Manuela
Barden
1
Is
Problem
Solving
the
Trojan
horse
in
your
classroom?
Manuela
Barden
“Oh!”
exclaimed
Chloe
excitedly,
“we
can
use
that
in
our
rule
(when
the
sequence)
doesn’t
start
with
1!”
“Can
we
use
variables?”
Sally
interjected
looking
at
their
writing
on
the
board.
Nodding
Chloe
continued:
“…so
the
next
step
is
to
find
half
of
whatever
number
is
here
(she
points
to
the
board)…so
if
that
was
49
and
that
was
27
we
would
know
that
(the
difference)
is
22
and
we
need
to
find
half
of
it.
How
do
we
show
that?
1
Oh!”
she
exclaimed,
and
wrote ( y ! x ) on
the
board.
2
Conversation
in
a
mathematics
classroom,
2011
In
this
extract
we
are
witness
to
the
girls’
engagement
with
the
problem
in
hand.
It
is
not
difficult
to
imagine
these
girls
as
successful
problem
solvers
in
a
real
life
situation.
Wiliam(2006)
argues
that
effective
participation
in
today’s
society
requires
higher
levels
of
critical
thinking
than
50
years
ago.
So
what
do
students
need
to
learn
to
become
effective
problem
solvers?
The
research
conducted
by
Project
Zero,
Harvard
University
graduate
School
of
Education,
identifies
high
level
thinking
as
the
precursor
to
problem
solving.
To
facilitate
thinking
in
the
classroom,
it
must
be
made
explicit
and
be
the
focus
of
learning.
It
becomes
evident
through
observing
students’
thinking
whether
they
are
simply
following
a
procedure
or
actively
engaging
in
problem
solving.
A
9
year
siege,
a
Trojan
horse
and
the
city
of
Troy
What
does
the
story
of
the
Trojan
horse
have
to
do
with
problem
solving
in
the
modern
classroom?
Traditionally
I
considered
problem
solving
to
be
a
topic
in
its
own
right
and
used
strategies
to
teach
it
to
my
class.
I
usually
presented
4
to
6
discrete
approaches,
from
which
students
selected
a
method
and
used
it
to
arrive
at
a
solution.
Although
the
expressions
‘guess
and
check’,
‘make
a
table’,
‘work
backwards’,
and
‘draw
a
diagram’
are
often
remembered
long
after
the
first
encounter,
this
superficial
understanding
is
not
useful.
Problem
solving
is
about
describing
real
life
events
in
numbers
so
that
connections
can
be
made
and
patterns
in
the
data
found.
Making
a
table
is
useful
as
it
can
illuminate
these
patterns,
drawing
a
diagram
and
working
backwards
can
identify
relationships
by
looking
at
the
information
from
different
angles.
Guess
and
check,
a
much
abused
strategy,
is
really
the
heuristic
test
and
verify,
and
is
a
useful
step
to
take
when
an
intuitive
sense
of
the
answer
is
held,
or
all
possible
solutions
have
been
reduced
to
a
few.
To
approach
a
problem
believing
that
the
application
of
a
strategy
in
its
own
right
is
the
goal
of
problem
solving
kills
off
any
opportunity
for
thinking.
The
students
are
separated
from
the
very
part
of
the
activity
that
is
engaging.
“If
you
simply
teach
students
how
Manuela
Barden
2
to
solve
problems
they
may
learn
very
little
other
than
the
sequence
of
steps
they
need
to
follow
in
order
to
solve
a
particular
type
of
problem”
Killen
(2006).
When
the
presentation
of
strategies
is
followed
by
short
oversimplified
examples,
Albert
(2001)
argues
students
are
only
involved
in
a
brief
recollection
of
facts
and
the
application
of
formulae
or
rules.
Using
strategies
to
teach
problem
solving
often
eventuates
in
a
classroom
where
very
little
thinking
is
happening.
The
Trojan
horse
left
at
the
gates
of
Troy
contained
enemy
soldiers.
When
the
horse
was
taken
inside
the
gates
after
the
Greeks
had
appeared
to
flee,
these
soldiers
stole
out
of
hiding
under
the
cover
of
dark
and
sacked
the
city.
In
the
same
way
a
dearth
of
thinking
is
encompassed
in
teaching
strategies
for
problem
solving.
What
kind
of
thinking
supports
problem
solving?
Much
has
been
written
about
thinking
in
educational
research
and
beyond.
In
their
book
Making
Thinking
Visible,
Ritchhart,
Church
and
Morrison
develop
the
definition
of
thinking
from
a
sequential
activity,
as
described
in
Bloom’s
taxonomy,
to
a
more
fluid
exchange
between
high
leverage
mental
moves.
Tishman
et
al
argues
that
thinking
is
something
that
needs
to
be
developed
and
sustained
in
students:
“…thinking
and
more
specifically
the
disposition
toward
thinking…must
be
nurtured
in
students
over
time”
(Tishman,
Perkins,
&
Jay,
1993).
Taken
together
with
Wiliam’s
belief
that
“Teachers
do
not
create
learning…teachers
create
the
conditions
in
which
students
learn”
(Wiliam,
2006),
the
paradigm
shifts
from
teaching
problem
solving
via
strategies,
to
creating
a
thinking
environment
conducive
to
the
solving
of
problems.
Accepting
this
shift,
the
obvious
next
question
is
“do
we
all
share
the
same
understanding
of
what
it
means
to
think?
Could
we
articulate
what
kinds
of
thinking
we
should
support
to
assist
problem
solving
in
the
classroom?”
Ritchhart
and
his
colleagues
David
Perkins,
Shari
Tishman
and
Patricia
Palmer
identify
the
following
eight
high
leverage
mental
moves
in
response
to
these
questions.
•
•
•
•
•
•
•
•
Observing
closely
and
describing
what’s
there
Building
explanations
and
interpretations
Reasoning
with
evidence
Making
connections
Considering
different
viewpoints
and
perspectives
Capturing
the
heart
and
forming
conclusions
Wondering
and
asking
questions
Uncovering
complexity
and
going
below
the
surface
Ritchhart,
Church,
Morrison
(2011)
Problem
solving
is
a
complex
activity
and
requires
a
combination
of
the
above
mental
moves.
In
Manuela
Barden
3
my
experience
a
lack
of
attention
to
detail
and
misinterpreting
information
are
common
stumbling
blocks
for
students
when
problem
solving.
These
issues
directly
connect
to
‘Observing
closely
and
describing
what’s
there’,
the
first
thinking
move
in
Ritchhart’s
list.
To
examine
this
relationship
more
closely,
I
needed
to
be
able
to
see
what
my
students
were
thinking.
In
2006
I
was
introduced
to
Thinking
Routines
through
my
involvement
with
The
Ithaka
Project,
a
network
of
teachers
in
Melbourne,
Australia
[http://ithakaprojecthistory.pbworks.com/w/page/34244604/The‐Ithaka‐Project].
Thinking
routines
require
students
to
write
down
and/or
discuss
their
thoughts,
enabling
teachers
to
“collect
evidence
about
where
the
students
are
(so)
as
to
make
adjustments
to
our
teaching
to
better
meet
our
students
learning
needs”
(Wiliam,
2006).
Used
within
the
International
Baccalaureate
Middle
Years
Programme
(IB
MYP),
whose
4
program
wide
assessment
criteria
reflect
the
8
high
leverage
thinking
moves,
thinking
is
placed
at
the
center
of
student’s
learning.
Presented
in
this
paper
are
my
insights
into
how
developing
these
eight
thinking
moves
with
a
class
of
25
Year
7
girls,
improved
their
willingness
to
engage
in
problem
solving,
especially
when
the
answer
was
no
longer
something
that
‘the
teacher
knew
and
the
student
had
to
guess’.
Students
also
became
much
more
open
to
the
idea
that
a
solution
could
be
reached
by
a
variety
of
pathways,
that
‘thinking’
involved
effort,
and
there
was
useful
thinking
and
not
so
useful
thinking.
Inside
the
walls
of
the
classroom
For
the
first
session,
my
central
aim
was
to
build
familiarity
with
the
main
routine
I
intended
to
use,
See‐Think‐Wonder
(STW)
(Ritchhart,
Church,
Morrison,
2011).
This
routine
addresses
the
first
move
in
the
list
of
‘Observing
closely
and
noticing
what’s
there’,
and
hence
my
concerns
with
students’
lack
of
attention
to
detail.
The
first
step
asks
students
to
note
what
they
see
in
the
presented
scenario
(what
details
are
you
paying
attention
to?),
then
what
they
are
thinking
(what
interpretations
are
you
making
based
in
these
details?)
and
finally
to
record
their
wonders
(what
limitations
are
you
considering?).
Having
worked
with
thinking
routines
in
previous
schools,
I
decided
to
use
the
approach
of
the
Micro
Lab
protocol
(Micro
Lab)
to
structure
the
ensuing
conversations.
The
first
step
is
for
the
group
of
three
or
four
students
to
spend
two
minutes
silently
reflecting
on
the
STW
that
came
up
for
them.
Each
student
then
has
two
minutes
to
present
their
thoughts
to
the
group,
who
can
make
notes
but
not
interrupt
the
speaker.
Lastly
the
group
has
five
minutes
in
which
to
collaboratively
build
a
solution
from
the
shared
data
and
interpretations.
As
well
as
developing
fluency
with
the
routine,
I
was
curious
to
see
if
any
girls
would
invoke
strategies
learnt
in
previous
years
in
the
first
few
sessions.
In
response
to
the
problem
posed
below,
I
was
interested
to
see
how
the
girls
would
deal
with
the
specific
terminology
presented
and
whether
they
could
express
the
concept
of
‘consecutive’
algebraically.
Manuela
Barden
4
The
sum
of
three
consecutive
numbers
is
57
and
their
product
is
6783,
what
are
the
numbers?
DIAGRAM
1
•
•
•
•
•
•
•
•
•
“How
will
I
work
this
out?”
“I
wonder
how
to
solve
this
problem”
“I
will
need
to
multiply,
I
might
need
to
add,
I
might
use
guess
and
check”
“will
I
have
trouble
with
this
question”
“I
wonder
if
this
is
a
trick
question”
“How
will
I
know
they
are
(the)
right
(numbers)?”
“57
divided
by
6783,
long
division”
“nxaxb=6783,
what
is
n,
a,
b?
Do
I
divide
or
minus?”
“Can
they
be
the
same
number?”
For
many
students
in
the
first
few
sessions,
their
responses
to
the
‘think’
prompt
were
not
interpretations
of
the
facts
but
rather
statements
of
what
they
were
thinking
at
the
time.
Unconsciously
my
response
to
this
was
to
repeat
prompts
that
were
interpretations
to
the
class,
and
ignore
those
that
weren’t.
This
implied
qualification
of
what
was
a
‘strong
think’
or
a
‘poor
think’
put
the
teacher
back
in
the
role
of
‘she‐who‐knows‐everything’
a
position
that
is
not
conducive
to
student
thinking.
At
this
point
I
was
unsure
of
how
to
change
this
in
the
Manuela
Barden
5
classroom.
I
was
more
content
with
the
one‐to‐one
conversations
where
I
deliberately
used
the
What‐Makes‐You‐Say‐That
(WMYST)
routine,
eliciting
justification
with
evidence.
Some
referred
to
the
strategy
guess
and
check,
but
it
was
interesting
to
note
that
in
the
five
minutes
for
problem
solving
the
students
didn’t
follow
the
strategy.
One
group
did
discuss
the
idea
that
they
could
try
values
less
than
20
as
3x20=60.
For
those
of
us
who
understand
what
we
are
doing,
this
could
be
argued
to
be
valid
use
of
the
strategy
guess
and
check.
However,
for
me
this
is
the
heuristic,
test
and
verify.
The
students
had
to
connect
the
problem
to
the
expression
3x20
and
then
identify
the
set
of
values
worth
checking.
Closing
in
on
interpretations
grounded
in
data
I
was
pleased
with
the
development
of
learning,
but
was
keen
to
push
thinking
further.
All
students
were
able
to
identify
the
majority
of
key
data,
and
even
those
with
poor
English
skills
completed
this
stage
well.
Students
were
invited
to
explain
the
meaning
of
difficult
words
prior
to
splitting
into
groups.
However,
I
wanted
more
students
to
make
useful
interpretations
of
the
data
presented.
The
focus
moved
to
the
second
high
level
thinking
move
in
Ritchhart’s
list
“building
explanations
and
interpretations”.
In
a
discussion
with
colleagues,
another
routine
called
Zoom‐In
was
suggested.
Like
STW
this
routine
asks
students
to
notice
the
details
and
make
interpretations,
however,
to
further
highlight
how
different
pieces
of
information
can
affect
a
solution,
the
information
is
presented
in
layers.
In
this
way
students
build
several
solutions
to
the
problem,
and
witness
how
possible
solutions
vary
as
new
data
is
taken
into
consideration.
I
continued
to
use
the
Micro
Lab
and
the
students,
without
direction,
continued
to
use
the
language
of
STW.
I
purposely
selected
the
following
problem
as
it
could
be
separated
into
layers
that
would
direct
students’
attention
to
key
aspects
of
the
question.
At
the
same
time
this
problem
addressed
the
concept
of
shape
and
area
the
current
topic
of
study.
Christie
made
a
cake
in
an
8in
x
8in
pan,
which
she
wants
to
share
with
her
3
friends.
She
wants
to
divide
the
cake
equally
among
all
4
people
while
giving
each
person
a
different
shaped
piece
of
cake.
She
wants
one
piece
to
be
a
triangle,
one
piece
to
be
a
quadrilateral,
one
piece
to
be
a
pentagon,
and
one
piece
to
be
a
hexagon.
Show
how
Christie
can
divide
the
cake.
Explain
how
you
know
that
each
person
received
the
same
amount
of
cake?
Mooney,
March
2007
The
first
layer
of
the
Zoom‐In
routine
is
shown
below
with
the
subsequent
layers,
revealing
the
need
for
different
shaped
pieces,
what
the
specific
shapes
should
be,
and
finally
the
invitation
to
‘show
how’.
Leaving
this
to
the
last
reveal,
I
was
interested
in
seeing
which
students
would
Manuela
Barden
6
choose
to
use
a
diagram
from
the
start.
A
diagram
allows
students
to
see
the
information
from
a
different
perspective
often
opening
up
their
interpretation.
Christie
made
a
cake
in
an
8in
x
8in
pan,
which
she
wants
to
share
with
her
3
friends.
She
wants
to
divide
the
cake
equally
amoung
all
4
people.
Explain
how
you
know
that
each
person
received
the
same
amount
of
cake.
As
expected,
during
the
two
minute
round
of
individual
thoughts,
I
heard
a
lot
of
girls
say
that
the
question
was
easy.
I
was
intrigued
to
see
that
most
students
had
represented
a
round
cake
before
dividing
it
into
quarters,
despite
the
phrase
“an
8inx8in
pan”.
The
full
question
had
been
presented
when
the
following
responses
to
the
‘think’
prompt
were
recorded.
DIAGRAM
2
•
•
•
•
“8in
x
8in
=
64
in
pan”
“Triangle
–
3
sides,
Quadrilateral
–
4
sides”
“Give
each
person
64/4=
16in
then
make
a
shape
out
of
it”
“What
shape
is
the
tin?”
The
two
stumbling
blocks
were
the
shape
of
the
pan
and
how
to
divide
a
shape
into
unequal
parts.
The
first
surprised
me
until
I
reflected
that
it
is
a
common
issue
in
the
classroom
where
a
teacher
unwittingly
only
ever
‘uses’
round
cakes.
The
second
was
another
reason
why
I
had
selected
this
question.
It
nicely
counterbalances
the
concept
of
fractions
always
having
equal
Manuela
Barden
7
parts,
which
was
reevaluated
by
students
in
a
discussion
on
what
to
call
the
pieces.
The
evidence
in
this
lesson
suggested
that
the
students’
thinking
was
becoming
more
‘high
level’.
The
students
were
embedded
in
building
explanations.
I
was
pleased
to
hear
many
of
the
conversations
from
the
small
groups
carried
on
between
lessons.
Using
problem
solving
to
uncover
misconceptions
“Write
down
these
decimals
in
order
of
size,
from
largest
to
smallest:
0.75
0.04
0.375
0.25
0.4
0.125
0.8.
Underneath,
describe
and
explain
your
method
for
doing
this”
Swan,
2006
I
used
Zoom‐In
at
the
start
of
the
next
topic,
fractions
and
decimals,
instead
of
the
more
standard
pre‐test.
I
was
hoping
the
discussions
would
help
students
overcome
their
misconceptions,
as
opposed
to
just
highlighting
them
to
me.
Equivalent
area
diagrams,
line
segments
and
percentage
cards
were
used
as
the
layers
in
the
routine.
Students
could
request
the
final
layer
of
area
diagrams
and
line
segments
divided
into
100
equal
parts
if
they
felt
they
needed
it.
Manuela
Barden
8
DIAGRAM
3
Sandra
ordered
her
fractions
as
follows:
“0.375,
0.125,
0.75,
0.25,
0.8,
0.4,
0.04”.
In
quite
a
fervent
discussion
with
Wendy
she
said
“0.375
is
smaller
than
0.250
because
it
was
further
away
from
0”.
Wendy
found
it
difficult
to
explain
why
she
disagreed
with
this,
although
she
had
placed
0.04
as
the
smallest
fraction.
After
the
first
and
second
reveals
of
the
area
diagrams
and
percentages
cards,
both
students
were
observed
ordering
the
area
diagrams
correctly
without
referring
to
the
ordered
decimals.
Asked
to
link
the
two
together,
Sandra
saw
that
her
decimal
arrangement
was
incorrect
but
had
difficulty
accepting
it.
In
their
group
Wendy
alternated
between
calling
the
fraction
0.375,
“zero
point
3
hundred
and
75
and
zero
point
three,
seven,
five”.
Discussion
between
the
two
and
myself
produced
an
‘ah‐ha’
moment
for
Sandra,
and
a
connection
between
the
final
layer
area
diagrams
and
decimals.
At
the
end
of
the
session
the
girls
were
asked
to
reflect
on
their
learning.
Sandra
commented
that
it
was
easier
to
arrange
the
area
diagrams
as
you
could
easily
see
which
were
smaller.
Her
final
reflection
captured
this
thinking.
“When
I
started
I
didn’t
understand
decimals
at
all,
I
thought
the
decimals
grew
smaller
and
larger
by
the
number
of
digits
they
had…the
green
(area)
diagrams
helped
a
lot
because
you
could
see
how
big
they
were”.
Using
problem
solving
to
develop
concepts
As
the
sessions
progressed,
solving
problems
didn’t
stay
within
the
confines
of
one
session
every
two
weeks.
The
more
I
looked
through
resources
for
problems
to
use,
the
more
I
envisioned
how
particular
problems
could
be
used
to
answer
questions
arising
from
previous
discussions.
The
next
problem
was
chosen
to
further
illustrate
the
necessity
of
defining
the
‘whole’
that
a
fractional
part
is
from,
before
being
able
to
name
it.
At
this
point
formal
algorithms
had
not
been
discussed
for
adding
fractions
with
different
size
denominators.
However,
some
girls
had
experienced
these
at
primary
school.
In
addition
to
using
the
STW
routine
and
Micro
Lab,
the
girls
recorded
their
STW
on
mini
white
boards.
Some
students
found
the
semi
permanence
of
the
white
boards
more
liberating
when
discussing
tentative
ideas.
1
of
the
pie
and
put
4
the
rest
back
in
the
fridge.
On
Monday
Jimmy
took
out
the
pie
from
the
fridge
and
gave
1
to
Kate.
Who
had
the
biggest
slice
of
pie?
3
On
Sunday
Jimmy’s
mother
made
a
pie.
Later
that
day
Jimmy
ate
Manuela
Barden
9
DIAGRAM
4
•
•
•
•
•
Teacher:
“Tell
me
about
your
board”
Student:
“Jimmy
has
¼
of
the
cake
and
he
gives
Kate
1
third.
So
that’s
sharing…
dividing,
so
¼
divided
by
1
third…3
twelfths”
Teacher:
“WMYST”
Student:
(turning
to
her
diagram
by
way
of
explanation)
“The
blue
is
the
quarter
of
the
cake
Jimmy
ate,
leaving
the
white
squares…which
is
the
new
amount
to
share
into
3…so
(Kate’s
share)
that’s
the
crosses…3
of
the
twelve
squares”
Student:
(after
further
examination
of
her
board)
“ohh…the
denominators…4
x
1
isn’t
12…
that’s
not
right….maybe
I
shouldn’t
be
dividing?”
This
student
was
trying
to
make
sense
of
the
problem
from
two
angles
‐
one
by
using
a
partially
remembered
algorithm,
and
one
by
the
use
of
an
area
diagram.
The
temptation
for
the
teacher
is
to
tell
students
where
they
have
misunderstood
when
looking
at
the
diagram.
The
WMYST
routine
forces
the
student
to
articulate
and
hence
follow
their
own
reasoning
steps.
This
aids
them
in
uncovering
any
issues
for
themselves,
and
moves
them
into
thinking
more
deeply
about
Manuela
Barden
10
the
question.
Assessing
students
with
regard
to
their
thinking
skills,
supported
by
their
solutions
to
problems,
seemed
to
level
the
playing
field.
Neither
end
of
the
ability
spectrum
conformed
to
expectations.
Those
at
the
top
were
not
guaranteed
to
be
competent
thinkers,
and
those
at
the
bottom,
when
freed
up
to
express
themselves,
proved
to
be
more
capable
than
expected.
One
student
who
seemed
to
be
disconnected
in
class
prior
to
this
intervention
couldn’t
wait
to
share
the
following
with
me.
She
references
a
problem
regarding
the
concept
of
proportion
that
the
class
tackled
towards
the
end
of
the
topic
on
fractions:
“It
was
so
funny
yesterday
miss.
We
were
out
to
dinner,
Niamh,
Ciara
and
I
and
we
were
discussing
the
‘Bear/Cookie’
problem
and
we
realised
the
people
at
the
table
next
to
us
were
eavesdropping!
It
was
so
embarrassing!
We’ve
got
a
solution
though,
can
we
show
you?”
Conversation
between
student
and
teacher,
8:30am
DIAGRAM
5
Recasting
the
role
of
the
learner
At
the
start
of
this
process
I
was
surprised
how
quickly
the
girls
took
to
using
the
routines.
As
the
sessions
progressed
and
problem
solving
became
more
embedded,
I
noticed
that
some
students
continued
to
have
difficulty
building
explanations.
Use
of
Zoom‐In
helped,
but
as
is
often
the
way
with
human
beings,
it
didn’t
work
for
everyone.
I
felt
that
some
students
in
particular
were
jotting
down
everything
that
was
on
their
mind
in
response
to
the
think
prompt,
and
allowing
others
to
evaluate
the
relevance
of
their
thoughts.
I
was
reluctant
to
do
this
so
as
Manuela
Barden
11
not
to
bring
the
‘control
of
ideas’
back
to
myself,
and
yet
I
wanted
these
students
to
develop
the
ability
to
appraise
their
own
ideas.
I
was
worried
that
these
students
had
fallen
into
a
pattern
of
behavior
that
required
little
thinking.
As
Wiliam
(2006)
argues
“Many
students
will
resist
(thinking)
because
thinking
is
hard.
Henry
Ford
used
to
say
that
it’s
extraordinary
the
lengths
people
will
go
to
avoid
doing
it.
Thinking
is
really
hard
and
that’s
the
challenge:
to
create
classrooms
where
it’s
not
optional”.
Whilst
attending
a
conference
in
Melbourne,
I
was
inspired
by
Mark
Church’s
key
note
speech
on
‘Instructional
Core’.
In
this
he
argues
how
it
is
not
enough
to
provide
well‐chosen
problems
to
solve
but
that
the
learner
had
to
be
empowered
to
reflect
on
their
own
thinking.
In
other
words
a
key
step
in
supporting
thinking
in
the
classroom
was
to
have
the
students
discuss
which
of
the
group’s
‘think’
statements
had
‘legs’,
what
these
statements
had
in
common
and
build
up
a
picture
together
of
the
qualities
and
features
of
‘powerful
thinks’
and
‘weak
thinks’.
The
following
problem
was
presented
in
the
now
fully
accepted
way
using
the
Micro
Lab
and
STW
routine.
The
students
engaged
with
the
task
straight
away
and
clearly
knew
what
was
expected
of
them.
They
were
keen
to
solve
the
problem
themselves
and
used
the
teacher
only
as
a
facilitator.
Routine
use
of
the
Micro
Lab
enabled
each
team
member
to
contribute
to
the
collaborative
solution.
Two
communities
in
North
Sydney,
Caroll
Gardens
and
Flatbush,
each
gather
to
make
plans
for
an
empty
plot
in
their
neighbourhood.
The
plots
are
identical
in
size,
measuring
50m
x
100m.
In
Caroll
Gardens,
the
community
group
decides
to
allocate
3
quarters
of
the
empty
plot
to
playground
and
cover
2
fifths
of
this
playground
with
asphalt.
The
Flatbush
neighbourhood
will
devote
2
fifths
of
the
plot
to
playground,
and
3
quarters
of
the
playground
will
be
covered
in
asphalt.
In
which
park
is
the
asphalt
greater?
(Imm,
Stylianou,
&
Chae,
April
2008)
Manuela
Barden
12
DIAGRAM
6
The
second
of
the
two
lessons
on
this
problem
starting
with
the
class
gathered
around
the
solutions
displayed
at
the
back
of
the
classroom.
Having
asked
one
student
from
each
group
to
recap
the
interpretations
they
had
made,
the
girls
were
invited
to
select
‘thinks’
that
they
saw
as
powerful.
Some
of
their
offerings
are
listed
below.
•
•
•
•
“linking
‘of’
to
times”
“drawing
a
digram
with
quarters
horizontally
and
fifths
vertically”
“seeing
the
pattern
in
3
quarters
of
2
fifths
and
2
fifths
of
3
quarters”
“making
a
diagram”
In
allowing
students
to
evaluate
their
own
thinking,
I
observed
many
‘ah‐ha’
moments,
especially
regarding
the
drawing
of
diagrams.
Having
used
diagrams
in
my
explanations
since
the
start
of
the
year
I
had
previously
felt
that
students
knew
that
was
what
I
wanted
of
them,
but
didn’t
value
the
reinterpretation
of
the
information
themselves,
much
like
teaching
strategies
for
problem
solving.
Summing
up
Over
the
past
semester
I
have
witnessed
my
students
blossoming
as
thinkers
and
problem
solvers.
I
have
thought
deeply
about
the
type
of
thinking
that
is
required
to
be
a
mathematician,
how
to
construe
questions
in
the
classroom
so
as
to
invite
thinking
to
take
place
and
recasting
the
role
of
the
learner
in
the
classroom.
Being
part
of
a
writing
group
provided
a
valuable
opportunity
for
reflection
and
feedback.
Inspired
by
the
research
of
Harvard
University
School
of
education
and
the
shift
in
paradigm
to
problem
solving
being
the
result
of
high
level
thinking
moves,
I
was
committed
to
bringing
this
into
my
classroom.
At
the
start
I
wasn’t
able
to
achieve
the
ideal
of
embedding
problem
solving,
but
made
a
start
anyway.
Manuela
Barden
13
I
was
pleasantly
surprised
when
the
problem
solving
escaped
from
its
imposed
boundaries
into
almost
every
lesson.
There
was
an
abundance
of
resources
both
old
and
new
in
the
staff
room
which
reminded
me
that
‘problem
solving’
is
not
a
new
approach
in
this
subject.
As
well
as
the
Mathematics
Teaching
in
the
Middle
School
(National
Council
of
Teachers
of
Mathematics)
journal
I
found
the
Questions
and
Prompts
for
mathematical
thinking
(Watson
&
Mason,
1998)
to
be
very
useful.
This
article
describes
a
step
in
one
teacher’s
journey
to
deepen
students’
understanding
via
problem
solving
in
the
classroom.
As
a
result
of
the
implementation
of
Thinking
Routines,
students
have
made
their
thinking
visible
to
both
teacher
and
themselves.
Naming
and
noticing
this
thinking
has
empowered
students
to
develop
and
improve
their
reasoning.
The
See‐Think‐
Wonder
and
Zoom‐In
routines
helped
students
connect
to
how
a
solution
changes
in
response
to
each
piece
of
data.
The
What‐Makes‐You‐Say‐That?
routine
prompted
students
to
build
explanations
grounded
in
the
data.
I
have
approached
the
Trojan
horse
of
‘teaching
strategies
for
problem
solving’
by
listening
to
Cassandra’s
warning,
that
the
Trojans
foolishly
ignored,
that
the
wooden
structure
was
not
actually
a
gift,
but
a
trap.
References
Devlin, K. J. (1998). The language of mathematics: making the invisible visible. New York:
W.H. Freeman.
Imm, K., Stylianou, D., & Chae, N. (april 2008). Student Representations at the Center:
Promoting Classroom Equity. Mathematics Teaching in the Middle School, 13(8), 458463.
Killen, R. (2003). Effective Teaching Strategies. Lessons from Research and Practice (3rd ed.).
Tuggerah, NSW: Social Science Press.
Mooney, E. (dec 2007). Solve It! The Three Bears Cookie Store. Mathematics Teaching in the
Middle School, 13(5), 283.
Mooney, E. (March 2007). Solve It!: Christie's Cake. Mathematics Teaching in the Middle
School, 12(7), 379.
Manuela
Barden
14
Ritchhart, R., Church, M., & Morrison, K. (2011). Making Thinking Visible: How to Promote
Engagement, Understanding, and Independence for All Learners. San Francisco, CA:
Jossey-Bass.
Space Shuttle. The Final Mission [Motion picture]. (2011). UK: BBC.
Swan, M. (2006). Collaborative Learning in Mathematics [Scholarly project]. In The Standards
Unit Materials. Retrieved August 29, 2011, from
http://www.nanamic.org.uk/Conference%202006/Keynote%20address%202006.pdf
Visible Thinking. (n.d.). Project Zero. Retrieved August 29, 2011, from
http://www.pz.harvard.edu/vt/VisibleThinking_html_files/VisibleThinking1.html
Watson, A., & Mason, J. (1998). Questions and prompts for mathematical thinking. Derby,
[England: Association of Teachers of Mathematics.