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[
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–
[
3
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T
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–
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1
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em
atics
teac
h
er
r
ea
d
in
ess
en
co
m
p
ass
es
a
v
ar
iety
o
f
s
k
ills
an
d
co
m
p
eten
cies,
in
clu
d
in
g
p
e
d
ag
o
g
ical
ex
p
e
r
tis
e,
tech
n
o
lo
g
ical
p
r
o
f
icien
cy
,
c
u
r
r
i
cu
lu
m
ad
a
p
tab
ilit
y
.
Fra
m
ewo
r
k
s
lik
e
tech
n
o
lo
g
ic
al,
p
ed
ag
o
g
ical,
an
d
co
n
ten
t
k
n
o
wled
g
e
(
T
PAC
K)
h
av
e
b
ee
n
in
s
tr
u
m
en
tal
in
ev
alu
atin
g
teac
h
e
r
s
’
ab
ilit
y
t
o
in
teg
r
ate
tech
n
o
lo
g
y
in
to
th
eir
teac
h
in
g
p
r
ac
tices,
en
h
an
cin
g
b
o
th
th
eir
in
s
tr
u
ctio
n
al
m
eth
o
d
s
a
n
d
s
tu
d
en
t
en
g
ag
em
e
n
t
[
1
5
]
.
Ho
we
v
er
,
m
an
y
m
ath
em
atics
teac
h
er
s
co
n
tin
u
e
to
r
el
y
h
ea
v
ily
o
n
tr
ad
itio
n
al
m
eth
o
d
s
,
lack
in
g
th
e
tailo
r
ed
r
eso
u
r
c
es
an
d
p
r
o
f
ess
io
n
al
d
ev
elo
p
m
en
t
n
ee
d
ed
to
f
u
lly
ad
ap
t
to
m
o
d
er
n
ed
u
ca
tio
n
al
d
em
an
d
s
[
1
6
]
.
T
h
is
p
o
in
ts
t
o
a
n
ee
d
f
o
r
f
o
c
u
s
ed
in
ter
v
e
n
tio
n
s
th
at
s
u
p
p
o
r
t
co
n
tin
u
o
u
s
p
r
o
f
ess
io
n
al
g
r
o
wt
h
an
d
t
h
e
ad
o
p
tio
n
o
f
in
n
o
v
ati
v
e
teac
h
in
g
s
tr
ateg
ies
[
1
7
]
,
[
1
8
]
.
Fu
r
th
er
m
o
r
e
,
th
e
in
te
g
r
atio
n
o
f
s
cien
ce
,
tec
h
n
o
lo
g
y
,
e
n
g
in
ee
r
in
g
,
an
d
m
ath
e
m
atics
(
STE
M)
ed
u
ca
tio
n
an
d
th
e
s
h
if
t
to
war
d
o
n
lin
e
an
d
tech
n
o
lo
g
y
en
h
an
ce
d
lear
n
in
g
en
v
ir
o
n
m
e
n
ts
p
r
esen
t
ad
d
itio
n
al
ch
allen
g
es
an
d
o
p
p
o
r
tu
n
ities
f
o
r
m
ath
em
atics
teac
h
er
s
.
R
esear
ch
in
d
icate
s
th
at
s
u
cc
ess
f
u
l
ad
ap
tatio
n
to
t
h
ese
ch
an
g
es
r
eq
u
ir
es
a
s
o
lid
f
o
u
n
d
atio
n
o
f
r
ea
d
in
ess
,
u
n
d
e
r
p
in
n
ed
b
y
o
n
g
o
in
g
tr
ain
i
n
g
,
ac
ce
s
s
to
r
eso
u
r
ce
s
,
an
d
s
u
p
p
o
r
tiv
e
p
o
licy
f
r
a
m
ewo
r
k
s
[
1
9
]
,
[
2
0
]
.
T
h
e
c
u
r
r
en
t
la
n
d
s
ca
p
e
n
ec
ess
itates
th
at
m
ath
em
atics
teac
h
er
s
n
o
t
o
n
ly
p
o
s
s
ess
d
ee
p
co
n
ten
t
k
n
o
wled
g
e
b
u
t
also
th
e
s
k
ills
to
d
eliv
er
in
ter
d
is
cip
lin
ar
y
ed
u
c
atio
n
th
at
p
r
e
p
ar
es
s
tu
d
en
ts
f
o
r
f
u
tu
r
e
ca
r
ee
r
s
in
a
tech
n
o
lo
g
ically
d
r
iv
en
wo
r
ld
.
T
o
ad
d
r
ess
th
e
g
ap
,
th
is
s
tu
d
y
co
n
d
u
cts
a
s
y
s
tem
atic
r
ev
iew
o
f
r
ec
en
t e
m
p
ir
ical
r
esear
ch
,
p
a
r
ticu
lar
ly
f
o
cu
s
in
g
o
n
s
tu
d
ies p
u
b
lis
h
ed
in
2
0
2
3
,
to
ev
al
u
ate
th
e
r
ea
d
in
ess
o
f
m
ath
em
atic
s
teac
h
er
s
as
ag
en
ts
o
f
ch
a
n
g
e.
B
u
ild
in
g
u
p
o
n
Fu
llan
’
s
m
u
ltid
im
en
s
io
n
al
f
r
am
ewo
r
k
o
f
c
h
an
g
e
r
ea
d
in
e
s
s
[
2
1
]
,
th
is
r
ev
iew
s
y
n
t
h
esizes
cu
r
r
en
t
f
in
d
in
g
s
to
i
d
en
tif
y
p
atter
n
s
,
tr
e
n
d
s
,
ch
allen
g
es,
an
d
b
est
p
r
ac
tices,
p
r
o
v
id
i
n
g
h
ig
h
lig
h
t
n
o
v
el
in
s
ig
h
ts
th
at
ca
n
in
f
o
r
m
f
u
tu
r
e
e
d
u
ca
tio
n
al
r
ef
o
r
m
s
an
d
p
r
o
f
ess
io
n
al
d
e
v
elo
p
m
en
t
in
itiativ
es
[
2
2
]
–
[
2
7
]
.
T
h
is
ap
p
r
o
ac
h
aim
s
to
a
d
v
an
ce
th
e
d
is
co
u
r
s
e
o
n
teac
h
er
r
ea
d
in
ess
b
y
an
s
wer
in
g
th
e
f
o
llo
win
g
r
esear
ch
q
u
esti
o
n
g
u
i
d
in
g
t
h
is
r
ev
iew
is
,
wh
at
ar
e
t
h
e
cu
r
r
en
t
p
atter
n
s
an
d
tr
en
d
s
in
m
ath
em
atics tea
ch
er
s
’
r
ea
d
in
ess
liter
atu
r
e
d
is
c
u
s
s
ed
in
th
e
r
ec
en
t stu
d
ies?
2.
RE
S
E
ARCH
M
E
T
H
O
D
I
n
ac
co
r
d
an
ce
with
p
r
ef
er
r
e
d
r
ep
o
r
tin
g
item
s
f
o
r
s
y
s
tem
atic
r
ev
iews
an
d
m
eta
-
an
al
y
s
es
(
PR
I
SMA
)
,
s
u
itab
le
r
esear
ch
m
eth
o
d
o
lo
g
ies
ar
e
s
u
g
g
ested
in
o
r
d
er
t
o
g
en
er
ate
s
y
s
tem
atic
liter
atu
r
e
h
ig
h
lig
h
ts
.
T
h
e
s
elec
tio
n
o
f
ar
ticles
f
o
r
th
is
s
tu
d
y
was
f
ac
ilit
ated
u
s
in
g
PR
I
SMA
as
a
g
u
id
e
[
2
8
]
.
Fu
r
th
e
r
m
o
r
e,
in
cl
u
d
in
g
a
s
u
b
s
tan
tial
am
o
u
n
t
o
f
d
ata
a
n
d
ad
h
er
en
ce
to
a
r
elativ
ely
r
i
g
o
r
o
u
s
an
d
m
eticu
lo
u
s
p
r
o
ce
d
u
r
e
c
o
n
tr
ib
u
tes
to
en
h
an
cin
g
th
e
q
u
ality
o
f
s
y
s
t
em
atic
liter
atu
r
e
r
ev
iews
(
SLR)
.
Per
tain
in
g
to
th
e
r
ea
d
in
e
s
s
o
f
m
ath
em
atics
teac
h
er
s
to
ac
t
as
ca
taly
s
ts
f
o
r
tr
an
s
f
o
r
m
atio
n
,
th
is
r
esear
c
h
em
p
lo
y
e
d
a
q
u
alitativ
e
m
et
h
o
d
o
lo
g
y
to
g
ath
er
co
m
p
r
eh
e
n
s
iv
e
d
ata.
Fu
r
th
er
m
o
r
e,
d
o
cu
m
en
ts
s
o
u
r
ce
d
f
r
o
m
th
e
d
atab
ase
ar
e
in
co
r
p
o
r
at
ed
in
to
th
is
s
tu
d
y
t
o
ad
d
r
ess
th
e
r
esear
ch
q
u
esti
o
n
s
.
2
.
1
.
I
dentif
ica
t
io
n
Nu
m
er
o
u
s
cr
u
cial
s
tag
es
o
f
th
e
s
y
s
tem
atic
r
ev
iew
wer
e
em
p
lo
y
ed
to
id
e
n
tify
a
s
u
b
s
tan
tial a
m
o
u
n
t
o
f
p
er
tin
en
t
liter
atu
r
e
f
o
r
th
is
s
tu
d
y
.
I
n
itially
,
k
e
y
wo
r
d
s
wer
e
c
h
o
s
en
,
f
o
llo
wed
b
y
th
e
e
x
p
lo
r
atio
n
o
f
ass
o
ciate
d
ter
m
s
th
r
o
u
g
h
d
ictio
n
ar
ies,
th
esau
r
u
s
es,
en
cy
clo
p
e
d
ias,
an
d
p
r
io
r
r
esear
ch
.
I
n
o
r
d
er
to
m
i
tig
ate
th
e
in
f
lu
e
n
ce
o
f
r
etr
ie
v
al
b
ias,
t
h
e
r
esear
c
h
er
s
d
ec
id
ed
to
u
tili
ze
m
u
ltip
l
e
d
atab
ases
[
2
9
]
.
On
ce
th
e
s
ea
r
ch
ter
m
s
f
o
r
th
e
Sco
p
u
s
an
d
W
eb
o
f
Scien
ce
d
atab
ases
wer
e
f
o
r
m
u
lated
,
as
in
T
ab
le
1
,
all
r
elev
an
t
ter
m
s
wer
e
ch
o
s
en
.
I
n
th
e
in
itial
p
h
ase
o
f
th
e
s
y
s
tem
atic
r
ev
iew
p
r
o
ce
s
s
,
a
to
tal
o
f
4
,
2
6
6
p
ap
e
r
s
wer
e
g
ath
er
ed
s
u
cc
ess
f
u
lly
f
o
r
th
e
p
r
esen
t stu
d
y
p
r
o
ject
f
r
o
m
b
o
t
h
d
atab
ases
.
T
ab
le
1
.
Ar
ticle
s
ea
r
ch
f
o
r
SL
R
D
a
t
a
b
a
s
e
Th
e
se
a
r
c
h
st
r
i
n
g
S
c
o
p
u
s
TI
TLE
-
A
B
S
-
K
EY
(
(
k
e
e
n
n
e
ss
O
R
p
r
e
p
a
r
a
t
i
o
n
O
R
w
i
l
l
i
n
g
n
e
ss
O
R
r
e
a
d
i
n
e
ss
O
R
d
e
x
t
e
r
i
t
y
)
A
N
D
m
a
t
h
e
mat
i
c
*
A
N
D
(
t
e
a
c
h
e
r
s
O
R
i
n
s
t
r
u
c
t
o
r
O
R
e
d
u
c
a
t
o
r
O
R
t
u
t
o
r
)
)
A
N
D
(
LI
M
I
T
-
TO
(
P
U
B
Y
EA
R
,
2
0
2
3
)
)
A
N
D
(
LI
M
I
T
-
TO
(
D
O
C
TY
P
E,
“
a
r
”
)
)
A
N
D
(
LI
M
I
T
-
TO
(
LA
N
G
U
A
G
E,
“
E
n
g
l
i
s
h
”
)
)
A
N
D
(
LI
M
I
T
-
TO
(
S
U
B
JA
R
EA
,
“
M
A
TH
”
)
O
R
LI
M
I
T
-
TO
(
S
U
B
JA
R
EA
,
“
S
O
C
I
”
)
)
W
e
b
o
f
S
c
i
e
n
c
e
(
k
e
e
n
n
e
ss
O
R
p
r
e
p
a
r
a
t
i
o
n
O
R
w
i
l
l
i
n
g
n
e
ss
O
R
r
e
a
d
i
n
e
ss
O
R
d
e
x
t
e
r
i
t
y
)
A
N
D
mat
h
e
m
a
t
i
c
*
A
N
D
(
t
e
a
c
h
e
r
s
O
R
i
n
s
t
r
u
c
t
o
r
O
R
e
d
u
c
a
t
o
r
O
R
t
u
t
o
r
)
(
T
o
p
i
c
)
a
n
d
2
0
2
3
(
P
u
b
l
i
c
a
t
i
o
n
Y
e
a
r
s)
a
n
d
A
r
t
i
c
l
e
(
D
o
c
u
me
n
t
T
y
p
e
s)
a
n
d
En
g
l
i
sh
(
La
n
g
u
a
g
e
s)
a
n
d
S
o
c
i
a
l
S
c
i
e
n
c
e
s O
t
h
e
r
T
o
p
i
c
s
o
r
M
a
t
h
e
m
a
t
i
c
s (
R
e
sea
r
c
h
A
r
e
a
s)
2
.
2
.
Scre
ening
T
h
e
s
cr
ee
n
in
g
p
r
o
ce
s
s
in
v
o
lv
e
s
ev
alu
atin
g
a
s
et
o
f
p
o
ten
tiall
y
r
elev
an
t
r
esear
ch
item
s
to
d
e
ter
m
in
e
if
th
ey
ar
e
in
lin
e
with
th
e
p
r
ed
e
ter
m
in
ed
r
esear
ch
t
o
p
ic
o
r
q
u
esti
o
n
s
.
Her
e,
th
e
s
cr
ee
n
in
g
p
h
ase
o
f
ten
em
p
lo
y
s
co
n
ten
t
-
r
elate
d
cr
iter
ia,
s
u
ch
as
ca
teg
o
r
izin
g
s
tu
d
y
item
s
a
cc
o
r
d
in
g
to
m
ath
em
atics
teac
h
er
s
’
r
ea
d
i
n
ess
as
ch
an
g
e
a
g
en
ts
.
Du
p
licate
p
a
p
e
r
s
will
b
e
r
em
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Evaluation Warning : The document was created with Spire.PDF for Python.
I
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:
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8
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I
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14
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1
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tag
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lted
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ased
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tlin
ed
in
th
is
s
tu
d
y
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as
s
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in
T
ab
le
2
.
T
h
ese
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elec
tio
n
cr
iter
ia
ar
e
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en
tial
f
o
r
en
s
u
r
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e
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ce
o
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t
h
e
ch
o
s
en
a
r
ticles
to
th
e
s
tu
d
y
[
3
0
]
.
Sin
ce
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ch
a
r
ticles
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e
th
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ain
s
o
u
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ce
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l
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e
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e
f
ir
s
t
cr
iter
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e
ap
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t
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clu
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r
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,
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en
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d
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r
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e
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was
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E
n
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h
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lan
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a
g
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b
licatio
n
s
.
T
h
is
s
tu
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y
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s
es
p
r
ev
i
o
u
s
s
tu
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ies
in
th
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r
2
0
2
3
.
M
o
s
t
r
e
ce
n
tly
,
th
ese
s
tu
d
ies
d
elin
ea
te
an
d
d
is
cu
s
s
th
e
r
ea
d
in
ess
o
f
m
at
h
em
atics
teac
h
er
s
to
ac
t
as
ch
a
n
g
e
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en
ts
,
ad
v
o
ca
tin
g
f
o
r
p
r
o
ac
tiv
e
m
ea
s
u
r
es.
T
h
is
p
er
io
d
is
s
u
f
f
icien
t
an
d
r
elev
an
t
f
o
r
id
en
tify
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g
co
n
ce
p
ts
an
d
s
tu
d
y
g
ap
s
r
elate
d
to
t
h
e
r
ea
d
in
ess
o
f
m
ath
em
atics
teac
h
er
s
as
ag
en
ts
o
f
ch
an
g
e
[
2
9
]
,
[
3
1
]
.
Du
e
to
d
u
p
licate
cr
it
er
ia,
o
n
e
p
a
p
er
was
r
ejec
ted
o
v
er
all.
2
.
3
.
E
lig
ibi
lity
I
n
th
e
th
ir
d
s
tag
e,
ter
m
ed
th
e
elig
ib
ilit
y
ass
ess
m
en
t,
a
to
tal
o
f
1
4
6
ar
ticles
wer
e
ass
em
b
le
d
.
Du
r
i
n
g
th
is
p
h
ase,
a
m
eticu
lo
u
s
ex
am
in
atio
n
o
f
b
o
th
titl
es
an
d
cr
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ci
al
co
n
ten
t
o
f
all
ar
ticles
o
cc
u
r
r
ed
to
co
n
f
ir
m
th
eir
alig
n
m
en
t
with
th
e
in
cl
u
s
io
n
cr
iter
ia
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d
th
eir
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er
tin
en
ce
to
th
e
r
esear
ch
g
o
als
o
f
th
e
c
u
r
r
en
t
s
tu
d
y
.
As
a
r
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lt,
1
1
5
a
r
ticles
wer
e
elim
in
ated
f
o
r
r
ea
s
o
n
s
s
u
ch
as
b
ein
g
o
u
t
o
f
s
co
p
e,
lack
in
g
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ig
n
i
f
ican
t
titl
es,
h
av
in
g
ab
s
tr
ac
ts
u
n
r
elate
d
to
th
e
s
tu
d
y
’
s
o
b
jectiv
es,
an
d
h
a
v
in
g
n
o
f
u
ll
-
tex
t
ac
ce
s
s
b
ased
o
n
em
p
i
r
ical
ev
id
en
ce
.
As
a
co
n
s
eq
u
en
ce
,
a
to
tal
o
f
3
1
a
r
ticles ar
e
r
etain
ed
f
o
r
th
e
f
o
r
th
c
o
m
in
g
r
e
v
iew.
2
.
4
.
Da
t
a
a
bs
t
ra
ct
io
n a
nd
a
na
ly
s
is
T
h
is
s
tu
d
y
u
tili
ze
d
an
in
teg
r
ativ
e
an
aly
s
is
as
o
n
e
o
f
its
ass
ess
m
en
t
s
tr
ateg
ies,
f
o
cu
s
in
g
o
n
ex
am
in
in
g
an
d
s
y
n
t
h
esizin
g
v
a
r
io
u
s
r
esear
ch
d
esig
n
s
,
p
ar
ticu
la
r
ly
u
tili
zin
g
q
u
an
titativ
e
m
eth
o
d
s
.
T
h
e
in
itial
p
h
ase
o
f
th
em
e
d
ev
elo
p
m
en
t
en
tailed
g
ath
er
in
g
d
ata.
Fig
u
r
e
1
d
ep
icts
th
e
au
th
o
r
s
’
th
o
r
o
u
g
h
ex
am
in
atio
n
o
f
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co
llectio
n
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f
3
1
p
u
b
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n
s
to
id
en
tif
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tatem
en
ts
o
r
co
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t
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n
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to
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e
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o
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ics
o
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r
esen
t
s
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y
.
C
o
n
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eq
u
en
tly
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th
e
au
th
o
r
s
a
s
s
es
s
ed
ex
is
tin
g
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ig
n
if
ican
t
s
tu
d
ies
r
elate
d
to
t
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e
r
ea
d
in
e
s
s
o
f
m
ath
em
atics
teac
h
er
s
.
Ad
d
itio
n
ally
,
an
ex
a
m
in
atio
n
was
co
n
d
u
cted
o
n
th
e
m
eth
o
d
o
lo
g
ies
em
p
lo
y
ed
ac
r
o
s
s
all
s
tu
d
ies,
as
well
as th
e
r
esear
ch
r
esu
lts
.
T
ab
le
2
.
T
h
e
s
elec
tio
n
cr
iter
io
n
is
s
ea
r
ch
in
g
C
r
i
t
e
r
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o
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I
n
c
l
u
s
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o
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Ex
c
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si
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Li
t
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r
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t
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r
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Jo
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l
(
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r
t
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)
C
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o
o
k
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e
v
i
e
w
La
n
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a
g
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En
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l
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sh
N
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-
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n
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Ti
me
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n
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2
0
2
3
<
2
0
2
3
P
u
b
l
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c
a
t
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g
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F
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B
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e
a
n
d
m
a
t
h
e
m
a
t
i
c
s
Fig
u
r
e
1
.
Flo
w
d
ia
g
r
am
o
f
th
e
p
r
o
p
o
s
ed
s
ea
r
ch
in
g
s
tu
d
y
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t
J
E
v
al
&
R
es E
d
u
c
I
SS
N:
2252
-
8
8
2
2
Th
e
r
ea
d
in
ess
o
f m
a
th
ema
tics
tea
ch
ers
a
s
a
g
en
ts
o
f c
h
a
n
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:
a
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(
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h
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r
id
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u
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lib
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1471
Su
b
s
eq
u
en
tly
,
we
i
d
en
tify
t
h
em
es
r
o
o
ted
in
th
e
s
tu
d
y
’
s
e
v
id
en
ce
with
in
its
co
n
te
x
t.
A
lo
g
was
co
n
s
is
ten
tly
m
ain
tain
ed
d
u
r
in
g
th
e
d
ata
a
n
aly
s
is
p
h
ase
t
o
d
o
cu
m
en
t
an
aly
s
es,
v
iewp
o
in
ts
,
q
u
esti
o
n
s
,
an
d
an
y
s
ig
n
if
ican
t
co
n
s
id
er
atio
n
s
r
ele
v
an
t
to
in
ter
p
r
etin
g
th
e
d
ata.
E
v
en
tu
ally
,
we
co
m
p
ar
ed
th
e
f
in
d
in
g
s
to
id
en
tif
y
an
y
in
co
n
s
is
ten
cies
in
th
e
th
em
e
d
esig
n
p
r
o
ce
s
s
.
T
h
e
p
r
o
d
u
ce
d
th
em
es
u
n
d
er
wen
t
ad
j
u
s
tm
en
ts
to
en
s
u
r
e
co
h
er
en
ce
.
Fo
r
th
e
v
alid
atio
n
o
f
th
e
an
al
y
s
is
,
two
ex
p
er
ts
wer
e
in
v
o
lv
e
d
—
o
n
e
s
p
ec
ializin
g
in
cu
r
r
icu
lu
m
a
n
d
p
ed
ag
o
g
y
,
with
a
f
o
c
u
s
o
n
tea
ch
er
co
m
p
ete
n
cy
,
an
d
th
e
o
th
er
in
m
ath
em
atics
ed
u
ca
tio
n
.
T
h
eir
co
llab
o
r
ativ
e
ef
f
o
r
t
d
eter
m
in
ed
th
e
is
s
u
e’
s
v
alid
ity
.
T
h
e
ex
p
e
r
t
r
ev
iew
p
h
ase
aim
ed
at
en
s
u
r
in
g
cla
r
ity
,
s
ig
n
if
ican
ce
,
an
d
ap
p
r
o
p
r
iaten
ess
with
r
esp
ec
t to
ev
er
y
s
u
b
th
em
e
a
n
d
estab
lis
h
in
g
d
o
m
ain
v
alid
ity
.
3.
RE
SU
L
T
S AN
D
D
I
SCU
SS
I
O
N
T
h
is
s
ec
tio
n
p
r
esen
ts
th
e
k
e
y
f
in
d
in
g
s
f
r
o
m
th
e
liter
atu
r
e
r
e
v
iew
an
d
th
em
atic
a
n
aly
s
is
,
h
i
g
h
lig
h
tin
g
th
e
s
ig
n
if
ican
ce
o
f
th
e
s
tu
d
y
’
s
co
n
tr
ib
u
tio
n
s
.
T
h
em
atic
an
aly
s
is
o
f
th
e
r
ev
iewe
d
liter
atu
r
e
id
en
tifie
d
th
r
e
e
m
ain
th
em
es:
m
ath
em
atics
tea
ch
in
g
s
tr
ateg
ies
a
n
d
p
ed
a
g
o
g
y
,
f
ac
to
r
s
af
f
ec
tin
g
m
ath
e
m
atics
lear
n
in
g
,
an
d
th
e
in
teg
r
atio
n
o
f
STE
M
ed
u
ca
tio
n
with
teac
h
er
r
ea
d
in
ess
.
T
ab
l
e
3
p
r
o
v
i
d
es
a
s
u
m
m
ar
y
o
f
th
ese
th
em
es
an
d
th
e
co
r
r
esp
o
n
d
in
g
s
tu
d
ies.
T
ab
le
3
.
T
h
em
es with
au
th
o
r
(
s
)
Th
e
m
e
s
S
t
u
d
i
e
s
M
a
t
h
e
ma
t
i
c
s t
e
a
c
h
i
n
g
st
r
a
t
e
g
i
e
s a
n
d
p
e
d
a
g
o
g
y
[
3
2
]
–
[
4
7
]
F
a
c
t
o
r
s a
f
f
e
c
t
i
n
g
ma
t
h
e
ma
t
i
c
s l
e
a
r
n
i
n
g
[
1
1
]
,
[
4
8
]
–
[
5
4
]
S
TEM
e
d
u
c
a
t
i
o
n
a
n
d
t
e
a
c
h
e
r
r
e
a
d
i
n
e
s
s
[
5
5
]
–
[
6
1
]
3
.
1
.
M
a
t
hema
t
ics t
ea
ching
s
t
ra
t
eg
ies a
nd
peda
g
o
g
y
Ma
th
em
atics
teac
h
in
g
s
tr
ate
g
ies
an
d
p
ed
ag
o
g
y
ar
e
cr
iti
ca
l
to
ac
h
iev
i
n
g
s
u
cc
ess
f
u
l
ed
u
ca
tio
n
al
o
u
tco
m
es.
Sev
en
teen
s
tu
d
ies
s
p
ec
if
ically
ad
d
r
ess
th
e
r
ea
d
i
n
ess
o
f
m
ath
em
atics
teac
h
er
s
,
em
p
h
asizin
g
th
e
im
p
o
r
tan
ce
o
f
ef
f
ec
tiv
e
in
s
tr
u
ctio
n
al
s
tr
ateg
ies
[
3
2
]
–
[
4
7
]
.
J
ó
zsa
et
a
l
.
[
3
2
]
u
s
ed
th
e
d
ia
g
n
o
s
tic
ass
ess
m
en
t
s
y
s
tem
s
f
o
r
d
ev
elo
p
m
e
n
t
(
DI
FER)
test
a
s
an
ass
e
s
s
m
en
t
t
o
o
l
to
g
au
g
e
r
ea
d
in
ess
at
th
e
s
ch
o
o
l
lev
el.
T
h
e
f
in
d
in
g
s
in
clu
d
e
co
m
p
r
eh
e
n
s
iv
e
d
ata
o
n
s
ev
er
al
asp
ec
ts
,
in
clu
d
in
g
c
o
u
n
tr
y
,
ag
e,
an
d
g
en
d
er
.
T
h
is
ex
ten
s
iv
e
s
tu
d
y
o
f
f
er
s
in
s
ig
h
ts
f
o
r
tea
ch
er
s
an
d
s
ch
o
o
l
lead
er
s
h
ip
o
n
en
h
an
cin
g
class
r
o
o
m
s
tr
ateg
ies.
I
n
a
d
d
itio
n
,
Sas
s
et
a
l
.
[
4
7
]
also
p
r
o
p
o
s
e
th
at
f
u
tu
r
e
s
tu
d
ies
s
h
o
u
ld
in
v
esti
g
ate
th
e
im
p
ac
t
o
f
teac
h
er
in
s
tr
u
ctio
n
al
s
tr
ateg
ies o
n
th
ese
ch
an
g
es a
n
d
ex
p
lo
r
e
th
e
ex
te
n
t to
wh
ich
t
h
ey
ar
e
im
p
ac
ted
.
So
m
e
s
ch
o
lar
s
ar
g
u
e
th
at
a
p
e
d
ag
o
g
ical
ap
p
r
o
ac
h
th
at
f
o
c
u
s
es
o
n
th
e
p
r
o
d
u
ct
o
f
m
ath
em
a
tics
r
ath
er
th
an
th
e
p
r
o
ce
s
s
ca
n
af
f
ec
t
s
tu
d
en
t
lear
n
in
g
.
T
h
e
way
teac
h
er
s
teac
h
af
f
ec
ts
th
e
lear
n
in
g
p
r
o
ce
s
s
o
f
s
tu
d
en
ts
[
1
9
]
.
T
h
e
s
elec
tio
n
a
n
d
c
o
n
s
id
er
atio
n
o
f
teac
h
in
g
tech
n
iq
u
es
an
d
m
ate
r
ials
ap
p
r
o
p
r
iate
f
o
r
s
tu
d
en
t
n
ee
d
s
m
u
s
t
b
e
r
ef
in
ed
.
T
h
is
co
n
d
itio
n
n
ec
ess
itate
s
th
e
d
ev
elo
p
m
en
t
o
f
s
u
p
p
o
r
t
m
ater
ials
th
at
ca
n
s
er
v
e
as
a
g
u
id
elin
e
f
o
r
m
ath
em
atics
teac
h
er
s
in
m
e
etin
g
ed
u
ca
tio
n
al
o
b
jectiv
es
wh
ile
d
ev
elo
p
in
g
th
eir
p
er
s
o
n
al
ab
ilit
ies
an
d
k
n
o
wled
g
e
[
3
3
]
.
T
h
e
cu
r
r
en
t
s
tu
d
y
’
s
f
in
d
in
g
s
f
u
r
th
er
s
u
g
g
est
th
at
s
u
p
p
o
r
t
f
o
r
m
ath
em
at
ics
teac
h
er
s
s
h
o
u
ld
in
clu
d
e
less
o
n
p
r
e
p
ar
atio
n
,
m
ed
ia
s
elec
tio
n
an
d
u
s
e,
m
ater
i
al
p
r
o
d
u
ctio
n
,
an
d
ac
tiv
it
y
d
e
s
ig
n
th
at
co
n
s
id
er
s
s
tu
d
en
ts
’
r
eq
u
ir
em
e
n
ts
[
4
0
]
–
[
4
2
]
.
A
f
u
r
th
e
r
co
n
s
id
er
atio
n
to
c
o
n
s
id
er
is
th
e
im
p
ac
t
o
f
o
n
g
o
in
g
p
r
o
f
ess
io
n
al
d
ev
elo
p
m
en
t
a
n
d
r
ef
lectio
n
o
n
th
e
q
u
ality
o
f
m
ath
em
ati
cs
teac
h
er
ed
u
ca
tio
n
.
T
h
is
d
y
n
am
ic
f
ield
o
f
ed
u
ca
tio
n
r
e
q
u
ir
es
co
llab
o
r
ativ
e
ef
f
o
r
ts
,
s
u
ch
as
co
n
tin
u
o
u
s
r
e
s
ea
r
ch
an
d
d
ev
elo
p
m
en
t,
esp
ec
ially
in
th
e
a
r
ea
s
o
f
p
ed
ag
o
g
y
an
d
cu
r
r
icu
lu
m
.
C
o
m
m
itted
m
ath
em
atics
teac
h
er
s
ar
e
alwa
y
s
p
r
o
ac
tiv
e
i
n
en
h
an
cin
g
t
h
eir
s
k
ills
th
r
o
u
g
h
o
n
g
o
in
g
p
r
o
f
ess
io
n
al
d
ev
elo
p
m
e
n
t,
wo
r
k
s
h
o
p
s
,
co
n
f
er
en
ce
s
,
an
d
p
a
r
ticip
atio
n
i
n
co
llab
o
r
ativ
e
lea
r
n
in
g
co
m
m
u
n
ities
[
6
2
]
,
[
6
3
]
.
B
y
co
n
s
id
er
in
g
m
u
ltip
le
p
o
in
ts
o
f
v
iew
a
n
d
e
x
p
er
ie
n
ce
s
,
teac
h
e
r
ed
u
ca
to
r
s
ca
n
b
etter
ed
u
ca
te
tr
ain
ee
teac
h
er
s
to
f
ac
e
th
e
ch
allen
g
es
o
f
teac
h
in
g
m
ath
em
atics
an
d
p
r
o
m
o
te
a
n
in
clu
s
iv
e
lear
n
in
g
en
v
ir
o
n
m
en
t.
Acc
o
r
d
i
n
g
t
o
s
tu
d
ies,
u
tili
zin
g
d
ig
ital
wo
r
k
s
h
ee
ts
an
d
in
s
tr
u
ctio
n
al
v
id
eo
s
in
m
ath
em
atics
class
es
ca
n
b
o
o
s
t
s
tu
d
en
t
en
g
ag
em
e
n
t
an
d
lead
to
a
d
e
ep
er
u
n
d
er
s
tan
d
in
g
o
f
m
ath
c
o
n
ce
p
ts
[
4
3
]
.
W
illi
am
s
et
a
l.
[
4
6
]
co
n
v
in
cin
g
ly
s
u
g
g
est
th
at
p
r
o
p
er
in
itiativ
e
s
f
o
r
teac
h
er
s
ca
n
p
o
s
itiv
ely
af
f
ec
t
th
eir
b
eh
av
i
o
r
an
d
m
o
tiv
atio
n
to
teac
h
m
ath
em
atics.
Fin
al
ly
,
b
r
id
g
in
g
th
e
g
ap
in
m
ath
e
m
a
ti
c
s
ed
u
c
at
io
n
b
e
t
wee
n
s
ec
o
n
d
ar
y
an
d
u
n
iv
er
s
i
ty
l
ev
el
s
s
h
o
u
ld
b
e
em
p
h
a
s
i
ze
d
.
T
o
d
ay
,
s
tu
d
e
n
t
s
n
ee
d
to
b
e
in
s
ti
l
led
wi
th
a
v
ar
ie
ty
o
f
k
n
o
w
led
g
e
an
d
s
k
il
l
s
th
at
ar
e
m
o
r
e
up
-
to
-
d
at
e.
Fo
r
ex
am
p
le,
p
r
o
v
id
in
g
s
tu
d
en
t
s
wi
th
h
ig
h
er
p
r
o
g
r
am
s
an
d
co
u
r
s
e
s
s
u
ch
a
s
ad
v
an
c
ed
m
ath
em
a
t
ic
s
co
u
r
s
e
s
[
4
5
]
.
T
h
e
im
p
o
r
t
an
c
e
o
f
im
p
r
o
v
in
g
th
e
co
m
p
et
en
c
e
o
f
te
ac
h
e
r
s
to
k
ee
p
p
ac
e
w
ith
th
e
ch
an
g
e
s
th
a
t
o
cc
u
r
.
T
h
e
s
tu
d
y
’
s
f
in
d
in
g
s
r
ev
e
al
th
a
t
th
e
o
u
t
co
m
e
s
o
f
th
e
an
aly
s
is
o
f
th
e
r
ela
t
io
n
s
h
ip
b
et
we
en
h
ig
h
s
ch
o
o
l
g
r
ad
e
s
an
d
u
n
iv
e
r
s
ity
ac
h
iev
em
en
t
h
ig
h
l
ig
h
t
t
h
e
im
p
o
r
t
an
c
e
o
f
co
m
p
r
eh
en
s
iv
e
a
s
s
es
s
m
en
t
an
d
co
l
lab
o
r
at
iv
e
p
r
ac
t
ice
s
a
m
o
n
g
ed
u
ca
to
r
s
a
t
a
ll
lev
el
s
[
3
7
]
,
[
3
9
]
.
I
n
co
n
clu
s
io
n
,
t
h
e
r
e
ad
i
n
es
s
o
f
m
ath
em
a
t
ic
s
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2252
-
8
8
2
2
I
n
t
J
E
v
al
&
R
es E
d
u
c
,
Vo
l
.
14
,
No
.
2
,
Ap
r
il
20
25
:
1
4
6
8
-
1
4
7
6
1472
tea
ch
er
s
to
ad
ap
t
to
th
e
ch
a
n
g
in
g
ed
u
ca
tio
n
al
p
ar
a
d
ig
m
an
d
m
a
in
t
ain
a
g
o
o
d
l
ea
r
n
in
g
en
v
ir
o
n
m
en
t
i
s
im
p
o
r
tan
t
t
o
im
p
r
o
v
e
m
a
th
e
m
at
ic
s
ed
u
c
at
io
n
an
d
p
r
ep
ar
e
s
t
u
d
en
t
s
to
s
u
cc
ee
d
in
an
in
cr
ea
s
in
g
ly
co
m
p
l
ex
wo
r
l
d
[
3
8
]
,
[
4
7
]
.
3
.
2
.
F
a
ct
o
rs a
f
f
ec
t
ing
m
a
t
he
m
a
t
ics lea
rning
Sev
er
al
f
ac
to
r
s
in
f
lu
e
n
ce
m
ath
em
atics
lear
n
in
g
,
wh
ich
in
tu
r
n
a
f
f
ec
ts
m
ath
em
atics
teac
h
er
s
’
r
ea
d
in
ess
.
Key
f
ac
to
r
s
in
clu
d
e
th
e
lev
el
o
f
s
tu
d
en
t
ab
ilit
y
[
4
8
]
,
[
4
9
]
,
t
h
e
tr
an
s
itio
n
to
o
n
lin
e
teac
h
in
g
an
d
lear
n
in
g
[
1
1
]
,
[
5
0
]
,
t
h
e
in
teg
r
atio
n
o
f
v
ar
io
u
s
lear
n
in
g
d
i
s
cip
lin
es
[
5
1
]
,
[
5
2
]
,
lim
ited
t
ea
ch
in
g
s
k
ills
an
d
p
r
o
f
ess
io
n
al
teac
h
er
tr
ai
n
in
g
[
5
3
]
,
[
5
4
]
.
T
h
e
s
tu
d
y
em
p
h
asiz
es th
e
cr
u
cial
d
em
an
d
f
o
r
m
at
h
em
atics tea
ch
er
s
to
en
h
an
ce
th
eir
s
k
ills
to
p
r
o
p
e
r
ly
ad
d
r
ess
k
id
s
’
ar
ith
m
etic
d
if
f
icu
lties
,
u
n
d
er
s
co
r
in
g
th
at
u
n
r
eso
lv
ed
m
at
h
an
x
iety
ca
n
s
ev
er
ely
in
f
lu
e
n
ce
ch
ild
r
en
’
s
m
o
tiv
atio
n
an
d
ac
ad
em
ic
p
er
f
o
r
m
a
n
ce
.
Utilizin
g
f
o
r
m
ativ
e
ass
es
s
m
en
ts
is
ad
v
is
ed
as
an
ef
f
ec
tiv
e
a
p
p
r
o
ac
h
f
o
r
tea
ch
er
s
to
d
eter
m
in
e
s
tu
d
en
ts
’
u
n
d
e
r
s
tan
d
in
g
a
n
d
ch
allen
g
es,
th
u
s
allo
win
g
th
e
m
to
cr
ea
te
cu
s
to
m
ized
s
u
p
p
o
r
t
s
y
s
tem
s
to
im
p
r
o
v
e
lea
r
n
in
g
o
u
tco
m
es
[
4
8
]
.
Hen
ce
,
u
s
in
g
th
is
ap
p
r
o
ac
h
h
elp
s
id
en
tify
ar
ea
s
o
f
d
if
f
i
cu
lty
an
d
g
u
id
es
th
e
d
ev
elo
p
m
en
t
o
f
s
p
ec
if
ic
in
s
tr
u
ctio
n
al
s
tr
ateg
ies to
ass
is
t stu
d
en
t le
ar
n
in
g
[
6
4
]
.
Fu
r
th
er
m
o
r
e
,
th
e
u
tili
za
tio
n
o
f
tech
n
o
l
o
g
y
in
m
ath
e
m
atics
ed
u
ca
tio
n
e
m
p
h
asizes
th
e
im
p
o
r
tan
ce
o
f
teac
h
er
r
ea
d
i
n
ess
wh
en
u
s
in
g
a
n
ew
in
s
tr
u
ctio
n
al
a
p
p
r
o
ac
h
.
T
ea
ch
er
s
m
u
s
t
b
e
p
r
o
ac
tiv
e
a
n
d
ad
a
p
tab
le
wh
en
u
s
in
g
v
is
u
al
aid
s
an
d
tr
an
s
iti
o
n
to
tech
n
o
lo
g
y
-
e
n
h
an
ce
d
teac
h
in
g
ap
p
r
o
ac
h
es
[
1
1
]
,
[
5
0
]
.
Fu
jita
et
a
l.
[
1
1
]
r
esear
ch
ed
th
e
p
r
o
b
lem
s
an
d
p
o
s
itiv
e
attitu
d
es
o
f
J
ap
a
n
ese
m
ath
em
atics
teac
h
er
s
to
war
d
o
n
lin
e
ed
u
ca
tio
n
d
u
r
in
g
t
h
e
p
an
d
e
m
ic,
em
p
h
as
izin
g
th
e
n
ec
ess
ity
o
f
r
ea
d
in
e
s
s
,
p
o
s
itiv
e
attitu
d
es,
an
d
s
u
p
p
o
r
t
s
y
s
tem
s
.
T
h
e
p
r
esen
t
f
in
d
in
g
also
s
u
p
p
o
r
ts
Fah
r
u
r
o
zi
et
a
l.
[
1
5
]
wh
o
co
n
clu
d
ed
th
at
to
n
a
v
ig
ate
th
e
ch
an
g
in
g
ed
u
ca
tio
n
al
lan
d
s
ca
p
e,
m
at
h
em
atics
teac
h
er
s
m
u
s
t
en
g
ag
e
in
o
n
g
o
in
g
p
r
o
f
ess
io
n
al
d
ev
el
o
p
m
en
t
an
d
s
e
lf
-
d
ir
ec
ted
lear
n
in
g
to
im
p
r
o
v
e
th
eir
tech
n
o
lo
g
ical
co
m
p
eten
cies
a
n
d
in
s
tr
u
ctio
n
al
ap
p
r
o
ac
h
es,
en
s
u
r
i
n
g
th
e
y
a
r
e
well
-
p
r
ep
a
r
ed
to
u
s
e
tech
n
o
lo
g
y
in
th
e
class
r
o
o
m
.
T
o
f
o
s
ter
ef
f
ec
tiv
e
m
ath
e
m
atics
in
s
tr
u
ctio
n
,
teac
h
er
s
m
u
s
t
b
len
d
cr
ea
tiv
ity
with
in
teg
r
ati
n
g
d
iv
e
r
s
e
d
is
cip
lin
es,
a
n
ec
ess
ity
u
n
d
er
s
co
r
ed
b
y
Kir
wan
et
a
l.
r
esear
c
h
[
5
2
]
,
wh
ich
id
en
tifie
s
s
ix
ed
u
ca
tio
n
al
ac
tiv
ities
cr
u
cial
f
o
r
in
teg
r
atin
g
in
f
o
r
m
atio
n
in
teac
h
er
d
ev
elo
p
m
en
t
p
r
o
g
r
a
m
s
.
T
h
is
in
teg
r
atio
n
is
f
u
n
d
am
en
tal
f
o
r
en
h
an
cin
g
m
ath
em
atica
l
u
n
d
er
s
tan
d
in
g
.
Ad
d
itio
n
ally
,
d
ev
elo
p
in
g
tailo
r
e
d
m
o
d
u
les,
m
o
d
els,
m
eth
o
d
s
,
an
d
s
tr
ateg
ies
is
ess
en
tia
l
f
o
r
o
p
tim
izin
g
s
tu
d
en
t
o
u
tco
m
es
in
m
ath
em
atics.
Ho
wev
er
,
a
g
a
p
ex
is
ts
in
p
r
ac
tice,
p
ar
ticu
lar
ly
in
s
en
io
r
h
ig
h
s
ch
o
o
ls
,
wh
er
e
m
an
y
teac
h
er
s
r
el
y
h
ea
v
ily
o
n
tex
tb
o
o
k
s
d
u
e
t
o
a
lack
o
f
p
r
e
p
ar
ed
m
ater
ial
[
1
9
]
.
A
d
d
r
ess
in
g
t
h
is
g
ap
b
y
d
e
v
elo
p
in
g
a
n
d
a
d
o
p
t
in
g
co
m
p
r
eh
en
s
iv
e
m
o
d
els
an
d
m
ater
ials
tailo
r
ed
to
m
ath
em
atics
ed
u
ca
tio
n
ca
n
s
ig
n
if
ican
tly
im
p
ac
t
teac
h
er
r
ea
d
in
ess
an
d
e
f
f
ec
tiv
en
ess
,
u
lt
im
ately
im
p
r
o
v
in
g
s
tu
d
en
t le
ar
n
in
g
o
u
tco
m
es.
3
.
3
.
ST
E
M
educa
t
io
n a
nd
t
e
a
cher
re
a
din
ess
T
ab
le
3
p
r
esen
ts
th
at
th
e
r
elatio
n
s
h
ip
b
etwe
en
STE
M
ed
u
ca
tio
n
a
n
d
teac
h
e
r
r
ea
d
i
n
ess
is
an
im
p
o
r
tan
t p
a
r
t o
f
e
d
u
ca
tio
n
to
d
ay
,
p
r
o
m
p
tin
g
th
e
in
v
esti
g
atio
n
o
f
t
h
is
to
p
ic
in
s
ev
er
al
s
tu
d
ies.
T
h
er
e
ar
e
s
ev
en
p
u
b
lis
h
ed
s
tu
d
ies
t
h
at
ex
p
r
ess
ly
ad
d
r
ess
th
is
th
em
e
[
4
4
]
,
[
5
3
]
–
[
5
8
]
.
Kaz
ak
h
s
tan
’
s
s
tu
d
y
in
teg
r
ates
d
ig
ital
ed
u
ca
tio
n
al
r
eso
u
r
ce
s
to
h
elp
t
ea
ch
er
s
lear
n
m
ath
em
atics
[
5
5
]
.
T
h
e
f
in
d
i
n
g
s
p
r
o
v
e
th
at
u
s
in
g
d
ig
ital
r
eso
u
r
ce
s
im
p
r
o
v
es
lea
r
n
in
g
ef
f
ec
tiv
e
n
ess
an
d
ac
a
d
em
ic
p
er
f
o
r
m
an
ce
,
in
d
icatin
g
t
h
e
n
ee
d
t
o
in
co
r
p
o
r
ate
th
is
tech
n
o
lo
g
y
in
to
class
r
o
o
m
p
r
ac
tices.
I
n
ad
d
itio
n
,
Ma
r
tin
ez
an
d
E
llis
[
5
6
]
u
n
d
er
lin
e
th
at
co
n
f
id
en
ce
in
o
n
e’
s
ac
ad
em
ic
ab
ilit
y
is
a
s
tr
o
n
g
p
r
ed
ict
o
r
o
f
ac
ad
em
ic
p
er
f
o
r
m
an
ce
.
Pre
v
io
u
s
s
ch
o
lar
s
al
s
o
em
p
h
asized
th
e
im
p
o
r
tan
ce
o
f
teac
h
e
r
s
’
attitu
d
es
an
d
b
elief
s
s
in
ce
th
ey
in
clu
d
e
ep
is
tem
ic
p
r
ac
tices
t
h
at
ar
e
d
if
f
icu
lt
to
ch
an
g
e
[
6
5
]
,
[
6
6
]
.
Ou
r
f
in
d
in
g
s
i
n
d
i
ca
te
th
at
r
e
s
e
ar
ch
i
s
b
ein
g
p
er
f
o
r
m
ed
t
o
in
v
e
s
t
ig
a
te
th
e
co
m
p
l
ex
i
t
ie
s
o
f
STE
M
tea
ch
er
id
en
ti
ty
an
d
h
o
w
it
in
f
lu
e
n
ce
s
c
la
s
s
r
o
o
m
p
r
ac
t
ice
.
T
o
in
c
lu
d
e
STE
M
in
ele
m
en
tar
y
s
ch
o
o
l
ed
u
ca
t
io
n
,
t
ea
ch
er
s
m
u
s
t
h
a
v
e
a
s
o
p
h
i
s
t
ic
at
ed
g
r
a
s
p
o
f
s
e
lf
-
ef
f
i
ca
cy
an
d
S
T
E
M
ca
r
ee
r
a
war
en
e
s
s
[
5
7
]
.
Fu
r
th
er
m
o
r
e,
th
e
i
s
s
u
e
o
f
s
e
c
o
n
d
ar
y
s
ch
o
o
l
t
ea
c
h
er
s
b
ein
g
p
r
ep
ar
ed
to
te
ac
h
ST
E
M
s
u
b
j
ec
t
s
em
p
h
a
s
i
ze
s
th
e
n
ec
e
s
s
ity
o
f
h
av
in
g
a
s
o
lid
tea
ch
in
g
k
n
o
w
led
g
e
b
a
s
e
a
n
d
b
ein
g
aw
ar
e
o
f
S
T
E
M
o
p
p
o
r
tu
n
it
ie
s
wh
en
d
ete
r
m
i
n
in
g
o
n
e’
s
r
e
ad
in
e
s
s
f
o
r
STE
M
ed
u
ca
t
io
n
[
5
8
]
.
T
o
te
ac
h
ef
f
ec
tiv
ely
,
tea
ch
e
r
s
m
u
s
t
c
u
l
tiv
at
e
th
e
ab
i
li
ty
to
s
tr
u
ctu
r
e
to
p
i
c
k
n
o
wl
ed
g
e
an
d
s
k
i
l
ls
ar
o
u
n
d
f
u
n
d
am
en
ta
l
p
r
in
c
ip
l
e
s
.
T
h
e
s
e
f
i
n
d
in
g
s
s
u
g
g
e
s
t
t
h
e
n
ee
d
f
o
r
p
er
s
o
n
al
iz
ed
p
r
o
f
e
s
s
i
o
n
a
l
d
ev
e
lo
p
m
en
t
an
d
in
s
t
r
u
ct
io
n
a
l
m
o
d
e
l
s
th
a
t
ca
n
h
elp
t
ea
c
h
er
s
b
e
tt
er
ex
ec
u
te
S
T
E
M
in
s
tr
u
ct
io
n
.
An
o
th
er
s
tu
d
y
e
v
al
u
a
tin
g
t
ea
ch
er
s
’
r
ea
d
in
e
s
s
to
d
e
l
iv
er
STE
M
ed
u
ca
tio
n
h
a
s
le
d
to
th
e
d
ev
e
lo
p
m
en
t
o
f
v
a
lid
at
ed
s
ca
l
e
s
,
s
u
ch
a
s
th
e
T
R
i
-
S
T
E
M
s
ca
l
e
[
5
9
]
.
T
h
i
s
a
s
s
e
s
s
m
en
t,
e
n
co
m
p
a
s
s
in
g
em
o
t
io
n
a
l
f
a
cto
r
s
,
m
en
ta
l
s
ta
t
es
,
b
el
ief
in
o
n
e’
s
ab
il
it
ie
s
,
a
n
d
d
ed
ic
at
io
n
to
STE
M
ed
u
ca
t
i
o
n
,
is
an
ex
c
el
len
t
in
s
tr
u
m
en
t
f
o
r
p
r
e
ci
s
e
ly
ev
a
lu
a
tin
g
tea
ch
e
r
r
ea
d
in
e
s
s
.
C
o
n
s
e
q
u
en
tly
,
s
u
r
v
ey
s
co
n
d
u
c
ted
am
o
n
g
S
T
E
M
ed
u
ca
to
r
s
i
n
T
a
iw
an
a
ls
o
in
d
i
ca
t
e
p
o
s
i
tiv
e
a
tt
itu
d
e
s
to
war
d
in
te
g
r
a
tiv
e
S
T
E
M
ed
u
c
at
io
n
[
6
0
]
.
T
h
e
f
in
d
in
g
s
o
f
th
es
e
s
u
r
v
ey
s
h
ig
h
lig
h
t
th
e
n
ee
d
f
o
r
s
u
f
f
ic
ien
t r
e
s
o
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r
c
e
s
a
n
d
s
u
p
p
o
r
t
to
p
r
o
m
o
te
in
te
r
d
i
s
cip
lin
ar
y
te
ac
h
in
g
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t
J
E
v
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&
R
es E
d
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c
I
SS
N:
2252
-
8
8
2
2
Th
e
r
ea
d
in
ess
o
f m
a
th
ema
tics
tea
ch
ers
a
s
a
g
en
ts
o
f c
h
a
n
g
e
:
a
r
ec
en
t c
o
mp
r
eh
en
s
ive
…
(
S
h
a
r
id
a
A
b
u
Ta
lib
)
1473
m
eth
o
d
s
.
M
o
r
eo
v
er
,
a
s
tu
d
y
in
C
h
in
a
a
s
s
e
s
s
e
s
th
e
in
f
l
u
en
ce
o
f
b
eh
av
io
r
al
an
d
p
s
y
ch
o
lo
g
ic
al
f
a
cto
r
s
co
n
ce
r
n
in
g
th
e
ch
il
d
r
en
a
ca
d
em
i
c
ac
h
iev
em
en
t
in
STE
M
d
i
s
ci
p
l
in
e
s
f
r
o
m
k
in
d
er
g
ar
ten
to
1
2
th
g
r
ad
e
[
6
1
]
.
T
h
i
s
s
tu
d
y
em
p
h
a
s
i
ze
s
th
e
s
i
g
n
if
i
ca
n
ce
o
f
u
t
il
iz
in
g
i
n
t
er
v
en
t
io
n
s
to
im
p
r
o
v
e
p
s
y
ch
o
lo
g
ica
l
c
h
ar
a
ct
er
is
t
ic
s
s
u
ch
a
s
co
n
f
id
en
c
e
in
u
s
in
g
th
e
in
ter
n
et
an
d
o
f
f
er
in
g
g
u
id
an
ce
to
s
tu
d
en
t
s
o
n
t
h
e
ef
f
ec
tiv
e
u
s
e
o
f
in
f
o
r
m
at
io
n
an
d
co
m
m
u
n
ica
t
i
o
n
s
t
ec
h
n
o
lo
g
ie
s
.
T
h
e
f
i
n
d
i
n
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o
f
t
h
i
s
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n
h
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n
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t
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t
s
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p
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f
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e
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n
S
T
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M
d
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c
i
p
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a
n
d
t
h
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p
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o
b
l
e
m
-
s
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n
g
s
k
i
l
ls
.
4.
CO
NCLU
SI
O
N
E
v
id
en
tly
,
th
e
r
ec
en
t
c
o
m
p
r
e
h
en
s
iv
e
r
ev
iew
s
y
n
th
esizes
ex
i
s
tin
g
k
n
o
wled
g
e
r
eg
ar
d
i
n
g
th
e
r
ea
d
in
ess
o
f
m
ath
em
atics
teac
h
er
s
to
a
ct
as
ag
en
ts
o
f
ch
a
n
g
e.
As
e
d
u
ca
tio
n
e
v
o
lv
es,
id
e
n
tify
in
g
an
d
ad
d
r
ess
in
g
th
e
ch
allen
g
es
th
at
m
ath
em
atics
teac
h
er
s
en
co
u
n
ter
is
cr
u
cial
to
p
r
o
v
id
i
n
g
a
d
y
n
am
ic
an
d
ef
f
ec
tiv
e
lear
n
in
g
en
v
ir
o
n
m
en
t
f
o
r
s
tu
d
e
n
ts
.
Mo
r
eo
v
er
,
th
is
s
tu
d
y
o
n
th
e
r
ea
d
in
ess
o
f
m
ath
em
atics
teac
h
er
s
as
ch
an
g
e
ag
en
ts
u
s
in
g
PR
I
SMA
is
u
n
iq
u
e
in
th
at
it
tak
es
a
co
m
p
lete
an
d
t
r
an
s
p
ar
en
t
ap
p
r
o
ac
h
t
o
g
ath
e
r
in
g
,
f
ilter
in
g
,
as
well
as
r
ep
o
r
tin
g
c
u
r
r
en
t
r
esear
ch
.
T
h
e
f
in
d
in
g
s
ar
e
in
ten
d
e
d
t
o
co
n
tr
ib
u
te
to
th
e
cu
r
r
en
t
d
i
s
cu
s
s
io
n
r
eg
ar
d
in
g
m
ath
em
atics
teac
h
er
r
ea
d
in
ess
an
d
p
r
o
m
o
te
ev
id
e
n
ce
-
b
ased
tech
n
iq
u
es
f
o
r
im
p
r
o
v
in
g
f
lex
ib
ilit
y
,
in
n
o
v
atio
n
,
an
d
r
esil
ien
ce
in
m
ath
em
atics
teac
h
er
s
.
B
ased
o
n
th
e
an
aly
s
i
s
o
f
3
1
r
ec
en
t
ar
ticles,
o
u
r
m
ain
k
e
y
f
i
n
d
in
g
s
ca
n
b
e
s
u
m
m
ar
ized
:
i
)
m
ath
em
ati
cs
teac
h
in
g
s
tr
ateg
ies
an
d
p
ed
ag
o
g
y
;
ii
)
f
ac
to
r
s
af
f
ec
tin
g
m
ath
em
atic
lear
n
in
g
;
an
d
iii
)
STE
M
ed
u
ca
tio
n
an
d
t
ea
ch
er
r
ea
d
i
n
ess
.
T
h
is
s
tu
d
y
is
ex
p
ec
ted
to
o
f
f
er
r
esear
ch
er
s
in
s
ig
h
ts
f
o
r
f
u
r
th
er
d
e
v
elo
p
in
g
th
is
u
n
d
er
s
tan
d
in
g
,
esp
ec
ially
with
in
th
e
co
n
tex
t
o
f
Ma
lay
s
ia.
Mo
r
e
b
r
o
ad
ly
,
r
esear
ch
is
also
n
ee
d
ed
to
f
o
c
u
s
o
n
m
ath
em
atics
teac
h
er
s
’
r
ea
d
in
ess
as
ch
an
g
e
ag
en
ts
,
wh
ich
m
ay
b
e
u
s
ed
as
g
u
id
es
an
d
ch
ec
k
lis
ts
s
p
ec
if
ic
ally
with
r
eg
ar
d
to
th
e
p
ed
a
g
o
g
ical
d
im
en
s
io
n
.
C
o
llectiv
ely
,
th
ese
r
esu
lts
in
d
icate
th
at
m
ec
h
a
n
is
m
s
m
ay
co
n
tr
ib
u
te
to
t
h
e
teac
h
er
’
s
p
r
o
f
ess
io
n
al
d
ev
elo
p
m
en
t
an
d
f
u
n
cti
o
n
as
an
in
i
tial
s
tep
to
war
d
d
ev
elo
p
i
n
g
p
ed
ag
o
g
ical
m
o
d
els,
n
o
tab
ly
with
in
th
e
Ma
lay
s
i
an
s
ettin
g
.
C
o
n
s
eq
u
en
tly
,
a
d
d
itio
n
al
i
n
v
esti
g
atio
n
a
n
d
d
ev
elo
p
m
e
n
t
o
f
a
n
in
teg
r
ated
m
o
d
el
p
er
tain
i
n
g
to
th
e
r
ea
d
in
ess
o
f
m
ath
em
atics
teac
h
er
s
to
ac
t
as
ag
e
n
ts
o
f
ch
an
g
e
f
o
r
ac
ad
em
ician
s
an
d
p
r
ac
titi
o
n
e
r
s
in
m
ath
em
atics e
d
u
ca
tio
n
ar
e
r
eq
u
ir
ed
.
O
v
er
all,
th
is
s
tu
d
y
h
as im
p
licatio
n
s
f
o
r
th
e
f
ield
o
f
k
n
o
wled
g
e
an
d
r
el
ev
an
t
p
a
r
ties
s
u
ch
as
s
tak
eh
o
l
d
er
s
an
d
m
ath
e
m
atics
teac
h
er
s
.
Stak
eh
o
ld
er
s
c
an
u
s
e
th
ese
f
in
d
in
g
s
to
f
o
r
m
u
late
p
o
licies
co
n
ce
r
n
in
g
c
o
n
tin
u
in
g
p
r
o
f
ess
io
n
al
d
e
v
elo
p
m
en
t
(
C
PD)
f
o
r
m
ath
em
atics tea
ch
er
s
an
d
teac
h
er
tr
ain
in
g
in
s
titu
tes to
d
ev
is
e
ef
f
ec
tiv
e
s
tr
ateg
ies f
o
r
g
r
ad
u
ate
r
ea
d
in
ess
.
RE
F
E
R
E
NC
E
S
[
1
]
C
.
M
a
k
a
m
u
r
e
a
n
d
M
.
Tsa
k
e
n
i
,
“
C
O
V
I
D
-
1
9
a
s
a
n
a
g
e
n
t
o
f
c
h
a
n
g
e
i
n
t
e
a
c
h
i
n
g
a
n
d
l
e
a
r
n
i
n
g
S
TE
M
s
u
b
j
e
c
t
s
,
”
J
o
u
rn
a
l
o
f
B
a
l
t
i
c
S
c
i
e
n
c
e
E
d
u
c
a
t
i
o
n
,
v
o
l
.
1
9
,
n
o
.
6
,
p
p
.
1
0
7
8
–
1
0
9
1
,
2
0
2
0
,
d
o
i
:
1
0
.
3
3
2
2
5
/
J
B
S
E/
2
0
.
1
9
.
1
0
7
8
.
[
2
]
F
.
M
e
y
e
r
a
n
d
K
.
S
l
a
t
e
r
-
B
r
o
w
n
,
“
Ed
u
c
a
t
i
o
n
a
l
c
h
a
n
g
e
d
o
e
s
n
’
t
c
o
me
e
a
sy
:
l
e
a
d
t
e
a
c
h
e
r
s’
w
o
r
k
a
s
c
h
a
n
g
e
a
g
e
n
t
s,”
Ma
t
h
e
m
a
t
i
c
s
Ed
u
c
a
t
i
o
n
Re
se
a
r
c
h
J
o
u
r
n
a
l
,
v
o
l
.
3
4
,
n
o
.
1
,
p
p
.
1
3
9
–
1
6
3
,
2
0
2
2
,
d
o
i
:
1
0
.
1
0
0
7
/
s
1
3
3
9
4
-
0
2
0
-
0
0
3
3
3
-
y.
[
3
]
J.
W
h
i
t
e
,
P
.
Jo
h
n
so
n
,
a
n
d
M
.
G
o
o
s
,
“
P
r
e
-
ser
v
i
c
e
t
e
a
c
h
e
r
s
a
s
a
g
e
n
t
s
o
f
c
h
a
n
g
e
i
n
t
h
e
m
a
t
h
e
m
a
t
i
c
s
c
l
a
ssr
o
o
m
:
a
c
a
se
st
u
d
y
,
”
Ma
t
h
e
m
a
t
i
c
s Te
a
c
h
e
r
E
d
u
c
a
t
i
o
n
a
n
d
D
e
v
e
l
o
p
m
e
n
t
,
v
o
l
.
2
3
,
n
o
.
1
,
p
p
.
5
4
–
7
3
,
2
0
2
1
.
[
4
]
L.
B
i
b
e
r
ma
n
-
S
h
a
l
e
v
,
“
M
o
t
i
v
a
t
i
o
n
a
l
f
a
c
t
o
r
s
f
o
r
l
e
a
r
n
i
n
g
a
n
d
t
e
a
c
h
i
n
g
g
l
o
b
a
l
e
d
u
c
a
t
i
o
n
,
”
T
e
a
c
h
i
n
g
a
n
d
T
e
a
c
h
e
r
Ed
u
c
a
t
i
o
n
,
v
o
l
.
1
0
6
,
p
.
1
0
3
4
6
0
,
2
0
2
1
,
d
o
i
:
1
0
.
1
0
1
6
/
j
.
t
a
t
e
.
2
0
2
1
.
1
0
3
4
6
0
.
[
5
]
P
.
N
.
B
l
a
c
k
,
“
A
r
e
v
o
l
u
t
i
o
n
i
n
b
i
o
c
h
e
mi
st
r
y
a
n
d
m
o
l
e
c
u
l
a
r
b
i
o
l
o
g
y
e
d
u
c
a
t
i
o
n
i
n
f
o
r
me
d
b
y
b
a
s
i
c
r
e
s
e
a
r
c
h
t
o
m
e
e
t
t
h
e
d
e
m
a
n
d
s
o
f
2
1
s
t
c
e
n
t
u
r
y
c
a
r
e
e
r
p
a
t
h
s,”
J
o
u
rn
a
l
o
f
B
i
o
l
o
g
i
c
a
l
C
h
e
m
i
s
t
ry
,
v
o
l
.
2
9
5
,
n
o
.
3
1
,
p
p
.
1
0
6
5
3
–
1
0
6
6
1
,
2
0
2
0
,
d
o
i
:
1
0
.
1
0
7
4
/
j
b
c
.
A
W
1
2
0
.
0
1
1
1
0
4
.
[
6
]
M
.
S
.
P
e
r
r
y
,
“
2
1
st
c
e
n
t
u
r
y
sk
i
l
l
s
t
h
r
o
u
g
h
f
i
l
m
p
r
o
d
u
c
t
i
o
n
i
n
t
e
r
t
i
a
r
y
e
d
u
c
a
t
i
o
n
:
a
t
r
a
n
s
f
o
r
m
a
t
i
v
e
a
s
s
e
s
sm
e
n
t
i
n
a
l
i
t
e
r
a
t
u
r
e
a
n
d
m
e
d
i
a
c
o
u
r
se
,
”
3
L
:
L
a
n
g
u
a
g
e
,
L
i
n
g
u
i
s
t
i
c
s
,
L
i
t
e
ra
t
u
re
,
v
o
l
.
2
4
,
n
o
.
4
,
p
p
.
2
1
4
–
2
3
2
,
2
0
1
8
,
d
o
i
:
1
0
.
1
7
5
7
6
/
3
L
-
2
0
1
8
-
2404
-
16.
[
7
]
R
.
B
a
n
e
r
j
e
e
a
n
d
P
.
S
e
s
h
a
i
y
e
r
,
“
C
h
a
l
l
e
n
g
e
s
i
n
sc
h
o
o
l
m
a
t
h
e
m
a
t
i
c
s
c
u
r
r
i
c
u
l
u
m
r
e
f
o
r
m
i
n
I
n
d
i
a
:
t
r
a
n
s
f
o
r
m
i
n
g
t
e
a
c
h
e
r
p
r
a
c
t
i
c
e
s
t
h
r
o
u
g
h
p
e
d
a
g
o
g
i
c
a
l
i
n
n
o
v
a
t
i
o
n
s
,
”
i
n
S
c
h
o
o
l
M
a
t
h
e
m
a
t
i
c
s
C
u
r
ri
c
u
l
a
,
2
0
1
9
,
p
p
.
2
2
7
–
237
,
d
o
i
:
1
0
.
1
0
0
7
/
9
7
8
-
9
8
1
-
13
-
6
3
1
2
-
2
_
1
2
.
[
8
]
S
.
G
ö
k
ç
e
a
n
d
P
.
G
ü
n
e
r
,
“
F
o
r
t
y
y
e
a
r
s
o
f
m
a
t
h
e
m
a
t
i
c
s
e
d
u
c
a
t
i
o
n
:
1
9
8
0
-
2
0
1
9
,
”
I
n
t
e
rn
a
t
i
o
n
a
l
J
o
u
rn
a
l
o
f
E
d
u
c
a
t
i
o
n
i
n
Ma
t
h
e
m
a
t
i
c
s,
S
c
i
e
n
c
e
a
n
d
T
e
c
h
n
o
l
o
g
y
,
v
o
l
.
9
,
n
o
.
3
,
p
p
.
5
1
4
–
5
3
9
,
2
0
2
1
,
d
o
i
:
1
0
.
4
6
3
2
8
/
I
JEM
S
T
.
1
3
6
1
.
[
9
]
S
.
M
u
h
a
mm
a
d
e
t
a
l
.
,
“
R
e
a
d
i
n
e
ss
o
f
S
TE
M
t
e
a
c
h
e
r
s
t
o
i
m
p
l
e
me
n
t
s
t
e
m
e
d
u
c
a
t
i
o
n
i
n
c
o
a
s
t
a
l
s
c
h
o
o
l
s,
”
I
n
t
e
r
n
a
t
i
o
n
a
l
O
n
l
i
n
e
J
o
u
rn
a
l
o
f
L
a
n
g
u
a
g
e
,
C
o
m
m
u
n
i
c
a
t
i
o
n
,
a
n
d
H
u
m
a
n
i
t
i
e
s
,
v
o
l
.
4
,
n
o
.
1
,
p
p
.
5
2
–
7
0
,
2
0
2
2
.
[
1
0
]
A
.
S
.
M
.
N
a
p
i
a
h
a
n
d
M
.
H
a
s
h
i
m,
“
Th
e
l
e
v
e
l
o
f
r
e
a
d
i
n
e
ss
o
f
t
r
a
i
n
e
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Evaluation Warning : The document was created with Spire.PDF for Python.