اختلاف التعليم في مهام الرياضيات

Professor
Glenda
Anthony

Massey
University

Feb
2013

1

Why
differen+ate
instruc+on?

— Engage
all
students
in
instruction
and

learning

— Enables
teachers
to
consider
the
different

ways
that
students
learn
without
pre-‐
defining
their
capacity
for
learning

— Values
to
diversity
of
students’
thinking

2

Diﬀeren+a+on
is

— “an
organized
yet
ﬂexible
way
of
proactively

adjusting
teaching
and
learning
to
met
kids

where
they
are
and
help
to
achieve

maximum
growth
as
learners”

(Tomlinson,
1999)

3

Valuing
the
diversity
of
students’

thinking

— In
one
cupboard
you
have
3

shelves
with
4
boxes
on
each

shelf.

— There
are
3
cupboards
in
the

room.

— How
many
boxes
are
stored
in

the
room?

4

— What
diﬀerent
ways
could

students
respond
to
the

problem?

5

How
can
you
make
use
of
the

diﬀeren+ated
responses?

— Anticipate
as
many
responses
as
possible
–

knowing
your
students.

— Understand
the
mathematics
represented

in
the
diﬀerent
student
responses.

— The
instructional
decision
and
interaction

with
students
must
be
responsive
to
their

mathematical
ideas,
strategies,
and

communication.

6

What
instruc+onal
response
is

appropriate
for
these
scenarios

—  Liam
raises
his
hand
and
waits
for
the
teacher
to
help
him.

—  Angela
draws
a
picture
of
one
of
the
cupboards
and
counts

the
boxes.

—  Dan
draws
a
picture
of
3
cupboards
and
counts
the
3

shelves
in
each
but
not
the
boxes.

—  Tara
uses
addition
and
writes
4+4
+4
+4
+4
+4
+4
+4+4.

—  John
uses
addition
and
writes
4
+4+4
=12;
12+12+12
=
36

—  Rebecca
uses
multiplication
and
addition
writing

3
x
4
=
12
and
12+
12+
12
=
36.

7

Principles
to
diﬀeren+a+ng
tasks

1.  The
focus
on
instruction
must
be
on

the
big
ideas
being
taught.

2.  There
must
be
some
aspect
of
choice

for
the
student:
in
content,
in

process,
or
product.

3.  Prior/formative
assessment
to

determine
the
need.

8

Two
core
strategies:

Open
ques+ons
and
parallel
tasks

— An
open
task/question

involves
a

variety
of
possible
responses
or

approaches.

9

Comparing
open
and
closed
task

Question
1:

—  To
which
fact
family
does
the
fact
3
x
4
=
12
belong?

Question
2:

—  Describe
the
picture
below
by
using
a
mathematical

equation
/
sentence?

X

X

X

X

X

X

X

X

X

X

X

X

10

How
can
we
create
open

ques+ons?

— Turning
around
the
question

— Asking
for
similarities
and
diﬀerence

— Replacing
a
number
with
a
blank

— Asking
for
a
number
sentence

— Changing
the
question

11

Turning
around
a
ques+on

— Give
the
answer
and
ask
for
the
question.

— How
could
you
turn
around
:

(i)

3
+
4

=
?

(ii)

What
is
half
of
20?

12

Asking
for
similari+es
and

diﬀerences

— How
is
the
number
85
and
100
alike/
diﬀerent?

— How
are
10
and
12
alike?

— How
is
the
number
√2
and
√5
alike
?

13

Asking
for
a
number
sentence

— Create
a
sentence
that
includes
the

numbers
3
and
4
along
with
the
words
‘and’

and
‘more.’

— Create
a
sentence
that
includes
the
words

“linear’
and
“increasing’
as
well
as
the

numbers
4
and
9.

14

Changing
the
ques+on
in
the
text

— Rae
has
4
boxes
of
pencils.
There
are
6

pencils
in
each
box.
How
many
pencils

does
Rae
have?

15

Rae
has
4
boxes

of
pencils.
There

are
6
pencils
in

each
box.

How
many

pencils
does
Rae

have?

Rae
has
some

boxes
of

pencils.
There

are
2
more

pencils
in
each

box
than
the

number
boxes.

How
many

pencils
does

Rae
have
in
all?

16

Try
changing

— A
biscuit
has
a
diameter
of
5.75
cm
.
Express

the
diameter
as
a
fraction
in
simplest
form.

17

What
about
other
areas
other
than

number?

— Big
idea
in
Measurement:
The
same
object
can
be

described
by
using
diﬀerent
measurements.

— Open
question:
Which
shape
is
bigger
how
do

you
know?

18

Early
algebra

— A
pattern
begins
like
this:

2,
6,
...

How

might
it
continue?

— How
could
you
adapt
this
problem?

19

Recap

— What
deﬁnes
an
open
question?

— What
is
important
for
the
teacher
to

consider?

— What
is
important
in
receiving
the
answers

to
open
questions?

— What
to
avoid
in
an
open
question

20

Parallel
tasks

— Sets
of
tasks,
usually
two
or
three,
that
are

designed
to
meet
the
needs
of
students
at

diﬀerent
developmental
levels
BUT
get
at

the
same
big
idea
and
are
close
enough
in

context
that
they
can
be
discussed

simultaneously.

— What
are
the
beneﬁts?

21

Big
idea:
recognising
when

mul+plica+on
is
appropriate

Option
1:

Create
a
word
problem

that
could
be
solved
by

multiplying
two
one-‐
digit
numbers.

Option
2:

Create
a
word
problems

that
could
be
solved
by

multiplying
two

numbers
between
10

and
100.

What
discussion
questions
might
follow?

22

Crea+ng
a
parallel
task

— Variations
that
allow
struggling
students
to
be

successful
and
proficient
students
to
be

challenged.

— Important
to
think
about
how
students
might

differ
developmentally
in
approaching
the
idea.

— Differences

in
task:

— what
operations
the
students
might
use
?

— size
of
numbers
they
can
handle?

— What
meanings
of
an
operation
make
sense
to

students?

23

A
parallel
task

Option
1:

There
were
583

students
in
Ira’s

school
in
the

morning.
199
of
the

Year
3
students

went
on
a
trip.

How
many

students
are
left
in

the
school?

Option
2:

There
are
61
Year

2
students
in
Ira’s

school.
19
of
them

are
in
the
library.

How
many
Year
2

students

are
left

in
their

classrooms?

24

Parallel
tasks:

583-‐199

and
61-‐
19

—  Attend
to
the
‘what
is
the
same
and
what
is
diﬀerent’

—  How
did
you
know
that
most
of
the
students
were
left?

—  How
did
you
decide
how
many
were
left?

—  I
notice
that
Tui
solved
it
by
subtracting.
Why
does

subtraction
make
sense?

—  I
notice
that
Lisa
solved
it
by
adding.
Why
might
adding

make
sense?

—  How
would
your
answer
have
changed
if
one
more
student

had
left?

—  How
would
your
answer
have
changed
if
there
had
been

one
extra
student
to
start
with?

—  How
would
your
answer
have
changed
if
there
was
an
extra

student
to
start
with,
but
one
extra
student
left?

—  Which
problem
is
easier
for
you
to
solve?

25

Measurement:
Knowledge
of
the
size
of

benchmarks
assists
in
measuring

A
table
is
5
pencils
long.

—  How
many
paper
clips
long
would
it
be?

—  How
many
centimetres
long
would
it
be?

—  Questions?

26

Big
idea
in
Number:

There
are
many
ways
to
represent
numbers

—  Draw
a
picture
to
show
two
equivalent
fractions
for

2/8.

—  Two
fractions
are
equivalent.
If
you
add
the

numerators,
the
result
is
22
less
than
if
you
add
the

denominators.
What
could
the
fractions
be?

27

Linear
versus
area
measurement

—  Someone
suggests
that
the
school
driveway
is

4,ooo,ooo
mm
long.
Is
it
a
long
driveway?

—  Someone
suggest
that
a
shopping
mall
might
be

4,000,000
cm2
in
area.
Do
you
think
that
is

reasonable?

28

Newspapers
:
Adap+ng
text
ques+ons

—  Suppose
4
students
were
delivering
176
newspapers

and
decided
to
share
the
task
evenly.
How
many

papers
would
each
deliver?

—  PARALLEL:

—  Suppose
2
students
were
delivering
24
newspapers

and
decided
to
share
the
task
evenly.
How
many

papers
would
each
deliver?

29

What
are
some
cri+cal
discussion
ques+ons?

—  What
operation
did
you
use
to
decide
how
many

papers
each
person
would
deliver?

—  Why
would
you
use
that
operation?

—  Is
there
another
way
you
could
have
worked
out
the

answer?

—  How
did
you
know
that
each
person
had
to
deliver

more
than
10
papers?

—  How
did
you
know
that
each
person
had
to
deliver

fewer
than
100
papers?

—  How
did
you
ﬁgure
out
how
many
papers
each

student
had
to
deliver?

30

What
about
choice?

—  Sometimes
the
teacher
should
decide
who
does
what

but
most
of
the
time
allow
the
students
to
choice.

—  Choice
is
very
empowering.

31

Summing
up
parallel
tasks

— Generated
from
a
single
original
task
by
changing

the
complexity
of
the
numbers,
shapers,
graphs,

patterns,
equations,
or
measurements
being

employed
or
the
complexity
of
the
situations

being
addressed.

— The
context
being
the
same
allows
common

discussion.

— It
is
important
to
set
up
the
situation
so
that
there

are
common
questions
beyond
simple
What
did

you
do?

32

1.  Moves
that
a
teacher
uses
before
a

students
arrives
at
a
correct
answer
.

2.  Extending
moves
that
a
teacher
uses

after
a
correct
answer.

—  Adapted
from
Jacobs,
V.,
&
Ambrose,
R.
(2008).
Making
the
most
of
story
problems.
Teaching
Children
Mathematics,
15,
260-‐266.

Teacher
ac+ons

to
support
and

extend
students’
thinking

— Ask
her
to
explain
what
she
knows
about

the
problem
–
possibly
with
the
use
of

manipulatives

-‐
“model
of”

— Rephrase
the
problem

— Use
a
more
familiar
or
personalized
context

(put
yourself
or
her
in
the
story)

— Suggest
a
diﬀerent
representation
-‐
acting

out
etc.

Ensure
that
the
child
understands

the
problem

— Easier
numbers

-‐
then
back
to
harder

numbers,
looking
for
generalisation
of

structure,
strategy.

— Fold
back
to
easier
mathematical
structure:

Jan
had
12
cookies.
Sue
had
3
cookies.
How

many
more
cookies
than
Sue
did
Jan
have?

— What
would
be
an
easier
problem
that
still

involves
12-‐3?

Change
the
mathema+cs
to
match

the
child’s
level
of
understanding

— What
sort
of
questions
can
you
use?

— Are
these
the
same
sorts
of
questions

that
children
can
use
in
their
groups?

Explore
what
the
child
has
already

done

Sometimes
need
to
give
permission
to

move
on
and
try
an
alternative
way

— changing
a
representation,

— trying
a
new
tool,

— reminding
of
a
strategy
used
in
the

past.

Remind
the
child
to
use
other

strategies

Promoting
reﬂection
on
the
strategy
just

completed
(i.e.,
view
the
problem
solving
as

a
context
for
having
a
mathematical

conversation)

— Asking
for
a
strategy
explanation
or

clariﬁcation.

Ways
to
respond
aer
a
correct

answer

—  This
morning
I
had
some
muesli
bars.
Then
I
gave
you

five
muesli
bars.
Now
I
have
six
muesli
bars
left.
How

many
bars
did
I
have
this
morning
before
I
gave
some

to
you?

“Five
pus
five,
if
you
took
one

away
,
is
ten
and
then
one
more

is
eleven,
so
you
had
eleven.”

Be
specific
to
the
details

What
would
you
ask
to
probe
the
child’s

thinking
?

— “I
don’t
know.
I
just
added
them

together.”

Promo+ng
reﬂec+on

—  Generate
another
way

—  Think
of
another
way
that
is
connected
to
the
ﬁrst

strategy.

—  Asking
for
a
mental
strategy
that
is
an
abstraction
of

work
with
manipulative
–
could
you
solve
it
in
your

head?

—  Explicitly
compare
and
contrast
strategies.

—  Comparing
a
successful
strategies
to
an
earlier

unsuccessful
strategy.

Explora+on
of
mul+ple
strategies

and
the
mathema+cal
connec+ons

— Record
the
strategy
that
you
used

to
solve
the
problem.

— Generate
a
number
sentence
that

goes
with
the
problem.

Connect
the
children’s
thinking
to

symbolic
nota+on

— Build
on
the
children’s
thinking.

Rene
was
collecting
tomatoes.
She
has
nine

baskets,
and
she
put
ten
tomatoes
in
each

basket.
So
how
many
tomatoes
did
she
have

altogether?

— How
could
you
extend/build
on
this
thinking?

Generate
follow
up

problems

—  Oh,
I
get
it,
Well
there’s
already
ten
in
each
basket
so

that’s
90.

—  So
I
count
up
nine,
one
more
nine.

—  I
mean
nine
ones.

—  I
‘m
going
to
add
nine
ones.

—  So
there’s
already
ninety,
so
ninety-‐one,
ninety-‐
two....ninety-‐nine.

—  What
big
idea
has
been
advanced

— 9
x
11
=
9
x
(10
+
1)
=
9
x
10
+
9

Response

—  Ensure
child
understands
the
problem

—  Change
the
maths
to
match
understanding

—  Explore
what
they
have
already
done

—  Remind
child
to
use
other
strategies

—  Promote
reﬂection
on
strategy
just
completed

—  Explore
multiple
strategies
and
connections

—  Connect
child’s
thinking
to
symbolic
notation

—  Generate
linked
follow-‐up
problems.

Teacher
moves

(Summary)

اختلاف التعليم في مهام الرياضيات

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