1. E.
Paul
Goldenberg,
June
Mark,
and
Al
Cuoco
The
algebra
of
little
kids
The
algebra
of
little
kids:
A
mathematical-­‐habits-­‐of-­‐mind
perspective
on
elementary
school1
E.
Paul
Goldenberg,
June
Mark,
and
Al
Cuoco
Education
Development
Center,
Inc.
(EDC)2
Asking
“When
should
algebra
be
taught?”
is
like
asking
“Is
technology
harmful
or
helpful?”
There
are
lots
of
technologies
and
lots
of
uses
of
them.
Some
are
harmful;
some
are
helpful.
Refining
the
question—asking
about
a
particular
use
of
a
particular
technology
for
a
particular
purpose
in
particular
contexts
and
at
particular
stages
in
one’s
learning—
makes
the
question
researchable
and
potentially
answerable.
Similarly,
there
are
many
“algebras”—algebra
the
course,
algebra
the
discipline,
algebraic
ideas,
algebraic
language,
early
algebra,
“patterns,
functions,
and
algebra”—and
many
different
takes
on
the
learning
and
teaching
of
each
of
these.
Treating
algebra
as
an
indivisible
whole
obscures
the
options.
It’s
more
useful
to
ask
what
ideas,
logic,
techniques,
and
habits
of
mind
algebra
entails,
and
then,
about
each
of
these,
ask
when
and
to
what
extent
that
one
item
can
be
learned
with
intellectual
integrity
and
how
a
coherent
whole
can
be
woven
out
of
these
learnings.
The
answers
we
get
are
that
some
of
these
ideas
do
have
to
wait
for
eighth
or
ninth
grade,
but
that
others—even
including
aspects
of
algebraic
language—are
already
there,
early
in
the
primary
grades.
Moreover,
children
who
get
to
apply,
refine,
and
strengthen
those
ideas
and
skills
as
they
emerge
gain
the
advantage.
Any
credible
claim
about
habits
of
mind
must
surely
accord
with
features
of
mind:
children’s
cognitive
development.
For
a
charmingly
written
scientific
account
of
the
ways
that
babies
and
young
children
think,
read
The
Scientist
in
the
Crib,
by
Gopnik,
Meltzoff,
and
Kuhl
(2000).
The
habits
of
mind
approach
to
curriculum
that
we
first
described
well
over
a
decade
ago
(Cuoco,
Goldenberg
&
Mark,
1996;
Goldenberg,
1996)
and
have
continued
to
refine
(Goldenberg
&
Shteingold,
2003
and
2007;
Cuoco,
Goldenberg
&
Mark,
2009;
Mark,
et
al.,
2009)
does
accord
well
with
children’s
thinking
and
became
a
central
design
principle
behind
Think
Math!
(2008),
the
newest
NSF-­‐supported
elementary
curriculum,
developed
at
EDC.
Recognizing,
enhancing,
and
building
on
developmentally
natural
habits
of
mind
lets
us
dissect
algebra
and
sort
the
resulting
bits
and
pieces
in
a
developmentally
natural
way,
while
preserving
the
content,
concepts,
and
skills
that
schools
(and
states,
parents,
workplaces,
and
colleges)
expect.
The
fact
that
it
is
possible
to
organize
algebraic
ideas,
logic,
and
techniques
around
the
development
of
mind
makes
clear
that
we
are
truly
talking
about
thinking—habits
of
mind—rather
than
“features
of
mathematics”
or
“idiosyncrasies
of
mathematicians.”
This
article
describes
two
of
these
natural
habits
of
mind.
Two
algebraic
ideas
that
precede
arithmetic
The
common
wisdom
is
arithmetic
first,
algebra
later.
The
truth
is
not
so
simple.
Some
algebraic
ideas—ideas
about
the
properties
of
binary
operations
apart
from
the
numbers
these
operations
may
“combine”—develop
naturally
before
children
learn
arithmetic.
1
An
adaptation
of
this
paper
has
been
submitted
for
publication
in
Teaching
Children
Mathematics,
NCTM.
2
This
work
was
supported
in
part
by
the
National
Science
Foundation,
grant
numbers
ESI-­‐0099093,
DRL-­‐
0733015,
and
DRL-­‐0917958.
The
opinions
expressed
are
those
of
the
authors
and
not
necessarily
those
of
the
Foundation.
©
Education
Development
Center,
Inc.
page
1

2. E.
Paul
Goldenberg,
June
Mark,
and
Al
Cuoco
The
algebra
of
little
kids
In
fact,
they
must
develop
before
arithmetic
can
make
sense!
For
example,
for
many
4-­‐
year-­‐olds,
even
those
who
appear
to
count
well,
seven
objects
spread
out
like
this
feel
like
“more”
than
the
same
objects
bunched
together
.
(Though
“conservation”
remains
the
familiar
name
for
this
stage
in
children’s
logic—so
we’ll
still
use
it—child
logic
is
more
nuanced
than
was
previously
thought.
It’s
known,
for
example,
that
for
small
enough
numbers
of
objects,
babies
at
eleven
months
have
not
only
stability
of
number
but
essentially
addition
as
well.
See,
e.g.,
Feigenson,
Carey,
and
Spelke,
2002.
So-­‐called
non-­‐conservers
aren’t
“enslaved
by
their
senses”
but
haven’t
yet
privileged
the
analytic
act
of
counting
over
other
ways
of
making
social
and
mathematical
sense
of
the
world.)
For
children
whose
logic
still
works
that
way,
the
claim
that
+
is
the
same
amount
as
can
hardly
make
sense.
Faced
with
the
requirement
to
assert
that
5
+
2
=
7,
“non-­‐conservers”
have
only
two
options.
Some
divorce
the
assertion
from
their
current
“common
sense”—after
all,
they
“know”
that
the
two
quantities
are
not
the
same—and
learn
“5
+
2
=
7”
as
an
arbitrary
but
learnable
fact,
the
same
way
they
learn
the
names
of
their
classmates.
For
them,
math
is
memory.
Others
find
it
hard
to
accept
what
their
logic
tells
them
is
“not
true”
and,
instead,
just
feel
like
they
“don’t
get
it.”
An
important
property
of
addition
before
addition,
itself
What
will
later
be
formalized
as
the
commutative
and
associative
laws
of
addition
begins
as
an
intuitive
sense
of
stability/invariance
of
quantity
under
rearrangement.
Piaget
(1952)
called
it
conservation
of
number;
Wirtz,
et
al.
(1964)
and
Sawyer
(2003)
called
it
the
“any
order
any
grouping
property.”
Prior
to
conservation,
while
arrangement
trumps
number,
may
not
have
a
fixed
number
associated
with
it.
Later,
the
new
conserver
may
not
yet
know
how
many
fingers
are
without
counting,
but
will
be
sure
that
the
number,
whatever
it
is,
stays
put
if
the
hands
are
moved
like
this
or
even
like
this,
.
That
algebraic
idea,
a
property
of
aggregation,
must
exist
before
the
arithmetic
fact—
knowing
what
number
2
+
5
is—can
make
sense.
In
a
similar
way,
if
a
bunch
of
coins
are
hidden
and
we
ask
“how
much
money
is
there?”
children
for
whom
the
question
makes
any
sense
will
be
absolutely
certain
that
there
is
an
answer,
and
that
only
one
answer
is
correct.
They
may
be
uncertain
about
methods
of
counting,
and
may
think
that
some
methods
might
give
incorrect
answers.
The
complexities
of
communication
may
even
make
it
seem
that
they
believe
that
the
amount,
itself,
could
vary
depending
on
what
method
they
use
as
they
count
but,
in
all
likelihood,
other
means
of
questioning
would
suggest
that
they’re
sure
that
the
amount
is
stable.
In
fact,
if
they
do
believe
the
amount
can
vary,
they’re
not
cognitively
ready
for
the
question
of
what
“the
amount”
is.
There
is
no
“the
amount”
if
it
can
vary.
Some
six
year
olds,
but
not
many,
do
not
yet
conserve
number;
by
seven,
nearly
all
do.
Having
confidence
that
and
represent
the
same
quantity
is
not
the
same
as
knowing
the
commutative
property
of
addition.
The
commutative
property
is
not
about
the
arrangement
of
physical
objects
in
space,
but
about
the
behavior
of
a
particular
element
(here,
the
+
sign)
in
a
formal
syntactic
system
of
written
symbols.
In
some
contexts,
children
can
make
perfect
sense
out
of
written
symbols—even
significant
parts
of
algebraic
notation—but
most
young
children
cannot
make
sense
of
formal
operations
on
a
string
of
©
Education
Development
Center,
Inc.
page
2

3. E.
Paul
Goldenberg,
June
Mark,
and
Al
Cuoco
The
algebra
of
little
kids
symbols.
So,
at
this
stage,
commutativity
remains
largely
an
intuitively
obvious
idea
about
the
“physics
of
mathematics”:
the
nature
of
aggregation,
not
the
nature
of
symbols.
Even
so,
we,
as
educators,
can
support
the
young
child’s
logic
better
if
we
recognize
that
it
is
already
relying
on
the
underlying
ideas
that
formal
mathematics
will
later
codify.
The
fact
that
children
see
that
the
principle
applies
regardless
of
the
numbers
means
that
it
captures
the
essential
algebraic
aspect
of
the
structure
of
addition
that
commutativity
is
about.
Logical
precursors
of
the
distributive
property
of
multiplication
over
addition:
Pick
a
number.
Multiply
it
by
5;
also
multiply
it
(your
original
number)
by
2;
now
add
those
results.
You
get
the
same
answer
you’d
get
if
you
multiplied
your
original
number
by
7.
The
distributive
property,
a
general
statement
of
that
fact,
is
possibly
the
most
central
idea
in
elementary
arithmetic,
key
to
understanding
the
algorithms,
at
the
core
of
fluent
mental
calculations
(e.g.,
102
×
27
can
be
computed
in
two
parts,
as
100
×
27
+
2
×
27),
and
the
logical
basis
for
many
“rules”
of
algebra
that
might
otherwise
seem
arbitrary.
This
property
relates
multiplication
and
addition,
but
children
“know
it”
long
before
they
even
meet
multiplication!
It’s
in
the
language
(and
logic)
they
use
when
they
say
that
5
(fingers,
pennies,
or
27s)
plus
2
(fingers,
pennies,
27s)
make
7
(fingers,
pennies,
27s).
These
dialogues
with
6-­‐year-­‐olds,
late
in
their
kindergarten
year,
give
a
sense
of
what
their
logic
does
and
does
not
handle.
What
distinguishes
the
questions
the
children
get
“right”
from
those
they
get
“wrong”?
What
logic
might
explain
the
particular
wrong
answers
they
get?
T
What’s
a
really
big
number?
Ne
(girl):
A
million!
T:
Suppose
I
said
“How
much
is
a
thousand
plus
a
thousand?”
What
would
you
say?
Ne:
I
have
no
idea!
(big
smile)
T:
And
suppose
I
said
“How
much
is
two
thousand
plus
three
thousand?”
Ne:
(thinks,
then
confidently)
Five
thousand!
T:
Suppose
I
said
“How
much
is
a
hundred
plus
a
hundred?”
What
would
you
say?
Gi
(girl):
A
hundred.
T:
What
about
“Two
hundred
plus
three
hundred”?
Gi:
Five
hundred.
T:
(playfully)
And
what
if
I
said
“how
much
is
a
thousand
plus
a
thousand?”
…
Gi:
A
million!
T:
Suppose
I
said
“How
much
is
a
hundred
plus
a
hundred?”
What
would
you
say?
De
(boy):
De
may
hear
“a
hundred”
as
one
word,
so
confidently
says:
Two
ahundred.
T:
And
suppose
I
said
“How
much
is
two
hundred
plus
three
hundred?”
De:
Five
hundred.
T:
Suppose
I
said
“How
much
is
a
thousand
plus
a
thousand?”
What
would
you
say?
Co
(boy):
A
thousand
two.
(Co
might
have
meant
“A
thousand,
too.”
We
don’t
know.)
T:
And
suppose
I
said
“How
much
is
two
thousand
plus
three
thousand?”
Co:
Two
three
a
thousand.
(Co
clearly
isn’t
yet
adding
naturally.)
As
soon
as
children
are
comfortable
with
the
idea
(and
language
and
knowledge)
to
answer
“what’s
three
sheep
plus
two
sheep?”
perhaps
late
in
K
or
early
in
first
grade,
they’ll
happily
apply
that
to
give
the
“correct”
answer
to
the
spoken
question
“what’s
three
eighths
plus
two
eighths?”
or
“what’s
three
hundred
plus
two
hundred?”
The
answer
is
“correct,”
but
what
they
have
in
mind
may
well
be
quite
different
from
what
we
have
in
mind
when
©
Education
Development
Center,
Inc.
page
3

4. E.
Paul
Goldenberg,
June
Mark,
and
Al
Cuoco
The
algebra
of
little
kids
we
give
the
same
answer.
We
can
see
how
different
their
ideas
are
when
we
ask
a
slightly
different
question:
“what’s
a
hundred
plus
a
hundred”
(with
no
audible
“small”
numbers
like
“two”
or
“three”).
To
this
question,
young
six-­‐year-­‐olds
may
well
repeat
“a
hundred”
or
say
something
like
“a
million.”
If,
instead,
we
ask
“what’s
an
eighth
plus
an
eighth,”
little
ones
may
just
give
a
puzzled
stare
and
not
answer
at
all;
or,
if
their
arithmetic
is
strong
enough,
they
might
possibly
count
and
answer
“sixteen”
(or,
sometimes
“nine”).
How
can
we
explain
such
different
responses
to
questions
that
adults
see
as
so
similar?
Again,
the
answer
rests
more
in
language
and
general
cognition
than
mathematics.
Kindergarteners
typically
have
hundred
and
half
as
vocabulary
items.
For
most
little
ones,
these
terms
don’t
represent
precise
or
fixed
amounts,
just
as
“a
zillion”
is
not
a
specific
fixed
amount
to
us,
but
the
children
do
know
that
“half”
means
only
part.
Most
even
know
that
halves
should
be
equal—
no
fair
if
yours
is
bigger!—though
they
might
not
know
that
they
must
be
equal
or
that
there
are
only
two
of
them.
And
they
almost
certainly
don’t
know
that
half
is
a
number.
Likewise,
they
know
that
“a
hundred”
is
big,
though
they
are
unlikely
to
know
how
big.
The
question
“what’s
a
hundred
plus
a
hundred”
is,
therefore,
more
or
less,
“what
is
a
big
amount
plus
another
big
amount?”
The
natural
response
is
“a
big
amount”
(“a
hundred”)
or
a
very
big
amount
(“a
million”),
not
“two
big
amounts”
(“two
hundred”).
But
when
fixed
specific
quantities
are
available,
children
use
them.
The
question
“what’s
two
hundred
plus
three
hundred”
is
linguistically
and
cognitively
like
“what’s
two
sheep
plus
three
sheep”—it
draws
attention
to
2
+
3,
not
to
the
nature
of
a
sheep
or
a
hundred.
Children
for
whom
2
+
3
makes
sense
answer
correctly.
Of
course,
children
for
whom
2
+
3
does
not
yet
make
sense
try
to
find
some
other
way
of
making
sense
of
the
task,
but
their
answers
don’t
reflect
addition.
(The
different
response
to
“what’s
an
eighth
plus
an
eighth”—the
puzzled
look—is
because
an
eighth
not
even
part
of
the
child’s
vocabulary,
and
thus,
with
no
meaning,
gives
the
child
less
of
a
context
for
responding.
Anna
Sfard,
2008,
suggests
that
a
child
might
well
treat
“hundred”
as
a
number,
rather
than
a
sheep,
and
still
treat
“three
hundred”
not
as
a
number,
but
as
an
expression
composed
of
two
number
words.
If
so,
our
kindergarteners
seem
to
treat
these
numbers
differently,
one
as
a
counter,
the
other
as
a
unit
or
object,
which
might
be
consistent
with
Sfard.)
Why
these
errors
are
made,
and
why
“hundred”
and
“eighth”
lead
to
different
errors,
is
a
diversion.
The
point
is
that
when
no
audible
small
numbers
like
“two”
or
“three”
are
given,
little
children
tend
to
give
wrong
answers.
But
when
we
say
how
many
eighths
or
hundreds,
and
the
numbers
are
not
too
large,
even
kindergarteners
tend
to
answer
correctly,
more
first
graders
do,
and
we
can
absolutely
count
on
it
in
second
grade.
Whatever
an
eighth
or
a
hundred
is,
the
children
are
sure
that
three
of
them
plus
two
of
them
is
five
of
them!
This
does
not
constitute
“knowing
the
distributive
property,”
but
it
does
tell
us
that
the
children
already
have
the
underlying
idea
that
the
distributive
property
will
later
encode
formally.
If
we
use
sevens
(a
fully
understood
fixed
quantity)
in
place
of
hundred
(which
may
still
be
a
nonspecific
“zillion”
for
young
children),
children
still
know
that
three
of
them
plus
two
of
them
makes
five
of
them,
but
that’s
of
little
use
if
“three
sevens”
does
not
(yet)
have
meaning.
Once
a
child
does
have
meaning
for
“three
sevens”
and
that
meaning
is
a
specific
number
(even
if
the
child
doesn’t
yet
remember
which
number),
the
child’s
long-­‐standing
logic/intuition/linguistic
knowledge
that
“three
sevens
plus
two
sevens
is
five
sevens”
becomes
arithmetically
usable.
©
Education
Development
Center,
Inc.
page
4

5. E.
Paul
Goldenberg,
June
Mark,
and
Al
Cuoco
The
algebra
of
little
kids
The
meaning
of
“three
sevens”
might
be
given
in
several
ways:
as
an
image
,
or
a
sum,
7
+
7
+
7,
or
a
product
3
×
7,
or
in
other
ways.
Each
way
has
something
threeish
and
something
sevenish
about
it.
Because
7
+
7
+
7
and
3
×
7
are
both
language,
such
expressions
are
best
introduced
as
(mathematical)
descriptions
of
a
situation—for
example,
the
array
image—that
communicates
partly
without
analyzing
the
language
formally.
The
image,
of
course,
requires
some
analysis,
too—visual
rather
than
linguistic—
to
see
the
three
sevens.
To
connect
“three
sevens”
with
21,
the
“normal”
name
for
that
number,
we
must
agree
that
what
makes
“seven”
is
its
seven
squares.
Then
is
21
because
of
its
21
squares,
but
it
is
also
a
picture
of
three
sevens:
a
multiplication
fact.
Similarly,
if
is
“seven,”
then
is
two
sevens.
The
picture
shows
that
three
sevens
and
two
sevens
make
five
sevens.
In
spoken
form,
“three
sevens
plus
two
sevens
make
five
sevens”
is
familiar.
The
pictures
support
the
semantics
of
the
situation,
helping
to
establish
the
role
of
sevens
and
preserve
its
numerical
meaning
rather
than
letting
it
degenerate
into
a
non-­‐numeric
object,
like
sheep.
But
the
classical
written
form—(3
×
7)
+
(2
×
7)
=
5
×
7—is
quite
another
story.
Spoken
symbols
vs.
written
symbols
Knowing
that
the
finger
collections
and
can
be
described
by
the
same
number
does
not
guarantee
that
a
child
will
know
that
the
print
statements
5
+
2
and
2
+
5
refer
to
the
same
number.
The
written
language
of
mathematics
presents
challenges
that
can
be
finessed
by
spoken
language
and
by
appropriate
visual
presentations.
Perhaps
the
most
glaring
example
is
the
canonical
wrong
fourth-­‐grade
response
to
8 + 8 = ? .
No
first
grader
3 2
would
ever
say
“five
sixteenths.”
It’s
uninformative—in
fact,
misleading—to
“explain”
such
errors
simply
by
claiming
that
these
expressions
are
“too
abstract”
or
that
children
“can’t
handle
symbols.”
Spoken
words
are
symbols,
too,
and
words
like
the—which
young
children
use
flawlessly—are
about
as
abstract
as
one
can
get.
It’s
worth
understanding
the
difference
between
=
and
5
+
2
=
2
+
5
to
see
why
the
challenge
of
print
for
children
may
not
be
a
mathematical
challenge.
Humans
have
evolved
to
be
quite
flexible
about
visual
order
and
orientation,
but
in
the
life
of
any
individual
human,
it
takes
some
learning.
Infants
who
have
come
to
recognize
a
bottle
when
it
is
handed
to
them
in
the
proper
orientation
do
not,
at
first,
reach
for
it
when
it
is
handed
to
them
in
some
unfamiliar
orientation
(e.g.,
with
the
nipple
visible,
but
facing
away
like
this
).
But
very
soon
they
do
learn
to
recognize
objects
regardless
of
their
orientation.
When
you
consider
the
visual
processing
required,
this
is
quite
an
impressive
accomplishment.
Even
if
the
bottle
is
presented
in
the
same
orientation
but
at
different
distances,
very
Figure
1:
In
this
photo,
different
images
are
projected
onto
the
retina.
The
distortion
of
the
distance
from
the
tip
of
the
nipple
to
the
parts
relative
to
each
other
can
be
extreme,
and
yet
the
baby
bottle
is
the
same
as
the
recognizes
all
of
these
projections—most
of
them
never
seen
length
of
the
entire
rest
before—as
the
same
object.
of
the
bottle.
Measure
to
see
for
yourself!
©
Education
Development
Center,
Inc.
page
5

6. E.
Paul
Goldenberg,
June
Mark,
and
Al
Cuoco
The
algebra
of
little
kids
Though
this
complex
neural
computation
needs
data
(learning)
to
tune
it
up,
the
ability,
itself,
is
wired
in.
This
evolutionary
gift
is
essential
for
survival.
Otherwise,
we’d
have
been
meals
for
tigers
we
didn’t
recognize
because
they
didn’t
happen
to
be
facing
exactly
the
same
way
as
first
we
saw
them!
For
our
ancestors,
it
was
necessary
to
“see”
the
same
object
despite
different
retinal
images,
as
long
as
those
images
could
be
made
“the
same”
under
rotation,
reflection,
dilation,
or
certain
projective
transformations,
and
so
our
brains
are
adept
at
them.
(The
spatial
tests
that
some
people
find
quite
difficult
are
a
very
different
sort
of
thing.
The
“look-­‐alike”
objects
on
these
tests
require
an
analysis
that
goes
beyond
what
was
evolutionarily
useful.
Our
ancestors
didn’t
care
if
the
tiger
was
left-­‐handed!)
But
those
ancestors
didn’t
read.
The
letters
d,
b,
q,
and
p
are
the
same
shape
and
differ
only
by
rotation
or
reflection.
To
read,
children
must
learn
to
see
them
as
different
objects,
not
as
the
same
object
in
different
orientations.
Young
children’s
letter
reversals
are
not
neurological
failures
at
all—seeing
that
way
is
one
of
evolution’s
gifts—but,
just
for
this
one
purpose
of
decoding
print,
children
must
unlearn
a
principle
that
applies
to
nearly
everything
else
they
will
encounter
during
their
entire
life.
They
must
treat
print
as
an
exception
to
the
usual
rules
of
seeing.
Moreover,
w as
and
s aw—each
just
three
print-­‐squiggles
arranged
in
a
different
order—must
not
be
recognized
as
“the
same.”
Alas,
then
come
2 +5
and
5 +2,
two
perfectly
good
examples
of
print-­‐squiggles
that
are
to
be
treated
as
“the
same.”
(As
always,
the
truth
is
not
so
simple.
On
a
number
line,
numbers
represent
addresses—the
names
of
specific
points/locations
along
the
line—and
also
distances
between
addresses.
The
child
who
“enacts”
2 +5,
perhaps
by
jumping
along
a
large
number
line
on
the
floor
would
enact
5 +2
differently.)
It
is
therefore
not
surprising
that
the
notation,
in
some
contexts,
can
cause
confusions,
but
this
is
an
issue
of
notation,
not
of
concept.
Print
is
just
plain
different!
Similarly,
the
picture
lets
children
see
what
written
descriptions
like
(3
×
7)
+
(2
×
7)
=
(3
+
2)
×
7
or
(3
×
7)
+
(2
×
7)
=
5
×
7
typically
leave
opaque,
unless
they
are
written
as
an
abbreviated
version
of
language
the
children
themselves
are
using
to
describe
the
picture.
But
the
difficulty
is
with
the
notation—a
difficulty
with
the
manner
in
which
the
underlying
mathematical
idea
is
being
communicated—not
a
lack
of
the
idea
itself.
In
fact,
the
way
that
teachers
of
kindergarten
and
early
first
grade
teach
writing
could
help
them
teach
this
symbolic
language,
too:
children
tell
stories,
and
the
teacher
encodes
their
language
in
writing.
Here,
children
might
describe
how
a
three-­‐by-­‐seven
array
can
be
put
with
a
two-­‐by-­‐seven
array
to
make
a
five-­‐by-­‐seven
array,
and
the
teacher
can
be
writing
(3
×
7)
+
(2
×
7)
=
(5
×
7)
as
the
children
speak.
Before
that
can
happen,
children
need
to
have
the
idea
that
we
can
name
the
arrays,
and
that
one
useful
name
for
is
(3
×
7).
Imagine
that
array
to
be
on
a
card
we
hold
in
our
hands.
That
card
can
be
held
in
any
position
at
all—vertically,
slantwise,
horizontally—and
is
still
the
same
card.
It
makes
sense
to
give
it
the
same
name
no
matter
which
way
we
hold
it.
We
could
also
have
called
it
(7
×
3),
or
even
21
(or
a
zillion
other
things,
like
“half
of
6
×
7”
if
we
had
a
6
×
7
array
that
we
had
already
named).
So
(3
×
7)
=
(7
×
3)
=…
The
visual
idea
and
the
symbols
that
describe
what
the
children
see
are
not
yet
fully
generic—not
yet
a
property
of
+
and
×
that
can
be
used
in
syntactic
manipulations
of
©
Education
Development
Center,
Inc.
page
6

7. E.
Paul
Goldenberg,
June
Mark,
and
Al
Cuoco
The
algebra
of
little
kids
strings
of
symbols
to
generate
(a
×
c)
+
(b
×
c)
from
(a
+
b)
×
c
or
vice
versa.
In
fact,
there
are
so
many
parts
to
keep
track
of
that
doing
so
is
not
trivial.
Getting
good
enough
to
recognize
and
use
this
valuable
property,
even
with
arrays
as
a
particularly
powerful
representation,
takes
time
and
practice.
But
the
underlying
idea
is
there
very
early,
as
part
of
the
child’s
cognitive
structure,
as
soon
as
the
child
can
meaningfully
make
statements
like
“two
sheep
plus
three
sheep
are
five
sheep.”
Again,
the
underlying
idea
must
be
there
before
any
practice
of
it
can
make
sense.
Written
symbols
often
present
major
challenges
that
the
spoken
symbols
do
not.
Possibly
because
of
print’s
special
status,
the
logic
that
children
apply
when
information
is
presented
in
spoken
symbols
may
not
be
applied
when
the
same
information
is
presented
in
print.
The
canonical
error
with
fractions
is
a
perfect
example:
The
spoken
question
“what’s
three
eighths
plus
two
eighths”
focuses
attention
on
“three
plus
two”
and
tends
to
evoke
the
correct
reasoning
and
get
the
correct
answer;
by
contrast,
the
written
question
3
8
+ 8 = ?
doesn’t
focus
attention
only
on
the
top
numbers.
Children
for
whom
the
meaning
is
2
not
already
strongly
established
tend
to
interpret
the
plus
sign
as
“add
everything
in
sight.”
In
fact,
mathematical
reading
and
writing
are
quite
different
from
prose
reading
and
writing.
For
prose,
we
proceed
in
a
line,
strictly
left
to
right.
Even
top-­‐to-­‐bottom
movement
just
accommodates
the
limited
width
of
a
page;
it
gives
no
information
that
would
not
have
been
present
if
the
writing
were
strung
out
in
one
dimension—a
line—on
a
very
wide
scroll
of
paper.
(The
real
story
is,
of
course,
more
complex.
Strict
left
to
right
reading
applies
only
at
the
very
earliest
stages,
if
at
all.
A
fluent
reader,
largely
without
conscious
awareness,
takes
in
much
more
of
the
sentence
than
a
strictly
left-­‐to-­‐right
approach
would
give.)
By
contrast,
bar
graphs,
coordinate
graphs,
histograms,
charts
and
tables,
and
the
like
are
two-­‐dimensional
records.
One
must
attend
to
horizontal
and
vertical
position
in
order
to
interpret
the
information
they
contain.
Even
symbolic
expressions
can
require
attention
to
vertical
as
well
as
horizontal
position:
32
is
not
the
same
as
32.
Moreover,
mathematical
writing
that
is
just
horizontal
are
not
to
be
read
strictly
left
to
right:
2
×
(3
+
5),
7
+
6
÷
2,
and
7
+
___
=
5
+
4
all
require
attention
to
the
right
side
before
attention
to
the
left.
In
fact,
7
+
6
÷
2
requires
both
left-­‐to-­‐right
and
right-­‐to-­‐left
analysis:
6
÷
2
must
be
evaluated
left-­‐to-­‐right
(because
Figure
2:
Bar
graphs,
among
the
2
÷
6
is
different),
and
yet
the
convention
about
order
of
earliest
graphs
children
make,
require
attention
to
two
operations
dictates
that
the
6
÷
2
part
must
be
evaluated
dimensions:
which
bar
(horizontal
before
the
addition
that
is
specified
by
“7
+
.”
position)
and
the
bar’s
height.
Algebra
as
a
language
for
expressing
what
we
know
Algebraic
notation
is
used
in
two
distinct
ways:
for
describing
what
we
know,
and
for
deriving
what
we
don’t
know.
In
the
first
use,
algebra
is
a
language
for
describing
the
structure
of
a
computation,
a
numerical
pattern
we’ve
observed,
a
relationship
among
varying
quantities,
and
so
on.
Young
children
are
phenomenal
language
learners!
©
Education
Development
Center,
Inc.
page
7

8. E.
Paul
Goldenberg,
June
Mark,
and
Al
Cuoco
The
algebra
of
little
kids
Exercises
like
the
one
in
Figure
3,
but
without
the
leftmost
column,
are
familiar
enough
in
many
curricula.
Children
look
for
a
pattern
in
the
inputs
and
outputs,
figure
out
a
rule,
and
complete
the
table.
Think
Math!
often
adds
a
“pattern
indicator”
(the
first
column)
to
problems
of
this
kind.
When
Michelle,
a
second
grader
in
a
Think
Math!
classroom
finished
filling
out
this
table
before
I
had
finished
handing
out
copies
to
all
the
children,
I
asked
her
how
she
had
done
it
so
fast.
She
said
“Well,
I
saw
it
was
take-­‐away
8
because
I
looked
at
the
28
and
20,
and
then
I
saw
that
10
and
2
was
take-­‐away
8
again,
and
then
I
saw
8
and
0.”
n
10
8
28
18
17
58
57
n
–
8
2
0
20
3
4
Figure
3:
A
“pattern
indicator”
gains
meaning
from
context
when
it
accompanies
a
“find-­‐a-­‐rule”
exercise.
And
then
she
grinned
as
if
I
had
left
the
“clue”
by
accident,
pointed
to
the
left
column
and
added
“Besides,
it
says
that
right
here!”
How
did
Michelle
know?
Though
the
algebraic
language
was
there,
nobody
ever
discussed
“variables”
or
“letters
standing
for
numbers”
or
even
mentioned
that
column.
Had
Michelle
seen
just
the
table
in
Figure
4,
with
no
examples
to
infer
from,
she
most
likely
would
not
have
felt
the
symbols
“said”
anything.
But
after
she
discovered
the
pattern,
the
symbols
looked
“close
enough”
to
mean
the
same
thing,
and
so
she
then
assigned
them
that
meaning.
n
18
17
58
57
n
–
8
3
4
Figure
4:
A
“pattern
indicator”
without
a
pattern
from
which
to
infer
its
meaning
would
just
be
more
to
learn.
In
other
words,
she
did
what
little
children
excel
at:
she
learned
language
(in
this
case
“n
–
8”)
from
context.
If
algebraic
language
is
part
of
the
environment,
used
where
context
gives
it
meaning,
children
can
apply
their
natural—and
extraordinary—language-­‐learning
prowess
to
it,
and
learn
to
use
it
descriptively.
Just
as
children
learning
their
native
language
understand,
at
first,
more
than
they
can
say,
Michelle
could
not
immediately
produce
such
descriptive
language,
but
she
and
others
try
these
interesting
ways
of
writing
down
what
they
know
and,
over
time,
become
good
at
it.
Fourth
graders
learn
a
number
trick:
Think
of
a
number;
add
3;
double
that;
subtract
4;
cut
that
in
half;
subtract
your
original
number;
aha,
your
result
is
1!
They
love
it
and
want
to
do
it
to
their
parents
and
friends.
They
also
want
to
know
how
it
works,
so
we
add
pictures.
When
we
say
Think
of
a
number,
we
picture
a
bag
with
that
number
of
grapes
in
it:
.
For
add
3,
we
picture
and
double
that
becomes
.
This
act
of
doubling,
which
most
fourth
graders
find
quite
natural
and
“obvious,”
is,
again,
the
distributive
property
in
action.
While
the
expression
2(b
+
3)
does
not
make
obvious
what
the
result
is,
children
do
readily
learn
to
describe
the
picture
as
“two
bags
plus
6”
and
abbreviate
that
description
as
2b
+
6.
We
don’t
talk
about
“variables”
or
“letters
standing
for
numbers”;
we
simply
describe
what
we
know,
and
write
it
down
as
simply
as
we
can.
(See
a
detailed
description
of
the
activity
with
children
at
http://thinkmath.edc.org/index.php/Algebraic_thinking
and
see
Sawyer
(1964)
for
the
original
source
of
this
idea.)
June
Mark,
et
al.,
(2009)
describe
yet
another
way
in
which
Think
Math!
gives
students
this
algebra-­‐as-­‐description-­‐of-­‐what-­‐you-­‐
know
experience.
©
Education
Development
Center,
Inc.
page
8

9. E.
Paul
Goldenberg,
June
Mark,
and
Al
Cuoco
The
algebra
of
little
kids
So
why
don’t
we
teach
algebra-­the-­course
in
grade
4?
Because
that
other
use
of
algebra—deriving
what
we
don’t
know—is
a
formal
syntactic
operation
on
a
set
of
symbols,
and
children
are
(generally)
not
able
to
divorce
symbols
from
meanings
before
roughly
age
12.
This
is
not
because
they
cannot
handle
“symbolic”
or
“abstract”
things—words
are
symbols;
pictures
are
symbols;
little
children
can
be
symbolic
and
abstract
from
babyhood—but
because
the
use
of
the
symbols
is
different.
Formal
operations
on
strings
of
algebraic
symbols—rearranging
them,
apart
from
their
semantics,
to
create
other
strings
of
symbols
that
“solve”
a
problem—are,
well,
formal
operations,
and
children
are
not,
by
and
large,
formal
operational
before
11,
and
not
reliably
so
before
about
13,
whence
the
common
need
to
wait
until
that
age
for
“algebra.”
But
only
that
part
of
algebra
that
requires
deduction
by
formal
rules
needs
to
wait
that
long.
The
part
of
algebra
that
is
expressive
of
what
we
already
know—that
is,
essentially,
a
shorthand
for
semantic
content
clearly
tied
to
a
context
we
already
understand—that
part
can
be
learned
earlier.
It
is
just
language
to
express
oneself,
and
children
are
excellent
language
learners.
They
don’t
learn
language
from
explanations
or
formal
lessons;
they
learn
it
from
use
in
context.
And,
if
is
it
learned
all
along,
as
it
becomes
developmentally
possible,
then,
when
the
child
is
in
late
middle
school,
the
transition
to
the
new
use
of
that
language
for
deductive
purposes
could,
presumably,
be
much
easier,
much
more
accessible
for
all
children,
much
less
of
a
brick
wall
of
a
million
seemingly
new
things
to
learn
all
at
once.
What
does
this
tell
us
about
elementary
school
teaching
and
learning?
Taking
advantage
of
children’s
natural
algebraic
ideas
and
honing
them
is
a
focus
on
habits
of
mind,
rather
than
on
rules
that
can
otherwise
seem
arbitrary.
The
precursors
of
commutative
and
distributive
properties
that
we
described
earlier
do
need
to
be
refined,
honed,
extended,
practiced,
codified,
and
generalized,
but
they
are
already
there
as
“natural”
logic,
the
child’s
natural
habits
of
mind
and
the
building
blocks
of
higher
mathematics.
If
children
are
to
become
competent
at
mathematics,
including
arithmetic,
those
habits
of
mind
must
take
precedence
over
rules,
formulas,
and
procedures
that
do
not
derive
from
logic
that
the
child
can
grasp.
In
fact,
children
can
grasp
a
lot
more
if
the
foundations
for
their
learning
are
grounded
in
their
logic,
which
gives
them
all
the
tools
to
understand,
not
just
memorize,
the
algorithms
for
arithmetic
with
whole
numbers
and
fractions.
But
we
all
see
the
dramatically
disappointing
results
of
“learning”
rules
without
understanding:
they
are
easy
to
mix
up
and
result
in
procedures
that
don’t
work.
Organizing
the
arithmetic
part
of
the
elementary
school
mathematics
curriculum
around
mathematical
habits
of
mind
would
not
shift
the
curriculum
dramatically
in
content,
except
to
give
more
attention
to
mental
arithmetic
than
is
usual.
Paper
and
pencil
methods
are
engineered
to
make
the
work
easy,
to
reduce
the
cognitive
load,
the
amount
of
thinking
one
needs
to
do,
of
calculation.
Judiciously
chosen
mental
arithmetic
both
exercises
and
depends
on
mathematical
ways
of
thinking
that
the
paper-­‐and-­‐pencil
algorithms
deliberately
try
to
avoid,
mathematical
ways
of
thinking
that
are
the
backbone
of
the
algebra
that
we
want
to
prepare
children
to
succeed
at.
What
would
shift
is
the
order
in
which
we
acquire
that
content.
Instead
of
being
the
preparatory
step
for
computing,
algorithms
become
the
culmination
of
understanding
how
the
computation
works,
another
case
of
describing
what
we
already
know,
and
abbreviating
that
description.
©
Education
Development
Center,
Inc.
page
9

10. E.
Paul
Goldenberg,
June
Mark,
and
Al
Cuoco
The
algebra
of
little
kids
References
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Math. Behav. 15(4):375-402. December, 1996.
Cuoco, A., Goldenberg, E. P., and J. Mark. “Organizing a curriculum around mathematical habits of mind.”
Mathematics Teacher. (submitted)
Education Development Center, Inc. (EDC). Think Math! comprehensive K-5 curriculum. Boston: Houghton
Mifflin Harcourt. 2008.
Feigenson, L., Carey, S., & Spelke, E. (2002). Infants’ discrimination of number vs. continuous extent. Cognitive
Psychology, 44, 33–66.
Goldenberg, E. Paul. “‘Habits of mind’ as an organizer for the curriculum” J. of Education 178(1):13-34, Boston
U. 1996. (Also “‘Hábitos de pensamento’ …”Educação e Matemática, 47 March/April, & 48 May/June, 1998.)
Goldenberg, E. Paul & N. Shteingold. “Mathematical Habits of Mind.” In Lester, F., et al., eds. Teaching
Mathematics Through Problem Solving: prekindergarten–Grade 6. Reston, VA: NCTM. 2003.
Goldenberg, E. Paul & N. Shteingold “The case of Think Math!” In Hirsch, Christian, ed., Perspectives on the
design and development of school mathematics curricula. Reston, VA: NCTM. 2007.
Gopnik, A., Meltzoff, A., and P. Kuhl. The scientist in the crib: what early learning tells us about the mind. New
York: HarperCollins. 2000.
Mark, J., Cuoco, A., and Goldenberg, E. P. “Developing mathematical habits of mind in the middle grades.”
Mathematics Teaching in the Middle School. (submitted)
Piaget, J. The child’s conception of number. London: Routledge and Kegan Paul. 1952.
Sawyer, W. W. Vision in elementary mathematics. New York: Dover Publications. 2003 (1964).
Sfard, A. Thinking as Communicating. New York, NY: Cambridge University Press. 2008.
Wirtz, R., Botel, M., Beberman, M., and W. W. Sawyer. 1964. Math Workshop. Encyclopaedia Britannica Press.
©
Education
Development
Center,
Inc.
page
10