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9
,
20]
s
o
th
at
a
n
y
o
n
e
w
h
o
w
an
ts
a
n
o
p
in
io
n
ca
n
r
esp
ec
t
o
th
er
s
.
B
esid
es,
it
ca
n
tr
ai
n
s
t
u
d
en
ts
to
w
o
r
k
w
i
th
o
t
h
er
s
tu
d
e
n
ts
.
T
h
e
ex
p
er
ien
ce
o
f
m
at
h
e
m
atic
s
,
ex
p
lan
atio
n
s
o
f
m
ath
e
m
atics,
m
a
th
e
m
atica
l
d
if
f
er
en
ce
s
,
an
d
m
at
h
e
m
atica
l
co
m
m
u
n
ica
tio
n
ca
n
f
o
r
m
s
o
cio
m
a
t
h
e
m
ati
c
n
o
r
m
s
[9
,
17
,
1
8
]
.
Ma
th
e
m
a
tical
ex
p
er
ien
ce
i
n
g
en
er
al
th
at
t
h
e
lear
n
in
g
p
r
o
ce
s
s
f
o
r
s
t
u
d
en
t
s
s
h
o
u
ld
b
e
d
o
in
g
ma
th
,
s
o
th
e
y
n
ee
d
a
lo
t
o
f
m
at
h
e
m
a
tic
al
ex
p
er
ien
ce
t
h
at
ch
ild
r
en
h
a
v
e
b
ef
o
r
e
en
ter
in
g
s
c
h
o
o
l
[9
,
21
,
22]
.
W
ith
th
ese
e
x
p
er
ien
ce
s
,
s
tu
d
e
n
ts
h
o
p
e
to
u
n
d
er
s
ta
n
d
t
h
at
an
af
f
ir
m
atio
n
o
r
ar
g
u
m
e
n
tatio
n
m
u
s
t
al
w
a
y
s
h
a
v
e
a
r
ea
s
o
n
[
1
7
,
23]
.
T
h
e
m
a
th
e
m
atica
l
ex
p
lan
atio
n
is
a
p
r
o
ce
d
u
r
e
f
o
r
s
o
lv
in
g
m
at
h
e
m
atica
l
p
r
o
b
le
m
s
[
2
4
]
.
W
ith
th
is
m
a
th
e
m
atica
l
e
x
p
lan
atio
n
,
s
tu
d
e
n
ts
ca
n
co
m
m
u
n
icate
e
v
er
y
p
r
o
b
lem
-
s
o
l
v
i
n
g
p
r
o
ce
s
s
t
h
at
i
s
p
r
esen
ted
i
n
t
h
e
cla
s
s
[
1
8
]
.
Ma
th
e
m
atica
l
d
if
f
er
e
n
ce
s
i
n
s
o
cio
m
at
h
e
m
at
ics
n
o
r
m
s
ar
e
p
r
o
ce
s
s
es
o
f
v
alu
i
n
g
f
r
ee
d
o
m
to
e
x
p
lo
r
e
alter
n
ati
v
e
m
e
th
o
d
s
,
s
h
ar
e
t
h
eir
th
o
u
g
h
t
s
,
an
d
tak
e
th
e
r
is
k
o
f
e
x
a
m
in
i
n
g
th
e
i
r
m
is
co
n
ce
p
tio
n
s
a
n
d
er
r
o
r
s
in
r
ea
s
o
n
i
n
g
[
2
5
]
.
Ma
th
e
m
atica
l
co
m
m
u
n
icatio
n
o
f
lear
n
i
n
g
m
at
h
e
m
atics
n
ee
d
s
to
b
e
d
ev
elo
p
ed
b
ec
au
s
e
th
r
o
u
g
h
m
a
th
e
m
atica
l
co
m
m
u
n
icatio
n
s
t
u
d
en
ts
ca
n
o
r
g
an
ize
m
a
th
e
m
atica
l
t
h
in
k
i
n
g
b
o
th
v
er
b
ally
an
d
in
w
r
it
in
g
[
2
6
]
.
I
n
ad
d
itio
n
to
m
at
h
e
m
a
tical
co
m
m
u
n
icatio
n
,
s
tu
d
e
n
ts
ca
n
p
r
o
v
id
e
ap
p
r
o
p
r
iate
r
esp
o
n
s
e
s
i
n
r
esp
o
n
s
e
to
an
ar
g
u
m
en
t
f
r
o
m
a
f
r
ie
n
d
w
h
o
p
r
ese
n
ts
a
p
r
o
b
le
m
-
s
o
lv
i
n
g
p
r
o
ce
d
u
r
e
[
2
7
]
.
R
elate
d
to
t
h
i
s
,
t
h
e
p
u
r
p
o
s
e
o
f
t
h
is
s
t
u
d
y
is
to
co
n
f
ir
m
Yac
k
e
l
'
s
f
o
u
r
asp
ec
ts
,
n
a
m
e
l
y
ex
p
er
ie
n
ce
o
f
m
at
h
e
m
a
tics
,
ex
p
la
n
atio
n
s
o
f
m
ath
e
m
at
ics,
m
at
h
e
m
a
tical
d
if
f
er
en
ce
s
,
an
d
m
at
h
e
m
a
tical
co
m
m
u
n
icatio
n
w
it
h
s
o
cio
m
at
h
e
m
a
tic
n
o
r
m
s
2.
RE
S
E
ARCH
M
E
T
H
O
D
T
h
e
m
et
h
o
d
e
m
p
lo
y
ed
in
t
h
i
s
r
esear
ch
is
q
u
a
n
ti
tativ
e
[
2
8
,
29]
.
T
h
e
s
u
b
j
ec
ts
in
t
h
is
s
t
u
d
y
w
er
e
7
th
-
g
r
ad
e
j
u
n
io
r
h
ig
h
s
ch
o
o
l
s
tu
d
e
n
ts
in
J
o
g
j
ak
ar
ta
as
m
a
n
y
as
2
6
4
s
tu
d
en
ts
tak
e
n
u
s
i
n
g
c
lu
s
ter
r
an
d
o
m
s
a
m
p
li
n
g
.
T
h
is
s
t
u
d
y
u
s
e
s
C
o
n
f
ir
m
ato
r
y
Fa
k
to
r
A
n
a
l
y
s
is
in
d
ata
an
al
y
s
is
tec
h
n
iq
u
es
o
b
tain
ed
r
elate
d
to
th
e
e
x
p
er
ien
ce
o
f
m
at
h
(
P
1
)
,
ex
p
lan
at
io
n
o
f
t
h
e
m
at
h
e
m
atics
(
P
2
)
,
m
a
th
e
m
atica
l
d
if
f
er
en
ce
s
(
P
3
)
,
an
d
m
at
h
e
m
a
tical
co
m
m
u
n
ica
tio
n
(
P4
)
.
T
h
es
e
f
o
u
r
asp
ec
ts
ar
e
o
b
tain
ed
b
y
u
s
i
n
g
th
e
o
b
s
er
v
atio
n
s
h
ee
t
o
f
s
o
cio
m
at
h
e
m
a
tics
n
o
r
m
s
.
I
n
th
e
asp
ec
t
o
f
th
e
ex
p
er
ien
ce
o
f
m
at
h
,
in
d
icato
r
s
u
s
ed
ar
e
(
1
)
h
o
w
to
co
n
tr
ib
u
t
e
ac
co
r
d
in
g
l
y
a
n
d
v
al
u
ab
le
(
ite
m
n
u
m
b
er
A
1
a
n
d
A
2
)
,
an
d
(
2
)
h
o
w
to
ca
lcu
late
ac
c
u
r
ate
l
y
,
ef
f
icie
n
tl
y
,
an
d
eleg
an
tl
y
;
i
n
t
h
e
asp
ec
t
o
f
e
x
p
lan
atio
n
o
f
t
h
e
m
at
h
e
m
a
tics
(
ite
m
n
u
m
b
er
A
3
to
A
7
)
.
I
n
th
e
e
x
p
lan
a
tio
n
o
f
th
e
m
ath
e
m
at
ics,
t
h
e
in
d
icato
r
u
s
ed
is
h
o
w
to
ac
ce
p
t
ar
g
u
m
en
ts
o
r
j
u
s
ti
f
icatio
n
s
(
ite
m
n
u
m
b
er
B
8
to
B
1
2
)
.
I
n
th
e
asp
ec
t
o
f
m
at
h
e
m
ati
c
al
d
if
f
er
en
ce
s
,
th
e
in
d
icato
r
s
u
s
ed
ar
e
(
1
)
h
o
w
to
ca
lcu
late
m
ath
e
m
at
ical
d
if
f
er
e
n
ce
s
(
ite
m
n
u
m
b
er
C
1
3
)
,
an
d
(
2
)
id
en
tify
,
as
s
es
s
s
i
m
ilar
itie
s
,
d
if
f
er
e
n
ce
s
b
et
w
ee
n
v
ar
io
u
s
s
o
lu
tio
n
s
(
ite
m
n
u
m
b
er
C
1
4
to
C
1
6
)
;
o
n
th
e
m
at
h
e
m
a
tical
asp
e
cts
o
f
co
m
m
u
n
icatio
n
,
th
e
i
n
d
icato
r
s
u
s
ed
ar
e
in
tellect
u
al
a
u
to
n
o
m
y
(
I
te
m
n
u
m
b
er
D1
7
an
d
D1
8
)
,
h
o
w
to
m
ak
e
s
e
n
s
e
(
ite
m
n
u
m
b
er
1
9
)
,
h
o
w
to
ta
k
e
-
as
-
s
h
ar
ed
u
n
d
er
lie
co
m
m
u
n
icatio
n
(
I
te
m
n
u
m
b
er
D2
0
)
.
C
o
n
f
ir
m
a
to
r
y
f
ac
to
r
an
al
y
s
i
s
in
v
o
lv
e
s
th
e
s
p
ec
if
icatio
n
a
n
d
esti
m
atio
n
o
f
o
n
e
o
r
m
o
r
e
p
u
tati
v
e
m
o
d
el
s
o
f
f
ac
to
r
s
tr
u
ct
u
r
e,
ea
ch
o
f
w
h
ic
h
p
r
o
p
o
s
es
a
s
et
o
f
laten
t
v
ar
iab
les
(
f
ac
to
r
s
)
to
ac
co
u
n
t
f
o
r
co
v
ar
ian
ce
s
a
m
o
n
g
a
s
et
o
f
o
b
s
er
v
ed
v
ar
iab
les.
L
I
S
R
E
L
V
I
I
I
is
u
s
ed
to
d
escr
ib
e
alter
n
a
tiv
e
m
o
d
els
a
n
d
to
test
th
e
f
it
o
f
ea
ch
h
y
p
o
t
h
esiz
ed
m
o
d
el
ag
a
in
s
t
t
h
e
s
a
m
p
le
d
ata
[
3
0
,
3
1
]
.
M
o
d
el
s
p
ec
if
icati
o
n
is
ac
co
m
p
li
s
h
ed
b
y
f
i
x
i
n
g
o
r
co
n
s
tr
ai
n
i
n
g
e
le
m
en
ts
i
n
t
h
r
ee
m
atr
ices
th
at
ar
e
an
alo
g
o
u
s
to
t
h
e
f
ac
to
r
p
atter
n
m
atr
ix
,
f
ac
to
r
co
r
r
el
atio
n
m
atr
i
x
,
an
d
co
m
m
u
n
alit
ies
f
r
o
m
co
m
m
o
n
f
ac
to
r
an
al
y
s
i
s
.
Hig
h
er
-
o
r
d
er
f
ac
to
r
m
o
d
els
r
eq
u
ir
e
th
e
s
p
ec
if
icatio
n
o
f
an
ad
d
itio
n
al
m
atr
i
x
co
n
t
ain
i
n
g
lo
ad
in
g
s
o
f
th
e
f
ir
s
t
-
o
r
d
er
f
ac
to
r
s
o
n
th
e
h
i
g
h
er
-
o
r
d
er
f
ac
to
r
s
.
I
n
th
is
s
tu
d
y
,
th
e
an
a
l
y
s
is
p
r
o
ce
ed
s
in
th
r
ee
s
tep
s
.
First,
b
as
ed
o
n
lo
g
ic,
th
eo
r
y
a
n
d
p
r
ev
io
u
s
s
t
u
d
ies,
p
lau
s
ib
le
alter
n
ati
v
e
m
o
d
els
o
f
an
u
n
d
er
l
y
i
n
g
d
ata
s
tr
u
ct
u
r
e
ar
e
p
r
o
p
o
s
ed
.
Usi
n
g
s
e
v
er
al
g
o
o
d
n
ess
-
of
-
f
i
t
in
d
ex
e
s
,
co
n
f
ir
m
ato
r
y
f
ac
to
r
an
al
y
s
is
is
u
s
ed
to
co
m
p
ar
e
t
h
e
d
ata
-
m
o
d
el
f
i
t
an
d
e
x
a
m
i
n
e
th
e
ev
id
en
ce
f
o
r
a
h
ig
h
er
-
o
r
d
er
co
n
s
tr
u
ct.
On
e
m
o
d
el
is
s
elec
ted
as
b
est
r
ep
r
esen
tin
g
th
e
u
n
d
er
l
y
in
g
f
ac
to
r
s
tr
u
ct
u
r
e
i
n
th
e
s
a
m
p
le
d
ata.
T
h
e
m
o
d
el
u
s
ed
in
th
is
s
t
u
d
y
ca
n
b
e
s
h
o
w
n
in
Fi
g
u
r
e
1
.
Seco
n
d
,
co
n
f
ir
m
at
o
r
y
f
ac
to
r
an
al
y
s
i
s
is
u
s
ed
to
ass
e
s
s
t
h
e
r
eliab
ilit
y
an
d
v
alid
it
y
o
f
t
h
e
f
ac
to
r
s
an
d
ite
m
s
i
n
t
h
e
s
elec
ted
m
o
d
el.
T
h
ir
d
,
to
c
r
o
s
s
-
v
alid
ate
th
e
s
e
r
es
u
lts
,
g
o
o
d
n
es
s
-
of
-
f
it i
n
d
ex
e
s
f
o
r
t
h
e
p
lau
s
ib
le
alter
n
ati
v
e
m
o
d
els.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2252
-
8822
I
n
t
.
J
.
E
v
al
.
&
R
es
.
E
d
u
c
.
Vo
l.
9
,
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NC
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S
[1
]
N.
D.
G
ü
v
e
n
a
n
d
Y.
De
d
e
,
“
Ex
a
m
in
in
g
so
c
ial
a
n
d
s
o
c
io
m
a
th
e
m
a
t
ica
l
n
o
rm
s
in
d
if
fe
re
n
t
c
las
sro
o
m
m
icro
c
u
lt
u
re
s:
M
a
th
e
m
a
ti
c
s te
a
c
h
e
r
e
d
u
c
a
ti
o
n
p
e
rsp
e
c
ti
v
e
,
”
Ku
ra
m ve
Uy
g
u
la
m
a
d
a
Eg
it
.
B
il
im.
,
v
o
l.
1
7
,
n
o
.
1
,
pp.
2
6
5
-
2
9
2
,
2
0
1
7
,
[2
]
S
.
A
.
W
id
o
d
o
,
J.
A
.
Da
h
lan
,
a
n
d
T
u
rm
u
d
i,
“
Ca
n
s
o
c
io
m
a
th
e
m
a
t
ica
l
n
o
rm
s
b
e
d
e
v
e
lo
p
e
d
w
it
h
lea
rn
in
g
m
e
d
ia
?
”
J
.
Ph
y
s.
Co
n
f.
S
e
r.
,
v
o
l
.
1
3
1
5
,
n
o
.
1
,
p
p
.
0
1
2
0
0
5
,
2
0
1
9
.
[3
]
F
.
W
.
T
u
n
g
a
n
d
Y.
S
.
De
n
g
,
“
De
sig
n
in
g
so
c
ial
p
re
se
n
c
e
in
e
-
lea
rn
in
g
e
n
v
iro
n
m
e
n
ts:
T
e
stin
g
th
e
e
ff
e
c
t
o
f
in
tera
c
ti
v
it
y
o
n
c
h
il
d
re
n
,
”
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ter
a
c
t
.
L
e
a
rn
.
E
n
v
iro
n
.,
v
o
l.
1
4
,
n
o
.
1
,
p
p
.
2
5
1
-
2
6
4
,
2
0
0
6
.
[4
]
K.
H.
Hu
a
n
g
a
n
d
Y.
S
.
De
n
g
,
“
S
o
c
ial
i
n
tera
c
ti
o
n
d
e
sig
n
i
n
c
u
l
tu
ra
l
c
o
n
tex
t:
A
c
a
se
stu
d
y
o
f
a
tr
a
d
it
io
n
a
l
so
c
ial
a
c
ti
v
it
y
,
”
In
t.
J
.
De
s.
,
v
o
l.
2
,
n
o
.
2
,
p
p
.
8
1
-
9
6
,
2
0
0
8
.
[5
]
H.
K.
Hv
id
e
a
n
d
P
.
Ös
t
b
e
rg
,
“
S
o
c
ial
in
tera
c
ti
o
n
a
t
w
o
rk
,
”
J
.
fi
n
a
n
c
.
e
c
o
n
.,
v
o
l.
1
1
7
,
n
o
.
3
,
p
p
.
6
2
8
-
6
5
2
,
2
0
1
5
.
[6
]
A
.
M
.
P
a
rtan
e
n
a
n
d
R.
Ka
a
sila,
“
S
o
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io
m
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m
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ti
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a
l
No
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m
s
Ne
g
o
ti
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ted
in
th
e
Disc
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ss
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o
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S
m
a
ll
G
ro
u
p
s
In
v
e
stig
a
ti
n
g
Ca
lcu
lu
s,”
In
t.
J
.
S
c
i.
M
a
th
.
E
d
u
c
.
,
v
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l.
1
3
,
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o
.
4
,
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p
.
927
–
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4
6
,
2
0
1
5
.
[7
]
L
.
M
.
L
o
p
e
z
a
n
d
L
.
A
ll
a
l,
“
S
o
c
io
m
a
th
e
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a
ti
c
a
l
No
rm
s
A
n
d
T
h
e
Re
g
u
latio
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o
f
P
ro
b
lem
S
o
lv
in
g
in
C
las
sro
o
m
M
icro
c
u
lt
u
re
s,”
In
t.
J
.
E
d
u
c
.
Res
.
,
v
o
l.
4
6
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o
.
5
,
p
p
.
2
5
2
–
2
6
5
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2
0
0
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.
[8
]
E.
Ya
c
k
e
l
a
n
d
C.
Ra
sm
u
ss
e
n
,
“
Be
li
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s
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n
d
n
o
rm
s
in
th
e
m
a
th
e
m
a
ti
c
s
c
las
sro
o
m
,
”
Beliefs
a
h
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n
V
a
r.
M
a
th
.
Ed
u
c
.
,
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l.
3
1
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p
p
.
3
1
3
–
3
3
0
,
2
0
0
3
.
[9
]
E.
Ya
c
k
e
l
a
n
d
P
.
Co
b
b
,
“
S
o
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io
m
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th
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a
ti
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l
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a
r
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m
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tatio
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,
a
n
d
a
u
to
n
o
m
y
in
m
a
th
e
m
a
ti
c
s
,
”
J
.
Res
.
M
a
th
.
Ed
u
c
.
,
v
o
l.
2
7
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o
.
4
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p
p
.
4
5
8
–
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7
7
,
1
9
9
6
.
[1
0
]
M
.
M
u
h
a
im
in
,
Ka
rto
n
o
,
a
n
d
B.
A
st
u
ti
,
“
A
n
a
n
a
l
y
sis
o
f
so
c
io
m
a
t
h
e
m
a
ti
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l
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o
rm
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o
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m
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o
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st
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ro
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h
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o
ll
a
b
o
ra
ti
v
e
p
r
o
b
lem
so
lv
in
g
lea
rn
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g
,
”
J
.
Prim.
E
d
u
c
.
,
v
o
l.
8
,
n
o
.
3
,
p
p
.
6
7
–
7
4
,
2
0
1
9
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
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2
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.
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.
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f
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d
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K.
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s c
o
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ten
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o
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rse
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n
v
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stig
.
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a
th
.
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rn
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l.
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o
.
2
,
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p
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3
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0
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[1
3
]
J.
K.
Dix
o
n
,
e
t
a
l
,
“
Do
th
e
y
re
a
ll
y
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ise
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h
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n
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s?
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ll
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tra
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ial
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se
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ra
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m
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a
ti
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s c
las
sro
o
m
,
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.
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.
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d
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[1
4
]
S
.
E.
Ka
stb
e
rg
a
n
d
R.
S
.
F
ry
e
,
“
No
rm
s
a
nd
m
a
th
e
m
a
ti
c
a
l
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ro
f
icie
n
c
y
,
”
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e
a
c
h
.
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h
il
d
.
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th
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ti
c
s
,
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l.
2
0
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o
.
1
,
p
p
.
2
8
-
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5
,
2
0
1
3
.
[1
5
]
S
.
M
.
Ka
n
g
a
n
d
M
.
K.
Kim
,
“
S
o
c
io
m
a
th
e
m
a
ti
c
a
l
n
o
rm
s
a
n
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c
h
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r’s
m
a
th
e
m
a
ti
c
a
l
b
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li
e
f
:
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c
a
se
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d
y
f
ro
m
a
Ko
re
a
n
in
-
se
rv
ice
e
le
m
e
n
tar
y
te
a
c
h
e
r,
”
Eu
ra
sia
J
.
M
a
th
.
S
c
i
.
T
e
c
h
n
o
l.
E
d
u
c
.
,
v
o
l.
1
2
,
n
o
.
1
0
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2
0
1
6
.
[1
6
]
R.
I.
I
.
P
u
tri
,
M
.
Do
lk
,
a
n
d
Zu
lk
a
rd
i,
“
P
r
o
f
e
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io
n
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l
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m
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o
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h
e
rs
f
o
r
in
tr
o
d
u
c
in
g
so
c
ial
n
o
rm
s,”
J
.
M
a
th
.
Ed
u
c
.,
v
o
l.
6
,
n
o
.
1
,
p
p
.
1
1
-
1
9
,
2
0
1
5
.
[1
7
]
P
.
C
o
b
b
,
E.
Ya
c
k
e
l,
a
n
d
T
.
W
o
o
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,
Af
fec
t
a
n
d
ma
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h
e
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ti
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l
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ro
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lem
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lvi
n
g
,
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ff
e
c
t
a
n
d
M
a
th
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m
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ti
c
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l
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lem
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o
lv
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g
,
D.
B
.
M
c
L
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o
d
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n
d
V
.
M
.
A
d
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m
s,
Ed
s.
Ne
w
Yo
rk
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9
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8
]
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Ya
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k
e
l,
P
.
C
o
b
b
,
a
n
d
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.
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o
o
d
,
“
S
m
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ll
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se
c
o
n
d
-
gr
a
d
e
m
a
th
e
m
a
ti
c
s
,
”
J
.
Res
.
M
a
th
.
Ed
u
c
.
,
v
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l.
2
2
,
n
o
.
5
,
p
p
.
3
9
0
-
4
0
8
,
1
9
9
1
.
[1
9
]
I.
Biza
,
E
.
Na
rd
i
,
a
n
d
G
.
Jo
e
l,
“
B
a
lan
c
in
g
c
las
sro
o
m
m
a
n
a
g
e
m
e
n
t
w
it
h
m
a
th
e
m
a
ti
c
a
l
lea
rn
in
g
:
u
si
n
g
p
ra
c
ti
c
e
-
b
a
se
d
tas
k
d
e
sig
n
in
m
a
th
e
m
a
ti
c
s tea
c
h
e
r
e
d
u
c
a
ti
o
n
,
”
M
a
t
h
.
T
e
a
c
h
.
E
d
u
c
.
De
v
.,
v
o
l.
1
7
,
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o
.
2
,
p
p
.
1
8
2
-
1
9
8
,
2
0
1
5
.
[2
0
]
A
.
P
in
t
o
a
n
d
R.
Ka
rse
n
ty
,
“
F
ro
m
c
o
u
rse
d
e
sig
n
t
o
p
re
se
n
tatio
n
s
o
f
p
ro
o
f
s:
Ho
w
m
a
th
e
m
a
ti
c
s
p
ro
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e
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rs
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tt
e
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t
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d
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t
in
d
e
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e
n
d
e
n
t
p
ro
o
f
re
a
d
in
g
,
”
J
.
M
a
th
.
Beh
a
v
.,
v
o
l.
4
9
,
p
p
.
1
2
9
-
1
4
4
,
2
0
1
8
.
[2
1
]
W
.
S
c
o
tt
,
P
.
J.
Da
v
is,
a
n
d
R.
He
rsh
,
“
T
h
e
m
a
th
e
m
a
ti
c
a
l
e
x
p
e
rien
c
e
,
”
M
a
th
.
Ga
z
.,
v
o
l.
6
6
,
n
o
.
4
3
5
,
p
p
.
7
3
-
7
4
,
1
9
8
2
.
[2
2
]
I.
M
a
sito
h
a
n
d
S
.
P
ra
b
a
w
a
n
to
,
“
I
m
p
ro
v
in
g
u
n
d
e
rsta
n
d
i
n
g
o
f
m
a
th
e
m
a
ti
c
a
l
c
o
n
c
e
p
ts
a
n
d
m
a
th
e
m
a
ti
c
a
l
c
rit
ica
l
th
in
k
in
g
a
b
i
li
ty
o
f
g
ra
d
e
v
stu
d
e
n
ts
in
p
rim
a
r
y
sc
h
o
o
ls
th
r
o
u
g
h
e
x
p
lo
ra
ti
v
e
lea
rn
in
g
(in
Ba
h
a
sa
)
,
”
Ed
u
Hu
ma
n
io
r
a
|
J
.
Pen
d
i
d
ik.
Da
s
a
r Ka
m
p
u
s Ci
b
ir
u
,
v
o
l.
7
,
n
o
.
2
,
p
p
.
1
8
6
-
1
9
7
,
2
0
1
6
.
[2
3
]
V
.
A
lb
e
,
“
W
h
e
n
sc
ien
ti
f
ic
k
n
o
w
led
g
e
,
d
a
il
y
li
f
e
e
x
p
e
rien
c
e
,
e
p
istem
o
lo
g
ica
l
a
n
d
so
c
ial
c
o
n
si
d
e
ra
ti
o
n
s
in
ters
e
c
t:
S
tu
d
e
n
ts’
a
rg
u
m
e
n
tatio
n
in
g
ro
u
p
d
isc
u
ss
i
o
n
s
o
n
a
so
c
io
-
sc
ien
ti
f
ic
Iss
u
e
,
”
Res
.
S
c
i.
Ed
u
c
.,
v
o
l.
3
8
,
n
o
.
1
,
p
p
,
6
7
-
9
0
,
2
0
0
8
.
[2
4
]
E.
A
.
F
o
rm
a
n
,
D.
E.
M
c
c
o
rm
ick
,
a
n
d
R.
Do
n
a
t
o
,
“
L
e
a
rn
in
g
w
h
a
t
c
o
u
n
ts
a
s
a
m
a
th
e
m
a
ti
c
a
l
e
x
p
lan
a
ti
o
n
,
”
L
in
g
u
ist.
Ed
u
c
.
,
v
o
l.
9
,
n
o
.
4
,
p
p
.
3
1
3
-
3
3
9
,
1
9
9
7
.
[2
5
]
E.
Yo
u
n
g
,
“
Un
p
a
c
k
in
g
m
a
th
e
m
a
ti
c
a
l
c
o
n
ten
t
t
h
ro
u
g
h
p
ro
b
le
m
so
lv
in
g
,
”
P
h
D
T
h
e
sis
,
T
h
e
Un
iv
e
rsity
o
f
Ok
lah
o
m
a
,
2002.
[
On
li
n
e
].
A
v
a
il
a
b
le
:
h
t
tp
s://
s
h
a
re
o
k
.
o
rg
/h
a
n
d
le/1
1
2
4
4
/
5
1
0
[2
6
]
R.
Driv
e
r,
P
.
Ne
w
to
n
,
a
n
d
J
.
Os
b
o
r
n
e
,
“
Estab
li
sh
i
n
g
th
e
n
o
rm
s
o
f
sc
ien
ti
f
ic
a
r
g
u
m
e
n
tatio
n
i
n
c
l
a
ss
ro
o
m
s,”
S
c
i.
Ed
u
c
.
,
v
o
l.
8
3
,
n
o
4
,
p
p
.
2
8
7
-
3
1
2
,
2
0
0
0
.
[2
7
]
B.
P
e
d
e
m
o
n
te,
“
Ho
w
c
a
n
th
e
re
latio
n
sh
ip
b
e
tw
e
e
n
a
rg
um
e
n
tatio
n
a
n
d
p
ro
o
f
b
e
a
n
a
ly
s
e
d
?
”
Ed
u
c
.
S
tu
d
.
M
a
t
h
.,
v
o
l.
6
6
,
n
o
.
1
,
p
p
.
2
3
-
4
4
,
2
0
0
7
.
[2
8
]
J.
W
.
Cre
s
w
e
ll
,
Res
e
a
rc
h
De
sig
n
:
Qu
a
li
ta
ti
v
e
,
Q
u
a
n
ti
t
a
ti
v
e
,
a
n
d
M
ixe
d
M
e
th
o
d
s A
p
p
r
o
a
c
h
e
s
S
e
c
o
n
d
Ed
it
i
o
n
,
2
0
1
2
.
[2
9
]
J.
W
.
Cre
s
w
e
ll
,
Ed
u
c
a
ti
o
n
a
l
R
e
se
a
rc
h
:
Pl
a
n
n
i
n
g
,
Co
n
d
u
c
ti
n
g
a
n
d
Eva
lu
a
ti
n
g
Q
u
a
n
ti
ta
ti
v
e
a
n
d
Qu
a
li
t
a
ti
v
e
Res
e
a
rc
h
,
L
o
n
d
o
n
:
P
e
a
rso
n
,
2
0
1
2
.
[3
0
]
K.
G
.
Jö
re
sk
o
g
a
n
d
D.
S
ö
rb
o
m
,
L
IS
RE
L
8
:
S
tru
c
t
u
ra
l
E
q
u
a
ti
o
n
M
o
d
e
li
n
g
wi
th
th
e
S
IM
PL
I
S
Co
m
ma
n
d
L
a
n
g
u
a
g
e
S
c
ien
ti
f
ic S
o
f
twa
re
In
tern
a
ti
o
n
a
l,
1
9
9
3
.
[3
1
]
K.
G
.
Jö
re
sk
o
g
a
n
d
D.
S
ö
rb
o
m
,
“
M
o
d
e
l
se
a
rc
h
w
it
h
T
ET
RAD
II
a
n
d
L
IS
REL
,
”
S
o
c
i
o
lo
g
ica
l
M
e
th
o
d
s
&
Res
e
a
rc
h
,
v
o
l.
1
9
,
n
o
.
1
,
p
p
.
9
3
–
1
0
6
,
1
9
9
0
.
[3
2
]
H.
G
a
ti
g
n
o
n
,
Co
n
fi
rm
a
t
o
ry
fa
c
t
o
r a
n
a
lys
is
,
i
n
S
t
a
ti
stic
a
l
A
n
a
lys
is
o
f
M
a
n
a
g
e
me
n
t
Da
ta
,
Ne
w
Yo
rk
:
S
p
rin
g
e
r,
2
0
1
0
.
[3
3
]
I.
G
h
o
z
a
li
a
n
d
F
u
a
d
,
S
tru
c
tu
r
a
l
e
q
u
a
t
io
n
mo
d
e
li
n
g
:
t
h
e
o
ry
,
c
o
n
c
e
p
ts
a
n
d
a
p
p
li
c
a
t
io
n
s
wit
h
th
e
L
I
S
RE
L
p
ro
g
ra
m
,
S
e
m
a
ra
n
g
:
Ba
d
a
n
P
e
n
e
rb
it
U
n
iv
e
rsitas
Dip
o
n
e
g
o
r
o
,
2
0
1
4
.
[3
4
]
E.
Riad
i,
S
EM
st
a
ti
stics
:
stru
c
tu
r
a
l
e
q
u
a
ti
o
n
mo
d
e
l
in
g
wit
h
L
I
S
RE
L
,
Yo
g
y
a
k
a
rta:
P
e
n
e
rb
it
A
n
d
i,
2
0
1
8
.
[3
5
]
J.
F
.
Ha
ir,
W
.
C.
B
lac
k
,
B.
J.
Ba
b
i
n
,
a
n
d
R.
E.
A
n
d
e
rso
n
,
M
u
lt
iv
a
ria
te Da
ta
An
a
lys
is
,
Ne
w
Je
rs
e
y
:
P
e
a
rso
n
,
2
0
1
4
.
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