Numeracy Interview with Year 2 student using think boards top identify student strengths and weaknesses and recommend activities.

1.
SEMESTER
1
2011
EDUC8502
TEACHING
MATHEMATICS
IN
EARLY
YEARS
Assignment
2
What
do
children
know
about
numbers?
Due:
Friday
May
13th
2011
Sharon
McCleary
19113469
Unit
Co-­ordinator:
Associate
Professor
Christine
Howitt
Tutor:
Ms.
Clair
Kipling

2. What
do
children
know
about
numbers?
Background
The
purpose
of
this
report
is
to
present
findings
and
recommendations
arising
from
an
interview
with
Gabi,
a
Year
2
female
student,
conducted
on
1st
April
2011
at
Hollywood
Primary
School
in
Perth,
Western
Australia.
The
duration
of
the
interview
was
approximately
40
minutes.
It
focussed
on
determining
what
the
child
knew
about
numbers,
using
Think
Boards
as
a
strategy
to
promote
communication
about
her
number
knowledge
and
connections
across
various
modes
of
representation.
The
Think
Board
is
a
recording
format
that
allows
the
student
to
express
their
understanding
of
a
concept
in
various
ways
(i.e.
using
stories,
symbols,
pictures
and
real-­‐life
representations).
It
gives
valuable
insight
into
the
connections
the
student
has
formed
between
enactive
(concrete
objects),
iconic
(pictures,
diagrams)
and
symbolic
(words,
symbols)
representations
of
mathematical
concepts
(Frid,
2004),
shedding
light
on
the
individual
student’s
process
of
mathematical
meaning-­‐making
and
providing
a
useful
method
of
identifying
future
learning
areas
within
the
child’s
zone
of
proximal
development
(Krause,
2010).
Introduction
The
interview
commenced
with
a
relaxed
discussion
about
numbers,
aimed
at
making
the
student
feel
comfortable,
developing
rapport,
and
determining
her
general
disposition
towards
numbers.
The
book
“10”
by
Vladimir
Radunsky
was
read,
and
the
child
was
introduced
to
the
pre-­‐made
Think
Boards.
She
was
asked
to
choose
3
numbers
within
set
ranges
to
represent
in
different
ways
on
the
Think
Boards,
and
guided
through
each
of
the
sections.
She
displayed
genuine
excitement
about
numbers
and
was
eager
to
participate,
initially
asking
if
she
could
use
larger
numbers
outside
the
given
ranges.

3. Teaching
Mathematics
in
Early
Years
EDUC8502
Gabi
selected
‘15’
(recommended
range
of
11-­‐19)
as
her
first
number,
and
80
as
her
second
(recommended
range
50-­‐100).
After
completing
the
second
Think
Board
she
realised
she
would
have
difficulty
representing
larger
numbers
and
chose
‘2’
as
her
third
number.
Attempts
to
persuade
her
to
choose
a
higher
number
resulted
in
her
choice
of
‘20’
as
her
third
number.
She
was
clearly
outside
her
comfort
zone
when
larger
numbers
were
suggested
and
it
was
not
appropriate
to
challenge
her
further
that
particular
day.
The
resources
provided
for
the
child
to
use
included
various
sets
of
counters,
environmentally
available
natural
materials
such
as
leaves
and
stones,
coloured
pencils,
stickers,
stamps,
and
lead
pencils.
The
three
Think
Boards
are
included
for
reference
in
Appendix
A,
B
and
C
respectively,
along
with
photographs
of
the
‘real’
items
used
on
each
Think
Board.
Student
Profile:
Analysis
Think
Board
One:
‘15’
Gabi
was
able
to
represent
the
number
‘15’
correctly
in
symbolic
form,
as
can
be
seen
in
the
“symbol”
section
of
Think
Board
One
in
Appendix
A,
where
she
wrote
‘15’
using
the
correct
pencil
grip
and
number-­‐writing
formation.
When
asked
to
represent
‘15’
using
real
objects,
she
hesitated,
asking
“15
of
anything?”
This
demonstrated
an
understanding
of
‘number’
as
an
idea
that
describes
things
in
a
group,
independent
of
what
is
being
counted
or
labelled
(Demant,
2008).
She
proceeded
to
collect
and
count
15
leaves
from
the
surrounding
gardens,
initially
counting
in
1’s,
then
collecting
groups
of
two
and
skip
counting
(“7,9,11”),
before
collecting
a
group
of
four
and
counting-­‐on
to
arrive
at
15.
She
then
verified
there
were
15
leaves
using
rational
counting
(Cathcart,
2011):
making
a
one-­‐to-­‐one
correspondence
between
each
leaf
and
the
sequential
number
name
as
she
placed
it
on
the
Think
Board.
This
revealed
a
solid
understanding
of
the
principles
of
counting
identified
by
Gelman
and
Gallistel
(1978),
namely
the
Stable
Order
Principle,
One-­‐to-­‐one
correspondence,
Cardinal
principal,
abstraction
principle
and
the
order-­‐irrelevance
principle
(Compton,
2007).
Sharon
McCleary
3

4. Teaching
Mathematics
in
Early
Years
EDUC8502
Gabi
initially
had
difficulty
representing
’15’
pictorially,
and
was
unable
to
respond
to
prompts
requesting
her
to
think
about
instances
of
this
number
in
everyday
life.
However,
after
further
explanation
(i.e.
‘3’
could
be
represented
by
three
little
pigs
or
a
triangle),
she
produced
an
example
relating
to
the
‘real’
section
on
her
Think
Board,
drawing
three
flowers
with
five
petals
each
(See
Picture
Section
of
Think
Board
One).
She
counted
each
petal
individually,
then
stated
“5+5+5
equals
15”.
This
shows
she
successfully
decomposes
and
recomposes
numbers,
and
has
an
internal
concept
of
multiplication
as
repeated
addition
of
equivalent
groups,
consistent
with
the
second
level
of
conceptual
development
for
multiplication
representations
given
by
Thomas
(Thomas,
1997).
The
Story
section
of
Think
Board
One
indicates
she
has
sound
knowledge
of
the
standard
classroom
number
practise
of
creating
and
representing
word
problems
using
conventional
symbols
(i.e.
14+1=15).
It
also
shows
she
was
building
meaningful
connections
for
the
context
of
this
particular
Think
Board
as
she
engaged
with
the
activity,
as
her
symbolic
(story)
and
enactive
(real)
representations
related
to
the
same
theme
(i.e.
garden).
During
this
part
of
the
interview,
Gabi
demonstrated
good
early
number
sense,
a
solid
understanding
of
counting
and
the
beginnings
of
calculation.
Think
Board
Two
‘80’:
Think
Board
Two
(Appendix
B)
shows
‘80’
represented
in
a
non-­‐standard
form
in
the
Picture
section:
seven
longs,
nine
units
and
one
separated
unit;
the
place
value
chart
was
suggested.
Gabi
was
unable
to
create
an
equivalent
representation
of
‘80’
when
requested.
She
did
not
recognise
‘9+1’
could
be
traded
for
a
‘10’
and
represented
by
an
additional
long,
displaying
confusion
even
when
this
was
explicitly
demonstrated
and
stated.
This
indicates
she
has
not
fully
abstracted
the
concept
of
a
unit
of
ten
(Gray,
1999);
she
is
operating
within
the
extended
stage
of
structural
development
described
by
Thomas
(2002),
using
the
sub-­‐system
of
units
to
form
her
understanding
of
the
base-­‐10
system.
This
developing
understanding
of
the
base-­‐10
system
is
also
apparent
in
the
Real
section
of
Think
Board
Two
where
she
has
used
seven
bananas
to
represent
seven
tens
and
ten
random
fruit
counters
for
the
remaining
ten.
Sharon
McCleary
4

5. Teaching
Mathematics
in
Early
Years
EDUC8502
Both
sections
indicate
she
can
partition
the
decade
and
represent
the
number
accurately,
but
reveal
a
limited
understanding
of
grouping
and
place
value
concepts.
They
also
indicate
strain
on
her
working
memory
since
she
finds
it
difficult
to
consider
the
discrete
parts
and
the
whole
number
simultaneously
in
part-­‐part-­‐whole
relationships
(Gray,
2000).
This
may
result
from
repeated
classroom
experiences
of
partitioning
ten,
and
shows
she
has
not
conceptualised
groups
of
ten
as
a
unit,
or
visualised
the
pattern
of
tens
making
up
100.
Gabi
did
not
provide
authentic
real-­‐world
connections
in
the
Story
section
of
Think
Board
Two,
indicating
her
limited
awareness
of
real-­‐world
contexts
for
this
number.
Think
Board
Three
‘20’:
Gabi
initially
chose
‘2’
for
this
Think
Board,
stating
“I’ll
pick
an
easier
number,
‘2’.
It’s
my
Birthday!”.
This
indicates
her
awareness
that
the
previous
representations
had
been
difficult,
and
shows
she
is
capable
of
building
authentic
real-­‐world
connections
for
numbers
with
which
she
is
familiar
and
comfortable.
She
proceeded
to
use
the
birthday
connection
with
the
number
‘20’.
Examination
of
the
Picture
section
of
Think
Board
Three
shows
that
she
drew
twenty
cupcakes
to
represent
the
number,
linking
this
drawing
to
her
Story
section
by
showing
the
17
cupcakes
separated
from
the
“3
new
cupcakes”.
The
Real
section
of
this
Think
Board
reinforces
this
link
by
representing
the
‘17’
using
bananas
and
differentiating
the
‘3’
using
bunches
of
grapes.
This
shows
her
understanding
of
part-­‐part-­‐whole
relationships,
however,
as
can
be
seen
in
the
Story
section
of
the
Think
Board,
she
represents
her
number
sentence
incorrectly
as
“19+1=20”,
again
indicating
some
confusion
with
part-­‐part-­‐whole
relationships.
When
asked
to
write
the
number
sentence
corresponding
to
her
story,
she
produced
the
“17+3=20”,
as
shown
on
Think
Board
Three.
Again,
she
did
not
use
a
place
value
chart.
This
demonstrates
it
is
not
a
natural
part
of
her
expressive
repertoire;
she
thinks
of
multidigit
numbers
in
terms
of
units
and
is
operating
within
the
first
layer
of
the
number
system
(Geist,
2009).
(842
words)
Sharon
McCleary
5

6. Teaching
Mathematics
in
Early
Years
EDUC8502
Recommendations:
The
main
areas
Gabi
requires
support
in
are:
• Developing
the
underlying
conceptualisations
involved
in
grouping
in
tens
and
place
value
operations.
• Consolidating
her
number
sense
for
multidigit
numbers,
initially
up
to
100.
These
areas
have
been
identified
using
observations
from
the
interview
and
prioritised
using
the
WA
Curriculum
Framework
(WA
Curriculum
Council,
2005),
First
Steps
Documents
(Willis,
2004)
and
The
Australian
Curriculum,
Mathematics
(Australian
Curriculum,
Assessment
and
Reporting
Authority
[ACARA],
2010).
They
represent
the
foundation
for
developing
understanding
of
our
numeration
system
and
higher-­‐level
concepts
of
number,
including
estimation
and
computation
(Cathcart,
2011).
WA
Curriculum
Framework:
(Curriculum
Council,
2005)
Gabi
has
predominantly
achieved
Level
2
of
the
WA
Mathematics
Curriculum
Framework
(Curriculum
Council,
2005):
“Understand
Numbers
(N6.a.2):
Reads,
writes,
says
and
counts
with
whole
numbers
beyond
100,
using
them
to
compare
collection
sizes
and
describe
order.”
Understand
Operations
(N7.2):
Understands
the
meaning
and
connections
between
counting,
number
partitions,
addition
and
subtraction;
uses
this
understanding
to
represent
situations
involving
all
four
basic
operations.
Calculate
(N8.2):
Counts,
partitions
and
regroups
in
order
to
add
and
subtract
one-­‐and
two-­‐digit
numbers,
drawing
mostly
on
mental
strategies
for
one-­‐digit
numbers
and
a
calculator
if
numbers
are
beyond
the
student’s
present
scope.”
(WA
Curriculum
Framework
Progress
Maps
Mathematics
Outcomes
Overview:
Number,
2009)
In
her
Think
Board
representations
(See
Think
Boards
Two
and
Three),
Gabi
partitioned
the
last
decade,
demonstrating
she
thinks
of
numbers
as
part-­‐part-­‐whole
relations.
There
was
little
evidence
of
her
understanding
the
regrouping:
she
seemed
to
create
the
seven
tens
from
procedural
knowledge
as
she
was
unable
to
explain
the
Sharon
McCleary
6

7. Teaching
Mathematics
in
Early
Years
EDUC8502
base-­‐10
grouping
concepts
behind
the
procedure,
demonstrating
a
lack
of
relational
understanding
(Cathcart,
2011).
She
also
revealed
limited
number
sense
for
larger
numbers,
‘20’
and
‘80’,
relying
on
counting
in
units
(rather
than
grouping
tens)
to
represent
these
numbers,
indicating
she
has
not
fully
internalised
the
concept
of
grouping
in
tens
to
facilitate
more
efficient
counting.
Therefore
activities
emphasising
counting,
grouping,
place
value
and
number
patterns
up
to
100
should
be
introduced.
The
Australian
Curriculum:
(ACARA,
2010)
The
Australian
Curriculum
Year
Two
elaboration
requires
students
to
“Recognise,
model,
represent
and
order
numbers
to
at
least
1000”,
and
“Group,
partition
and
rearrange
collections
up
to
1000
in
hundreds,
tens
and
ones
to
facilitate
more
efficient
counting.”
(Australian
Curriculum,
Assessment
and
Reporting
Authority
[ACARA],
The
Australian
Curriculum,
Mathematics,
2010).
It
would
be
difficult
for
Gabi
to
build
number
sense
for
numbers
up
to
1000
as
required
by
The
Australian
Curriculum
since
she
has
not
yet
consolidated
grouping,
place
value
and
number
patterns
for
numbers
under
100.
Grouping
by
tens
is
fundamental
to
the
place
value
system,
and
a
thorough
understanding
of
place
value
is
necessary
for
the
development
of
higher-­‐order
number
sense
and
operations
(Reys,
1989).
Therefore,
Gabi
would
benefit
from
consolidation
of
the
Year
1
outcome
“Count
collections
to
100
by
partitioning
numbers
using
place
value.”
(ACARA,
The
Australian
Curriculum
Mathematics,
2010).
This
is
the
earliest
curriculum
outcome
which
she
is
not
confidently
able
to
demonstrate,
and
it
has
therefore
been
prioritised
in
order
to
minimise
misconceptions
and
build
a
solid
foundation
for
future
work.
First
Steps
in
Mathematics
Documents:
(Willis,
2004)
During
the
interview,
Gabi
displayed
several
characteristics
typical
of
the
First
Steps
Quantifying
Phase
(Willis,
2004),
automatically
selecting
counting
as
a
strategy,
skip
counting
leaves
when
constructing
Think
Board
One
and
realising
it
would
give
the
same
result
as
counting
by
ones.
She
was
able
to
write
number
sentences
matching
Sharon
McCleary
7

8. Teaching
Mathematics
in
Early
Years
EDUC8502
the
semantic
structure
for
each
of
the
Think
Boards,
producing
small
number
addition
problems.
Each
Think
Board
demonstrated
her
tendency
to
think
about
number
in
terms
of
part-­‐
part-­‐whole
relations
(e.g.
19+1=20
Think
Board
Three),
typical
of
a
child
in
the
Quantifying
Phase
(Willis,
2004).
A
key
element
of
this
phase
is
conservation
of
number,
which
Gabi
demonstrated
when
re-­‐arranging
counters
without
having
to
re-­‐count
them.
This
indicates
she
is
developmentally
able
to
deal
with
abstract
symbolic
activities
and
can
mentally
manipulate
numbers
represented
by
symbols
with
a
real
understanding
of
what
she
is
doing
(Charlesworth,
2007).
She
demonstrated
this
confidently
for
smaller
numbers
(i.e.
‘15’
Think
Board
One),
but
did
not
display
an
understanding
of
the
place
value
symbols
used
to
represent
larger
numbers
and
would
benefit
from
more
concrete
experiences
constructing
systems
of
10’s.
This
would
consolidate
her
understanding
of
the
Base-­‐10
patterns
and
place
value
representations
up
to
100,
and
eventually
translate
to
larger
numbers.
Recommended
Activities:
The
following
two
activities
have
been
designed
to
give
exposure
to
these
outcomes:
Activity
1:
Build
a
100’s
Chart
using
Tens-­Frames.
First
Steps
in
Mathematics
-­
Number:
(Willis,
2010):
Understand
Whole
and
Decimal
Numbers
Key
Understandings
(Willis,
2010,
pg
52):
“KU5
There
are
patterns
in
the
way
we
write
whole
numbers
that
help
us
remember
their
order.”
Reason
About
Number
Patterns
(Willis,
2010,
pg
242)
“KU
5
Our
numeration
system
has
a
lot
of
specially
built-­‐in
patterns
that
make
working
with
numbers
easier.”
Materials:
Lead
Pencil,
coloured
pencils,
paper,
die,
two
different
coloured
counters,
10
‘tens
frames’,
a
100’s
chart
cut
into
strips
of
10
(i.e.
1-­‐20,
11-­‐20,
21-­‐30,
etc).
Sharon
McCleary
8

9. Teaching
Mathematics
in
Early
Years
EDUC8502
1.
Roll
the
die
and
use
coloured
counters
to
fill
in
the
tens
frame.
Alternate
the
colour
of
counters
used
for
each
roll
of
the
die.
2.
When
the
first
tens
frame
is
completely
full,
write
the
corresponding
number
sentence
using
the
coloured
counters
to
assist.
4.
Trade
the
completed
tens
frame
for
the
first
row
of
the
100’s
chart,
and
colour
the
numbers
corresponding
to
the
coloured
counters.
5.
Continue
this
process
until
all
10
tens
frames
have
been
completely
filled,
and
the
entire
100’s
chart
has
been
generated.
This
activity
capitalises
on
Gabi’s
ability
to
partition
ten
(shown
on
Think
Boards
Two
and
Three),
integrating
visualisation
to
assist
recognition
of
the
part-­‐part-­‐whole
relationships
within
the
tens
frame
(McIntosh,
1997),
but
extending
her
thinking
to
the
next
level
of
counting,
where
ten
units
are
grouped
together
and
‘ten’
becomes
the
iterable
unit
(Jones,
1994).
Studies
show
that
imagery
is
used
extensively
in
the
construction
of
mathematical
meaning,
with
Presmeg
(1986)
identifying
five
main
types
of
visual
imagery:
concrete,
pattern
(relationships),
memory,
kinaesthetic
(involving
muscular
activity)
and
dynamic
(Thomas,
2002).
This
activity
utilises
concrete,
pattern,
kinaesthetic,
and
memory
imagery
to
reinforce
connections
between
verbal,
imagistic
and
formal
notation
systems
of
representation
(Goldin,
1987).
It
strengthens
the
connections
between
the
concrete
counters
and
the
conventional
symbolic
representation
by
writing
the
corresponding
number
sentence,
providing
explicit
links
which
encourage
mathematical
learning.
It
also
introduces
the
idea
of
trading
ten
units
for
a
single
entity
of
ten,
allowing
the
student
to
use
their
previous
constructs
of
the
system
of
1’s
to
develop
an
understanding
of
the
Base-­‐10
system
and
the
patterns
within
the
100’s
chart.
This
concept
of
grouping
is
a
crucial
part
of
the
numeration
system
and
understanding
base-­‐10
and
place
value.
Sharon
McCleary
9

10. Teaching
Mathematics
in
Early
Years
EDUC8502
The
activity
gradually
builds
multi-­‐digit
number
sense
by
reinforcing
the
structure
and
order
of
the
100’s
chart.
This
is
an
essential
pre-­‐requisite
for
understanding
larger
numbers
up
to
1000.
Activity
Two:
Jelly-­Bean
Party
Bag
Game
(Estimation,
Counting
and
Grouping
using
Place
Value
Mats)
First
Steps
in
Mathematics
–
Number:
(Willis,
2010):
Understand
Whole
and
Decimal
Numbers
Key
Understandings
(Willis,
2010,
pg12&60):
“KU1
We
can
count
a
collection
to
find
out
how
many
are
in
it.
KU6
Place
value
helps
us
to
think
of
the
same
whole
number
in
different
ways
and
this
can
be
useful.”
Materials:
Large
bag
of
painted
beans
(e.g.97),
party
bags,
place
value
charts,
die.
1.
Ask
student
to
estimate
how
many
jelly
beans
are
in
the
large
bag.
2.
Roll
the
die,
explaining
the
place
value
chart
by
representing
single-­‐digit
numbers
as
individual
beans
in
the
1’s
column.
2.
Explain
once
there
are
10
jelly-­‐beans
they
can
be
put
into
a
party
bag
in
the
10’s
column:
this
will
help
us
count
faster.
3.
When
the
bag
is
empty,
ask
student
to
count
using
the
party
bags,
and
write
the
number
in
the
place
value
chart,
comparing
it
with
their
estimate.
4.
Ask
how
many
jelly-­‐beans
in
each
column
to
encourage
partitioning
of
this
number
and
demonstrate
the
difference
between
face
value
and
complete
value.
(i.e.90+7=97).
This
activity
balances
challenge
and
success,
providing
a
meaningful,
real-­‐life
context
for
counting.
Sharon
McCleary
10

11. Teaching
Mathematics
in
Early
Years
EDUC8502
It
encourages
estimation,
which
is
an
effective
way
of
developing
number
sense
(Reys,
1989),
and
allows
direct
links
from
the
enactive
processes
of
counting
and
grouping
to
the
written/symbolic
place
value
representations
of
number.
Failure
to
understand
place
value
systems
often
stems
from
an
inability
to
differentiate
between
face
value
and
complete
value,
since
the
same
number
can
represent
several
values
(Varelas,
1997).
Focusing
on
the
semiotics
aspects
of
the
written
place
value
system
during
the
activity
helps
students
differentiate
between
face
value
and
complete
value,
and
assists
conceptual
understanding
of
place
value,
which
is
crucial
for
developing
higher-­‐
level
number
concepts.
Rubin
and
Russell
(1992)
state
counting,
grouping,
estimating
and
notating
are
essential
in
developing
representations
of
the
number
system
(Thomas,
1994).
The
activity
utilises
a
game
format
for
problem
solving
to
promote
automaticity
and
consolidate
key
concepts
by
encourage
justification
of
mathematical
ideas
in
a
social
context,
thereby
improving
mathematical
fluency
(Geist,
2009).
Fluency,
Reasoning
and
Problem
Solving
are
Proficiency
Strands
within
the
Content
Structure
of
The
Australian
Curriculum
(ACARA,
2010).
(1378
Words)
Critical
Reflection:
The
interview
process
made
me
realise
children’s
mathematical
thinking
is
highly
personal
and
very
different
to
adult
thinking
(McIntosh,
1997;
Sfard,
2005);
their
number
concepts
are
limited
by
their
developmental
stage
and
real-­‐world
experiences,
their
own
ability
to
make
mathematical
sense
out
of
these
experiences,
and
to
communicate
these
understandings
effectively.
Gabi’s
interview
responses
indicated
her
conscious
exposure
to
numbers
was
mostly
confined
to
the
classroom,
since
she
had
difficulty
providing
authentic
examples
for
the
Think
Board
representations.
I
was
surprised
how
difficult
it
was
for
Gabi
to
make
real-­‐world
connections
and
realised
the
links
between
classroom
mathematics
and
the
real-­‐world
need
to
be
regularly
and
explicitly
made
to
become
meaningful
for
children.
The
Think
Boards
challenged
her
to
consider
number
concepts
using
different
modes
of
representation,
and
enabled
her
to
explore
connections
between
these
representations.
I
observed
Gabi
actively
trying
to
make
meaning
in
true
Sharon
McCleary
11

12. Teaching
Mathematics
in
Early
Years
EDUC8502
constructivist
fashion
as
she
engaged
in
the
activities
-­‐
similar
to
Wilkerson-­‐Jerde
&
Wilensky’s
(2011)
description
of
mathematical
learning
as
the
process
of
building
a
network
of
mathematical
resources
by
establishing
relationships
between
different
components
and
properties
of
mathematical
ideas.
I
realised
children
are
only
able
to
reveal
their
knowledge
if
they
are
given
the
opportunity
to
do
so,
and
classroom
activities
need
to
be
open-­‐ended
to
allow
them
to
demonstrate
and
explore
their
own
mathematical
thinking
without
placing
limitations
on
it.
The
teacher’s
role
is
to
provide
opportunities
for
deep
understanding
of
concepts
and
make
clear
links
between
the
concept
and
the
conventional
mathematical
symbols,
allowing
semiotic
meaning
making
without
stifling
their
inherent
mathematical
thought
processes.
In
order
to
achieve
a
deep
understanding
of
mathematical
concepts
and
achieve
autonomous
learning,
children
must
be
allowed
to
reinvent
mathematical
concepts
in
their
own
minds
(Kamii,
1984).
During
the
interview,
communication
was
pivotal
in
encouraging
further
learning.
Talking
about
the
Real
and
Story
sections
of
the
Think
Boards
made
Gabi’s
thinking
visible
(Whitin,
2000)
and
generated
further
learning
opportunities;
it
motivated
learning
in
the
child,
and
teaching
in
the
adult
(Sfard,
2005).
This
was
a
clear
demonstration
of
Gabi
constructing
knowledge
in
a
social
context
(Vygotsky,
1978),
where
communication
clarified
and
consolidated
her
thinking.
It
also
demonstrated
how
meaning
arises
from
the
tension
(Radford,
2011)
between
the
child’s
inner
understanding
of
mathematical
ideas
and
their
functioning
in
a
shared
sociocultural
world
of
semiotic
systems
(Fried,
2011).
Given
that
very
young
children
are
still
developing
knowledge
of
mathematical
language
and
conventions,
and
the
limitations
of
interpreting
their
external
representations,
it
is
important
to
observe
them
on
multiple
occasions,
through
various
representational
modes,
using
active
listening,
and
interpreting
gesture,
pictures
and
symbols
to
determine
their
mathematical
understandings
and
assist
them
towards
achieving
autonomous
learning.
(446
words)
Sharon
McCleary
12

13. Teaching
Mathematics
in
Early
Years
EDUC8502
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EDUC8502
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McCleary
14

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Mathematics
in
Early
Years
EDUC8502
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15