for inquiry
•model thinking and problem solving, and challenge
children's existing ideas about how things work.
Intentional teaching and the Early
Years Learning Framework

12. Intentional teaching and the Early Years
Learning Framework
Intentional educators:
•know the content—concepts, vocabulary, skills and
processes—and the teaching strategies that support
important early learning in mathematics
•carefully observe children so that they can thoughtfully
plan for children’s next-stage learning and emerging
abilities
•take advantage of spontaneous, unexpected teaching and
learning opportunities.

14. Numeracy or Mathematics?
“Numeracy is the
capacity,
confidence and
disposition to use
mathematics I
daily life”
EYLF, 2009 p.38

15. “Outcome 4: Children are confident
and involved learners.
Children develop dispositions for learning
such as curiosity, cooperation,
confidence, creativity, commitment,
enthusiasm, persistence, imagination and
reflexivity. Children develop a range of
skills and processes such as problem
solving, inquiry, experimentation,
hypothesising, researching and
investigating”.
(EYLF, 2009)

16. Disposition of children
Encourage young children to see
themselves as mathematicians by
stimulating their interest and ability in
problem solving and investigation
through relevant, challenging,
sustained and supported activities
(AAMT and ECA 2006)

17. Low mathematical skills in the earliest years
are associated with a slower growth rate –
children without adequate experiences in
mathematics start behind and lose ground
every year thereafter.
(Clements and Sarama, 2009, p. 263)
Interventions must start in pre K and
Kindergarten (Gersten et al 2005). Without
such interventions, children in special need
are often relegated to a path of failure
(Baroody, 1999)

18. 2. Early mathematical ideas

19. Outcome 5: Children are
effective communicators.
Spatial sense, structure and pattern,
number, measurement, data,
argumentation, connections
and exploring the world
mathematically are the
powerful mathematical ideas children
need to become numerate.
(EYLF, 2009 p38)
Research….
Perry, Dockett & Harley (2007) - powerful ideas and professional
development

20. Critical concepts
underpinning number
understanding.
Counting
Subitising
More – less
Part part whole

21. Children who understand number
relationships develop multiple
ways to represent them.

22. Interpreting quantity
Symbol
three
Number word
3

23. Principles of Counting
• Each object to be counted must be touched or ‘included’
exactly once as the numbers are said.
• The numbers must be said once and always in the
conventional order.
• The objects can be touched in any order and the starting
point and order in which the objects are counted doesn’t
affect how many there are.
• The arrangement of the objects doesn’t affect how many
there are.
• The last number said tells ‘how many’ in the whole collection,
it does not describe the last object touched.

24. Principles of Counting
• Which of the principles of counting
does Charlotte understand?

26. Principles of Counting
• Each object to be counted must be touched or ‘included’
exactly once as the numbers are said.
• The numbers must be said once and always in the
conventional order.
• The objects can be touched in any order and the starting
point and order in which the objects are counted doesn’t
affect how many there are.
• The arrangement of the objects doesn’t affect how many
there are.
• The last number said tells ‘how many’ in the whole collection,
it does not describe the last object touched.

27. Intentional opportunities for
counting
• Model counting experiences in meaningful
contexts, for example, counting girls, boys as
they arrive at school, counting out pencils at the
art table.
• Involving all children in acting out finger plays
and rhymes and reading literature, which models
the conventional counting order.
• Seize upon teachable moments as they arise
incidentally. “Do we have enough pairs of
scissors for everyone at this table?”

28. Seize teachable moments as
they occur

29. Ten frames

30. Pick up chips :
• Take a card from
the pile and pick up
a corresponding
number of counters.
• Play until all the
cards have been
taken.
• The winner is the
person with the
most chips at the
end of the game.

31. Estimating
1 10
Round about what
would this number be ?

32. Guess my number :
• The leader thinks of a secret
number. The children may
assist the teacher in drawing
a line on the white board to
indicate the range in which
the secret number lies. The
leader asks the group to try
and guess the secret
number. The group asks
questions of the leader to try
and ascertain the number.
The leader may only answer
yes or no to the questions.
(A process of elimination)

33. Sandwich boards
• Add string to numeral cards
so they can be hung around
the students necks. Provide
each student with a numeral
card. Students move
around the room to music.
Once the music stops, the
children arrange themselves
into a line in a correct
forward or backward
number sequence.
Ask students why
they lined up the
way they did.

34. Understanding is
encouraged through
sharing our
thinking.
Listening to each other
facilitates learning.
It makes us think.
Talk is vital in
building
understanding.

35. More-less relationships
More-less relationships are not
easy for young children.
Which group has more?
How many more?

36. More – less
relationships
• How many?
• What is two more?
• One less?

37. More-less relationships
Four-year-olds may
be able to judge
which of two
collections has more,
but determining how
many more (or less)
is challenging, even
when they count.

38. More-less relationships
• Young children must arrive at the
important insight that a quantity (the
less) must be contained inside the
other (the more) instead of viewing
both quantities as mutually exclusive.
The concept requires them to think of
the difference between the two
quantities as a third quantity, which is
the notion of parts-whole.

39. Stages in comparison
1. There are more blue than red and there are less
red than blue
2. There are seven more blue than red and seven
less red
3. Ten is seven more than three and three is seven
less than ten
© Catholic Education Office Tasmania 2012

40. Whilst students need many
counting experiences, teaching
should also emphasise equally
decomposing or partitioning
collections into parts.

41. Subitising
(suddenly recognising)
• Seeing how many at a
glance is called
subitising.
• Attaching the number
names to amounts that
can be seen.

42. Subitising
(suddenly recognising)
• Promotes the part part
whole relationship.
• Plays a critical role in the
acquisition of the concept
of cardinality.
• Children need both
subitising and counting to
see that both methods give
the same result.

44. 10 bead string
• They enable children to subitise up to five and learn
the number combinations which make ten.

45. Peek and say :
• Have a different
number of containers
with different numbers
of objects under each.
Ask the children to find
the container with
2,5,3… objects.
• Take a number ticket
and try to find the
container hiding the
matching number of
objects.

46. Speedy dominoes:
• Share the domino
pieces. Play the game
in the same way as
regular dominoes,
except in this game
there is no turn taking.
• As soon as players see
the opportunity to place
a domino in the game,
they may do so. The
winner is the first player
to correctly place all the
dominoes.

47. Part whole
relationships
Partitioning numbers into part-part-
whole forms the basis for children
coming to understand the meaning
of addition and subtraction.

48. Parts – whole relationships
• The parts – whole relationship refers to
the notion that you can break up
(partition) a quantity and move bits
from one group to another without
changing the overall quantity. (e.g. 5
can be thought of as 3 and 2 or 1 and
4 etc)

49. A ten frame is effective in teaching
parts /whole relationships, as in
this example of combinations that
total six.

50. For a true understanding of number
relationships,
Teachers must encourage young
children to work with quantity in a
variety of situations using different
math manipulatives over an
extended period of time.

51. Mathematical concepts
do not inherently lie in
manipulatives. Children
must construct the
understanding.

52. 3. The role of the Early
childhood educator

53. Role of the educator
Planning and resourcing challenging
learning environments.
Supporting children’s learning through
planned play activity.
Extending and supporting children’s
spontaneous play.
Extending and developing children’s
language and communication through
play.

54. Role of the educator
Model mathematical language.
Ask challenging questions.
Build on children’s interests and natural
curiosity.
Provide meaningful experiences.
Scaffold opportunities for learning &
model strategies.
Monitor children’s progress and plan for
learning.

55. Assessment methods
Collect data by observation and or/listening to
children, taking notes as appropriate
Use a variety of assessment methods
Modify planning as a result of assessment

56. Effective teachers are inclusive of
all learners

57. 4. Learning opportunities for
numeracy - play spaces

58. Play spaces
Outdoors (climbing, tunnels,
tents, riding, construction, sand & water,
gardening, dance and gymnastics)
Puzzles (spatial puzzles, number games,
sorting)
ICT (computer games, creative graphics
software, programmable toys, digital
cameras, calculators, interactive whiteboard

60. Bobby Bear NCTM Illuminations

61. Calculator counting
Calculator counting contributes to
a better grasp of large numbers,
thereby helping to develop
students number sense.
“It is a machine
to engage children
in thinking about
mathematics”
(Swan and Sparrow 2005)

62. Cultivate an interest in
number
“Is googolplex a number?
Can you make the calculator count
until it gets to googolplex?
What other big numbers are there?”
Harry aged 5

63. Play spaces
Role play (home, shop, dress-
up, puppets)
Construction (blocks, tracks,
linking materials)
Display area (peg line,
pinboards, magnet board)
Play trays (sand, water, multiple
objects e.g. buttons, pasta, shells,
leaves)
Mini-worlds (story/drama,
cloth or sand tray environments,
small toy animals, people,
vehicles

64. Play spaces
Modelling & painting
Graphics (drawing,
writing, recording,
shapes)
Reading and listening
areas (story-telling,
picture books, rhymes,
songs, CDs, music &
percussion

65. Charlotte aged 3

66. • (2010)
• SAGE Books UK
• Distributed in
Australia by
• Footprint Books

67. References
AAMT & ECA. (2006). Position paper on Early Childhood Mathematics.
www.aamt.edu.au
www.earlychildhoodaustralia.org.au
DEEWR. (2009). Belonging, Being & Becoming: The Early Years Learning Framework
for Australia.
http://www.deewr.gov.au/earlychildhood/policy_agenda/quality/pages/earlyyearslearningf
ramework.aspx
Papic, M. & Mulligan, J. (2007). The Growth of Early Mathematical Patterning: An
Intervention Study. In J.
Perry, B, Dockett, S, Harley, E. (2007) Preschool Educators’ Sustained Professional
Development in Young Children’s Mathematics Learning. Mathematics Teacher
Education and Development
Special Issue 2007, Vol. 8, 117–134. Available at:
http://www.merga.net.au/documents/MTED_8_Perry.pdf
Tucker, K. (2010) (2nd. Ed.). Mathematics through play in the early years. London:
Sage.
Hunting, R. et al. Mathematical Thinking of Preschool Children in Rural and Regional
Australia: Research and
Practice. Report & video clips at: http://www.latrobe.edu.au/earlymaths/resources.html