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1.
I
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D
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Du
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m
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in
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s
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s
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th
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an
s
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s
to
q
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n
s
th
at
ar
is
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[
1
]
–
[
3
]
.
T
h
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ac
tiv
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ar
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wn
as
ex
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m
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in
k
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k
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[
5
]
,
[
6
]
.
I
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T
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k
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x
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f
in
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s
[
7
]
–
[
1
0
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.
I
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f
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m
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ex
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a
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T
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lv
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k
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wh
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ar
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s
till
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elativ
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lo
w
[
4
]
,
[
1
1
]
,
[
1
2
]
.
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T
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ath
em
atica
l e
x
p
lo
r
at
io
n
[
5
]
ar
e
in
clu
d
e
:
i
)
u
n
d
er
s
tan
d
in
g
th
e
p
r
o
b
lem
;
ii
)
ex
am
in
in
g
p
atter
n
s
;
iii
)
s
ea
r
ch
in
g
in
f
o
r
m
ally
;
iv
)
clar
if
y
in
g
th
e
s
o
lu
tio
n
ef
f
o
r
t
;
an
d
v
)
s
y
m
b
o
lizatio
n
an
d
g
e
n
er
aliza
tio
n
.
T
h
ey
in
d
icate
d
t
h
at
th
ey
wer
e
u
n
ab
le
to
p
r
o
p
er
ly
u
n
d
er
s
tan
d
t
h
e
p
r
o
b
lem
c
o
n
tain
ed
i
n
th
e
q
u
esti
o
n
,
d
eter
m
in
e
t
h
e
n
u
m
b
er
p
atter
n
th
at
m
u
s
t
b
e
u
s
ed
,
s
o
lv
e
t
h
e
p
atter
n
u
s
in
g
th
e
n
u
m
b
er
p
atter
n
f
o
r
m
u
la
an
d
u
s
e
o
r
wr
ite
th
e
s
y
m
b
o
ls
u
s
ed
in
th
e
n
u
m
b
er
p
atter
n
.
Fu
r
th
er
m
o
r
e,
f
r
o
m
a
g
en
d
e
r
p
er
s
p
ec
tiv
e,
th
e
r
e
ar
e
g
e
n
er
al
ly
d
if
f
e
r
en
ce
s
in
m
ath
em
atic
al
ab
ilit
ies
am
o
n
g
m
ale
an
d
f
em
ale
s
tu
d
en
ts
in
ex
p
lo
r
in
g
an
d
s
o
lv
in
g
m
ath
e
m
atica
l
p
r
o
b
lem
s
,
alth
o
u
g
h
n
o
t
s
ig
n
if
ican
t.
So
m
e
s
tu
d
ies
m
en
tio
n
th
at
m
ale
s
tu
d
en
ts
’
ab
ilit
y
to
ex
p
l
o
r
e
is
lo
wer
th
an
f
em
ale
s
tu
d
en
ts
an
d
s
o
m
e
s
tu
d
ies m
en
tio
n
th
e
o
p
p
o
s
ite
[
1
4
]
–
[
1
8
]
.
T
h
e
lo
w
m
ath
em
atica
l e
x
p
lo
r
a
tio
n
ab
ilit
y
ca
n
b
e
in
f
lu
e
n
ce
d
b
y
th
e
af
f
ec
tiv
e
asp
ec
ts
o
f
s
tu
d
en
ts
.
T
h
is
is
b
ec
au
s
e
s
tu
d
en
ts
at
titu
d
es
a
n
d
ch
ar
ac
ter
s
af
f
ec
t
th
eir
ab
ilit
y
to
ca
p
tu
r
e
an
d
u
n
d
er
s
tan
d
m
ath
em
atics
lear
n
in
g
[
1
9
]
–
[
2
2
]
.
I
n
lear
n
in
g
m
at
h
em
atics,
o
n
e
o
f
th
e
af
f
ec
tiv
e
asp
e
cts
o
f
s
tu
d
e
n
ts
wh
o
ar
e
tr
ain
ed
is
to
h
av
e
h
a
b
its
o
f
m
in
d
[
2
3
]
–
[
2
5
]
.
Hab
its
o
f
m
i
n
d
wer
e
f
ir
s
t
d
e
v
elo
p
e
d
b
y
C
o
s
ta
an
d
Kallick
in
1
9
8
5
a
n
d
co
n
s
is
t
o
f
1
6
h
ab
its
,
n
am
ely
:
i
)
p
er
s
is
tin
g
;
ii
)
m
an
ag
in
g
im
p
u
ls
iv
ity
;
iii
)
lis
ten
in
g
to
o
th
er
s
-
with
u
n
d
e
r
s
tan
d
in
g
an
d
em
p
ath
y
;
iv
)
th
in
k
in
g
f
lex
ib
ly
;
v
)
th
in
k
i
n
g
ab
o
u
t
th
in
k
in
g
;
vi
)
s
tr
iv
in
g
f
o
r
ac
cu
r
ac
y
an
d
p
r
ec
is
io
n
;
v
ii
)
q
u
esti
o
n
in
g
an
d
p
o
s
in
g
p
r
o
b
lem
s
;
v
iii
)
ap
p
ly
in
g
p
ast
k
n
o
wled
g
e
to
n
ew
s
itu
at
io
n
s
;
ix
)
th
i
n
k
in
g
an
d
co
m
m
u
n
icatin
g
with
clar
ity
an
d
p
r
ec
is
io
n
;
x
)
g
ath
er
in
g
d
a
ta
th
r
o
u
g
h
all
s
en
s
es;
xi
)
cr
ea
tin
g
,
im
ag
i
n
in
g
,
an
d
in
n
o
v
ati
n
g
;
x
ii
)
r
esp
o
n
d
in
g
with
wo
n
d
er
m
en
t
a
n
d
awe
;
x
i
ii
)
tak
in
g
r
esp
o
n
s
ib
le
r
is
k
s
;
x
iv
)
f
in
d
i
n
g
h
u
m
o
r
;
xv
)
th
i
n
k
in
g
in
ter
d
ep
e
n
d
en
tly
;
an
d
xvi
)
lear
n
in
g
co
n
tin
u
o
u
s
ly
.
T
h
e
h
a
b
it
o
f
in
tellig
en
t
th
i
n
k
i
n
g
is
o
n
e
o
f
th
e
p
o
s
itiv
e
h
ab
i
ts
p
o
s
s
ess
ed
b
y
h
u
m
an
s
[
2
6
]
.
Hab
its
o
f
m
in
d
p
o
s
itiv
ely
in
f
lu
e
n
ce
s
tu
d
en
ts
'
m
ath
em
atica
l
ab
ilit
y
an
d
ca
n
s
h
ap
e
th
ei
r
th
in
k
in
g
s
k
ills
,
wh
ich
ar
e
im
p
o
r
tan
t
f
o
r
t
h
em
i
n
s
o
lv
in
g
m
ath
em
a
tical
p
r
o
b
lem
s
[
2
5
]
,
[
2
7
]
.
I
t
i
s
m
ea
n
,
h
ab
its
o
f
m
i
n
d
ca
n
m
o
tiv
ate
s
tu
d
en
ts
t
o
co
n
n
ec
t m
ath
em
atica
l id
ea
s
an
d
h
elp
t
h
em
in
o
v
er
c
o
m
in
g
o
r
r
eso
lv
in
g
th
e
p
r
o
b
lem
s
th
ey
f
a
ce
with
p
r
o
d
u
ctiv
e
ac
tio
n
s
an
d
way
s
.
T
h
ese
p
r
o
d
u
ctiv
e
way
s
ca
n
b
e
in
th
e
f
o
r
m
o
f
e
x
p
lo
r
atio
n
o
f
g
iv
e
n
m
at
h
em
atica
l
p
r
o
b
l
em
s
.
Stu
d
en
ts
wh
o
ap
p
ly
th
ese
h
ab
its
wil
l
k
n
o
w
h
o
w
to
s
tar
t
s
o
lv
i
n
g
th
e
p
r
o
b
lem
an
d
wh
at
s
tep
s
to
tak
e.
I
n
ad
d
itio
n
,
th
ey
also
r
ea
lize
wh
at
d
ata
n
ee
d
s
to
b
e
co
llected
an
d
g
en
er
ated
to
s
o
lv
e
th
e
p
r
o
b
lem
.
T
h
ey
will
ex
p
lo
r
e
b
y
tr
y
in
g
to
f
in
d
o
th
er
alter
n
ativ
e
s
o
lu
tio
n
s
an
d
ass
ess
in
g
wh
eth
er
th
e
s
o
lu
tio
n
is
ap
p
r
o
p
r
iate
o
r
n
o
t
[
2
8
]
,
[
2
9
]
.
T
h
e
co
n
tr
ib
u
tio
n
o
f
h
ab
its
o
f
m
in
d
to
m
ath
em
atica
l
ab
ilit
y
h
as
b
ee
n
in
v
esti
g
ated
b
y
s
ev
er
al
r
esear
ch
er
s
p
r
ev
io
u
s
ly
.
As
a
r
esu
lt,
h
ab
its
o
f
m
in
d
p
o
s
itiv
ely
af
f
ec
t
o
n
s
tu
d
en
ts
’
m
ath
em
atics
ab
ilit
y
[
3
0
]
–
[
3
3
]
.
B
ased
o
n
th
e
ex
p
lan
atio
n
s
an
d
th
e
o
p
in
i
o
n
s
o
f
th
e
r
esear
ch
r
esu
lts
th
at
h
as
b
ee
n
p
r
esen
ted
,
it
is
im
p
o
r
tan
t
to
k
n
o
w
h
o
w
th
e
d
if
f
e
r
en
ce
s
b
etwe
en
m
ale
an
d
f
em
ale
s
tu
d
e
n
ts
’
m
ath
e
m
atica
l
ex
p
lo
r
atio
n
a
b
ilit
y
an
d
to
s
ee
wh
eth
er
th
e
in
f
lu
en
ce
o
f
h
ab
its
o
f
m
in
d
o
n
th
e
d
if
f
er
en
ce
s
th
at
m
a
y
ap
p
ea
r
in
o
r
d
e
r
to
d
esig
n
a
n
ef
f
ec
tiv
e
lear
n
in
g
p
r
o
ce
s
s
,
b
ec
au
s
e
s
o
m
e
r
ele
v
an
t
r
esear
ch
o
n
l
y
s
tu
d
ies
ea
ch
v
ar
ia
b
le
s
ep
ar
ately
e
ith
er
m
ath
em
atica
l
ex
p
lo
r
atio
n
ab
ilit
y
o
r
h
a
b
its
o
f
m
in
d
.
T
h
er
ef
o
r
e,
th
e
o
b
jectiv
e
o
f
th
is
s
tu
d
y
is
to
an
aly
ze
an
d
d
escr
ib
e
h
o
w
th
e
d
if
f
er
en
ce
s
in
m
ath
em
atica
l
ex
p
lo
r
atio
n
ab
ili
ty
b
etwe
en
m
ale
an
d
f
em
ale
s
tu
d
en
ts
wh
ich
is
th
en
s
tu
d
ied
f
r
o
m
th
eir
h
ab
its
o
f
m
in
d
wh
ile
lear
n
in
g
o
r
s
o
lv
in
g
m
ath
em
atics p
r
o
b
lem
s
in
ju
n
io
r
h
ig
h
s
ch
o
o
l le
v
el.
2.
M
E
T
H
O
D
2
.
1
.
Resea
rc
h desi
g
n
T
h
is
s
tu
d
y
u
s
ed
co
m
b
in
atio
n
o
r
m
ix
e
d
r
esear
ch
.
T
h
is
is
a
co
m
b
in
atio
n
o
f
two
r
esear
ch
m
eth
o
d
s
,
n
am
ely
q
u
a
n
titativ
e
an
d
q
u
alit
ativ
e
m
eth
o
d
s
wh
ich
ar
e
th
en
u
s
ed
in
to
g
eth
er
in
a
r
esear
ch
to
ac
q
u
ir
e
d
ata
th
at
is
m
o
r
e
co
m
p
lete,
v
alid
,
r
eliab
le,
an
d
o
b
jectiv
e
[
3
4
]
.
T
h
is
m
i
x
ed
r
esear
ch
m
et
h
o
d
u
s
es
a
s
e
q
u
en
tial
ex
p
lan
at
o
r
y
m
o
d
el.
T
h
e
s
eq
u
en
tial
e
x
p
la
n
ato
r
y
m
o
d
el
is
a
c
o
m
b
in
ati
o
n
b
etwe
en
q
u
a
n
titativ
e
an
d
q
u
alitativ
e
m
eth
o
d
s
s
eq
u
en
tially
(
s
er
ially
)
,
wh
er
e
r
esear
ch
er
s
s
tar
t
b
y
c
o
n
d
u
ctin
g
a
q
u
a
n
titativ
e
an
aly
s
is
an
d
t
h
en
c
o
n
tin
u
e
with
a
q
u
alitativ
e
an
aly
s
is
to
h
elp
ex
p
lain
th
e
p
r
ev
io
u
s
q
u
a
n
titativ
e
p
h
ase
[
3
4
]
,
[
3
5
]
.
I
n
th
is
s
tu
d
y
,
th
e
r
esear
ch
d
esig
n
b
eg
an
with
co
llectin
g
d
ata
an
d
an
aly
zin
g
q
u
an
titativ
ely
,
th
en
co
n
tin
u
e
d
with
co
llectin
g
d
ata
an
d
an
aly
zin
g
q
u
alitativ
ely
,
wh
ich
is
p
r
esen
t
ed
in
Fig
u
r
e
1
.
2
.
2
.
P
a
rt
icipa
nts
T
h
is
s
tu
d
y
was
co
n
d
u
cted
at
SMP
Neg
er
i
1
0
Po
n
tian
ak
(
p
u
b
l
ic
m
id
d
le
s
ch
o
o
l
)
d
u
r
in
g
1
2
M
ay
-
1
2
J
u
n
e
2
0
2
3
.
T
h
e
s
tu
d
y
d
ata
was
co
llected
f
r
o
m
2
7
s
tu
d
en
ts
in
g
r
a
d
e
VI
I
I
(
s
ec
o
n
d
y
ea
r
)
,
1
2
wer
e
m
ale
s
tu
d
en
ts
an
d
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
Ma
th
ema
tica
l e
xp
lo
r
a
tio
n
a
b
i
lity a
n
d
s
tu
d
en
ts
’
h
a
b
its
o
f m
i
n
d
b
a
s
ed
o
n
g
en
d
er:
a
s
tu
d
y
i
n
…
(
V
ev
en
)
3675
1
5
wer
e
f
em
ale
s
tu
d
en
ts
wh
o
h
ad
lear
n
ed
th
e
n
u
m
b
e
r
p
atter
n
m
ater
ial.
T
h
e
s
u
b
ject
was c
h
o
s
en
as a
r
esu
lt o
f
a
d
is
cu
s
s
io
n
b
y
t
h
e
r
esear
ch
e
r
a
n
d
th
e
m
ath
em
atics
teac
h
e
r
o
f
SMP
Neg
er
i
1
0
Po
n
tian
a
k
b
a
s
ed
o
n
th
e
h
ig
h
est
av
er
ag
e
m
ath
em
atics
s
co
r
e.
T
h
is
n
u
m
b
er
was
also
b
ased
o
n
an
ad
eq
u
ate
s
am
p
le
s
ize
o
f
2
0
-
3
0
s
am
p
les
ac
co
r
d
in
g
to
p
r
e
v
io
u
s
s
tu
d
ies
[
3
5
]
,
[
3
6
]
.
A
to
tal
o
f
2
7
s
tu
d
en
t
s
wer
e
g
iv
e
n
an
ex
p
lo
r
ato
r
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a
b
ilit
y
test
,
th
en
two
s
tu
d
en
ts
ea
ch
wer
e
s
elec
ted
c
o
n
s
is
tin
g
o
f
m
ale
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d
f
em
ale
s
tu
d
en
ts
to
r
ep
r
esen
t
th
e
h
ig
h
,
m
o
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ate
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d
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w
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teg
o
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ies
to
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e
g
iv
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n
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h
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i
ts
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m
in
d
q
u
esti
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air
e
f
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ll
o
wed
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y
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n
d
u
ctin
g
in
ter
v
ie
ws
with
th
em
.
T
h
is
s
elec
tio
n
was b
ased
o
n
s
tu
d
en
t
s
test
r
esu
lts
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at
wer
e
n
o
t sig
n
if
ican
tly
d
if
f
er
e
n
t in
ea
ch
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teg
o
r
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.
Fig
u
r
e
1
.
Seq
u
en
tial e
x
p
lan
at
o
r
y
p
h
ase
2
.
3
.
I
ns
t
rum
ent
s
T
h
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in
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m
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ed
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s
is
te
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o
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test
o
f
m
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m
atica
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esti
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air
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ab
it
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o
f
m
in
d
,
an
d
in
ter
v
iew
g
u
id
elin
es.
T
h
e
t
est
u
s
ed
was
a
test
o
f
n
u
m
b
er
p
atter
n
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ater
ial
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d
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ip
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n
ty
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e
o
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u
esti
o
n
.
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h
e
test
q
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esti
o
n
s
co
n
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is
ted
o
f
two
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u
esti
o
n
s
with
ea
ch
q
u
e
s
tio
n
co
n
s
is
tin
g
o
f
f
iv
e
item
s
.
T
h
e
test
q
u
esti
o
n
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wer
e
v
alid
ated
f
ir
s
t
b
y
o
n
e
o
f
th
e
Ma
th
em
atics
E
d
u
ca
tio
n
Stu
d
y
Pro
g
r
a
m
lectu
r
er
s
.
Af
te
r
th
e
test
q
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esti
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wer
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o
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,
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test
was
co
n
d
u
cted
o
n
3
0
s
tu
d
e
n
ts
to
m
ea
s
u
r
e
v
alid
ity
a
n
d
r
eliab
ilit
y
.
T
h
e
r
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lts
o
f
th
e
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wer
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n
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361
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at
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ig
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0
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≤
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.
000
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.
B
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th
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icate
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h
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700
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.
900
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.
T
h
e
q
u
esti
o
n
n
air
e
f
o
r
th
is
s
tu
d
y
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a
clo
s
ed
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u
esti
o
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n
ai
r
e
co
n
s
is
tin
g
o
f
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2
s
tatem
en
ts
ac
co
r
d
in
g
to
t
h
e
1
6
h
ab
its
o
f
m
in
d
in
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icato
r
s
.
E
ac
h
in
d
icato
r
c
o
n
s
is
ts
o
f
tw
o
s
tatem
en
ts
co
n
s
is
tin
g
o
f
p
o
s
itiv
e
an
d
n
e
g
ativ
e
s
tatem
en
ts
.
T
h
is
q
u
esti
o
n
n
air
e
aim
s
to
f
in
d
o
u
t
h
o
w
s
tu
d
en
ts
'
h
ab
its
o
f
m
in
d
ar
e
in
s
o
lv
in
g
m
ath
e
m
atics
p
r
o
b
lem
s
.
Gu
ttm
an
Scale
is
th
e
s
ca
le
u
s
ed
i
n
th
is
in
s
tr
u
m
en
t
.
T
h
e
q
u
esti
o
n
n
air
e
was
v
alid
ated
f
ir
s
t
b
y
o
n
e
o
f
th
e
Gu
id
an
ce
an
d
C
o
u
n
s
ellin
g
Stu
d
y
Pro
g
r
am
lectu
r
er
s
.
Af
ter
th
e
q
u
esti
o
n
n
air
e
was
d
ee
m
ed
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o
d
,
it
was
test
ed
o
n
3
0
s
tu
d
en
ts
to
m
ea
s
u
r
e
v
al
id
ity
an
d
r
eliab
ilit
y
.
T
h
e
3
2
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t
atem
en
ts
wer
e
in
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v
alid
with
(
0
.
362
−
0
.
769
)
>
(
0
.
361
)
at
a
s
ig
n
if
ica
n
t
lev
el
o
f
5
%
a
n
d
wer
e
in
ca
te
g
o
r
ies
th
at
v
ar
i
ed
f
r
o
m
lo
w
to
h
ig
h
.
T
h
e
q
u
esti
o
n
n
air
e
was
also
in
d
icate
d
r
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with
11
(
0
.
875
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>
(
0
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361
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a
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ig
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if
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t
lev
el
o
f
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ig
h
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0
.
700
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.
T
h
e
in
ter
v
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s
ed
in
t
h
is
s
tu
d
y
was
a
s
em
i
-
s
tr
u
ctu
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n
ter
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I
n
its
im
p
lem
en
tati
o
n
,
s
em
i
-
s
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u
ctu
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ter
v
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ar
e
f
r
ee
r
th
an
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tr
u
ctu
r
ed
in
ter
v
iews.
T
h
is
is
b
ec
au
s
e
s
em
i
-
s
tr
u
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r
ed
in
ter
v
iews
aim
to
f
in
d
p
r
o
b
lem
s
m
o
r
e
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p
en
ly
b
y
p
r
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g
o
p
p
o
r
t
u
n
ities
f
o
r
r
e
s
p
o
n
d
en
ts
to
ar
g
u
e
[
3
7
]
.
T
h
e
i
n
ter
v
iew
g
u
id
elin
es
wer
e
also
v
alid
ate
d
b
y
o
n
e
o
f
th
e
Ma
th
em
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ca
tio
n
Stu
d
y
Pro
g
r
a
m
lectu
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s
.
T
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in
ter
v
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i
d
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co
n
s
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t
o
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q
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th
at
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u
s
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to
clar
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ex
p
lo
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m
o
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4
.
Da
t
a
a
na
ly
s
is
Data
an
aly
s
is
tech
n
iq
u
es
u
s
ed
in
th
is
s
tu
d
y
,
we
r
e
ad
j
u
s
ted
t
o
th
e
f
o
r
m
u
latio
n
o
f
p
r
o
b
lem
s
an
d
m
ix
ed
r
esear
ch
m
eth
o
d
s
s
o
th
at
v
alid
d
ata
wer
e
o
b
tain
e
d
.
Data
an
a
ly
s
is
b
eg
an
with
q
u
an
titativ
e
d
ata
an
aly
s
is
o
f
th
e
m
ath
em
atica
l
ex
p
lo
r
atio
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ab
ili
ty
test
an
d
h
ab
i
ts
o
f
m
in
d
q
u
e
s
tio
n
n
air
e
r
esu
lts
,
th
en
co
n
tin
u
ed
with
q
u
alitativ
e
d
ata
an
aly
s
is
o
f
th
e
in
ter
v
iew
r
esu
lts
.
Qu
an
titativ
e
an
aly
s
is
u
s
ed
in
th
is
r
esear
ch
is
d
escr
ip
tiv
e
s
ta
tis
tical
an
aly
s
i
s
ty
p
e.
T
h
e
d
at
a
an
aly
ze
d
with
d
escr
ip
tiv
e
s
tatis
tics
wh
i
ch
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e
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e
test
r
esu
lts
o
f
th
e
m
ath
em
atica
l
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ab
ilit
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h
ab
its
o
f
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i
n
d
q
u
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air
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r
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ted
t
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r
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g
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r
a
m
s
an
d
tab
les.
T
h
e
ca
teg
o
r
izatio
n
o
f
th
e
m
at
h
e
m
atica
l
ex
p
lo
r
atio
n
ab
ilit
y
test
r
esu
lts
tak
es in
to
ac
co
u
n
t th
e
a
v
er
ag
e
v
alu
e
(
̅
)
an
d
s
tan
d
ar
d
d
e
v
iatio
n
(
)
as
s
ee
n
in
T
ab
le
2
.
T
h
e
q
u
alitativ
e
d
ata
a
n
aly
s
is
p
r
o
ce
s
s
f
o
llo
ws
th
e
p
r
ev
io
u
s
q
u
an
titativ
e
d
ata.
B
ased
o
n
th
e
q
u
an
titativ
e
d
ata
th
at
h
as
b
ee
n
r
e
d
u
ce
d
p
r
e
v
io
u
s
ly
,
d
ata
an
aly
s
is
co
n
tin
u
es
with
d
ata
p
r
esen
tatio
n
b
y
c
o
n
n
ec
tin
g
h
ab
its
o
f
m
in
d
an
d
ex
p
lo
r
atio
n
s
k
ills
in
ea
ch
ca
teg
o
r
y
.
T
h
r
o
u
g
h
th
is
s
tag
e,
an
ex
p
lan
atio
n
ab
o
u
t
h
o
w
th
e
s
tu
d
en
ts
’
h
ab
its
o
f
m
in
d
in
th
e
h
ig
h
,
m
o
d
er
ate
an
d
lo
w
ca
teg
o
r
ies
o
f
m
ath
em
atics
ex
p
lo
r
atio
n
a
b
ilit
y
is
o
b
t
ain
ed
.
T
h
e
an
aly
s
is
p
r
o
ce
s
s
is
also
f
o
llo
wed
b
y
f
u
r
th
er
d
ata
c
o
llectio
n
co
n
d
u
c
ted
b
y
s
em
i
-
s
tr
u
ct
u
r
e
d
in
ter
v
i
ews
with
s
tu
d
en
ts
.
I
n
ter
v
iew
d
ata
was
o
b
tain
e
d
f
r
o
m
r
e
p
r
esen
tativ
es
o
f
s
tu
d
en
t
s
at
ea
ch
lev
el
o
f
th
eir
m
ath
e
m
atica
l
ex
p
lo
r
atio
n
ab
ilit
y
.
T
h
e
in
ter
v
iew
d
ata
was
an
aly
ze
d
with
th
e
aim
o
f
r
ev
ea
lin
g
in
d
e
p
th
th
eir
m
ath
e
m
atica
l
ex
p
lo
r
atio
n
ab
ilit
y
t
h
at
was n
o
t d
ir
ec
tly
o
b
s
er
v
ed
.
T
ab
le
2
.
C
lass
if
icatio
n
cr
iter
ia
f
o
r
m
at
h
em
atica
l e
x
p
lo
r
atio
n
ab
ilit
y
C
r
i
t
e
r
i
a
S
t
u
d
e
n
t
s’
c
a
t
e
g
o
r
y
≥
(
̅
+
)
H
i
g
h
(
̅
−
)
≤
<
(
̅
+
)
M
o
d
e
r
a
t
e
<
(
̅
−
)
Lo
w
3.
RE
SU
L
T
S AN
D
D
I
SCU
SS
I
O
N
3
.
1
.
Student
s
’
m
a
t
hem
a
t
ica
l e
x
plo
ra
t
io
n a
bil
it
y
ba
s
ed
o
n g
en
der
T
h
e
test
g
iv
en
aim
s
to
clas
s
if
y
in
g
th
e
lev
el
o
f
m
ath
em
atic
al
ex
p
lo
r
atio
n
ab
ilit
y
.
B
ased
o
n
th
e
test
r
esu
lts
o
f
th
e
m
ath
em
atica
l
ex
p
lo
r
atio
n
a
b
ilit
y
,
th
e
f
o
llo
win
g
d
ata
wer
e
o
b
tain
e
d
in
T
a
b
le
3
.
Fu
r
th
er
m
o
r
e
,
b
ased
o
n
th
e
class
if
icatio
n
cr
iter
ia
in
T
ab
le
3
,
th
e
class
if
icatio
n
o
f
s
tu
d
en
ts
'
m
ath
em
atica
l
ex
p
lo
r
atio
n
ab
ilit
y
s
co
r
es
is
p
r
esen
ted
th
r
o
u
g
h
a
f
r
eq
u
en
cy
g
r
ap
h
in
th
e
f
o
r
m
o
f
a
b
ar
c
h
ar
t a
s
Fig
u
r
e
2
.
B
ased
o
n
Fig
u
r
e
2
,
it
r
ev
ea
le
d
th
at
th
e
f
r
eq
u
en
cy
o
f
m
at
h
em
atica
l
ex
p
lo
r
ati
o
n
ab
ilit
y
i
n
th
e
h
ig
h
ca
teg
o
r
y
is
1
4
.
8
% o
r
as m
a
n
y
as 4
s
tu
d
en
ts
.
Fu
r
th
er
m
o
r
e,
th
e
f
r
eq
u
en
cy
o
f
m
ath
e
m
atica
l e
x
p
lo
r
atio
n
ab
ilit
y
in
th
e
m
o
d
er
ate
ca
teg
o
r
y
,
wh
ic
h
is
7
4
.
1
% o
r
as m
an
y
as 2
0
s
tu
d
en
ts
.
T
h
e
f
r
eq
u
en
cy
o
f
m
ath
e
m
atica
l e
x
p
lo
r
atio
n
ab
ilit
y
in
th
e
lo
w
ca
teg
o
r
y
,
wh
ich
is
1
1
.
1
% o
r
as m
an
y
as 3
s
tu
d
en
ts
.
T
h
is
r
esu
lt h
as sh
o
wn
th
at
m
o
s
t stu
d
en
ts
h
av
e
m
o
d
er
ate
m
ath
em
atica
l
e
x
p
lo
r
atio
n
ab
ilit
ies.
Ho
wev
er
,
if
v
iewe
d
b
ased
o
n
g
en
d
er
,
th
e
s
tu
d
en
t
a
b
ili
ty
test
r
esu
lts
av
er
ag
e
ar
e
as
p
r
esen
t
ed
in
T
a
b
le
4
.
B
ased
o
n
T
ab
l
e
4
,
it
r
ev
ea
led
th
at
f
em
ale
s
tu
d
en
ts
h
av
e
h
ig
h
er
av
er
ag
e
ac
h
iev
em
e
n
t
test
s
co
r
es
th
an
m
ales.
T
h
is
s
h
o
ws
th
a
t
th
e
test
s
co
r
es
o
b
tain
ed
b
y
f
em
ale
s
tu
d
en
ts
ar
e
m
o
s
tly
h
ig
h
er
th
an
th
e
s
co
r
es o
f
m
ale
s
tu
d
en
ts
.
T
ab
le
3
.
Stu
d
e
n
ts
’
m
ath
em
atic
al
ex
p
lo
r
atio
n
ab
ilit
y
test
r
esu
lts
I
n
f
o
r
mat
i
o
n
M
a
t
h
e
ma
t
i
c
a
l
e
x
p
l
o
r
a
t
i
o
n
a
b
i
l
i
t
y
t
e
s
t
r
e
su
l
t
̅
(
a
v
e
r
a
g
e
)
49
.
63
(
st
a
n
d
a
r
d
d
e
v
i
a
t
i
o
n
)
16
.
81
(
l
o
w
e
st
s
c
o
r
e
)
5
(
h
i
g
h
e
s
t
sc
o
r
e
)
25
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
u
c
I
SS
N:
2252
-
8
8
2
2
Ma
th
ema
tica
l e
xp
lo
r
a
tio
n
a
b
i
lity a
n
d
s
tu
d
en
ts
’
h
a
b
its
o
f m
i
n
d
b
a
s
ed
o
n
g
en
d
er:
a
s
tu
d
y
i
n
…
(
V
ev
en
)
3677
Fig
u
r
e
2
.
C
ateg
o
r
y
b
a
r
d
iag
r
a
m
o
f
s
tu
d
en
ts
’
m
ath
em
atica
l e
x
p
lo
r
atio
n
ab
ilit
y
T
ab
le
4
.
Av
e
r
ag
e
o
f
m
ath
em
at
ical
ex
p
lo
r
atio
n
ab
ilit
y
test
r
esu
lts
b
ased
o
n
g
e
n
d
er
S
t
u
d
e
n
t
s
A
v
e
r
a
g
e
o
f
m
a
t
h
e
mat
i
c
a
l
e
x
p
l
o
r
a
t
i
o
n
a
b
i
l
i
t
y
t
e
st
r
e
s
u
l
t
M
a
l
e
s
t
u
d
e
n
t
s
48
.
22
F
e
mal
e
s
t
u
d
e
n
t
s
51
.
05
3
.
2
.
Student
s
’
ha
bit
s
o
f
m
ind
ba
s
ed
o
n g
ender
Af
ter
b
ein
g
g
iv
e
n
th
e
m
ath
em
atica
l
ex
p
lo
r
atio
n
ab
ilit
y
test
,
2
s
tu
d
en
ts
co
n
s
is
tin
g
m
ale
s
tu
d
en
ts
an
d
f
em
ale
s
tu
d
en
ts
s
elec
ted
f
r
o
m
ea
ch
ca
teg
o
r
y
o
f
m
ath
em
atic
al
ex
p
lo
r
atio
n
ab
ilit
y
we
r
e
as
k
ed
to
f
ill
o
u
t
th
e
q
u
esti
o
n
n
air
e.
O
n
ea
ch
ca
teg
o
r
y
f
r
o
m
h
i
g
h
to
lo
w,
m
ale
s
tu
d
en
ts
co
d
ed
as
M
-
1
to
M
-
3
a
n
d
f
em
ale
s
tu
d
en
ts
co
d
ed
F
-
1
t
o
F
-
3
.
T
h
e
p
u
r
p
o
s
e
o
f
th
e
q
u
esti
o
n
n
ai
r
e
is
to
f
in
d
o
u
t
ab
o
u
t
th
e
h
a
b
its
o
f
m
in
d
i
n
ea
ch
ca
teg
o
r
y
o
f
m
ath
em
atica
l
ex
p
lo
r
atio
n
ab
ili
ty
.
T
h
e
s
co
r
e
o
f
th
e
s
tu
d
en
ts
'
h
ab
its
o
f
m
in
d
q
u
esti
o
n
n
air
e
r
esu
lts
ar
e
p
r
esen
ted
th
r
o
u
g
h
a
f
r
eq
u
e
n
cy
g
r
ap
h
in
t
h
e
f
o
r
m
o
f
a
b
ar
c
h
ar
t a
s
Fig
u
r
e
3
.
B
ased
o
n
Fig
u
r
e
3
,
it
r
ev
ea
le
d
th
at
in
th
e
h
i
g
h
m
ath
em
atic
al
ex
p
lo
r
atio
n
ab
ilit
y
ca
te
g
o
r
y
,
m
ale
an
d
f
em
ale
s
tu
d
en
ts
b
o
th
o
b
tain
ed
th
e
s
am
e
q
u
esti
o
n
n
air
e
s
co
r
e
p
er
ce
n
tag
e,
wh
ich
m
ea
n
s
in
lear
n
in
g
m
ath
em
atics,
th
ey
t
e
n
d
t
o
s
h
o
w
t
h
e
s
a
m
e
n
u
m
b
e
r
o
f
h
a
b
i
t
s
o
f
m
i
n
d
o
u
t
o
f
t
h
e
t
o
t
a
l
.
I
n
t
h
e
m
o
d
e
r
a
t
e
a
n
d
l
o
w
m
a
t
h
e
m
at
i
c
al
e
x
p
l
o
r
a
t
i
o
n
a
b
i
l
i
t
y
c
a
te
g
o
r
y
,
t
h
e
p
e
r
c
e
n
t
a
g
e
o
f
h
a
b
i
t
s
o
f
m
i
n
d
s
h
o
w
n
b
y
f
e
m
a
l
e
s
t
u
d
e
n
t
s
i
s
h
i
g
h
e
r
t
h
a
n
m
a
l
e
s
t
u
d
e
n
ts
.
A
l
t
h
o
u
g
h
t
h
e
d
i
f
f
e
r
en
c
e
s
h
o
w
n
i
s
n
o
t
t
o
o
s
i
g
n
i
f
i
c
an
t
,
b
u
t
h
i
s
r
e
f
l
e
c
ts
t
h
a
t
f
e
m
al
e
s
t
u
d
e
n
ts
s
h
o
w
m
o
r
e
h
a
b
i
t
s
o
f
m
i
n
d
d
u
r
i
n
g
t
h
e
l
e
a
r
n
i
n
g
p
r
o
c
e
s
s
o
r
a
p
p
l
i
e
d
t
h
e
m
o
n
p
r
o
b
l
e
m
s
o
l
v
i
n
g
p
r
o
c
e
s
s
.
I
n
g
e
n
e
r
a
l
,
f
e
m
a
l
e
s
t
u
d
e
n
ts
t
e
n
d
t
o
a
p
p
l
y
m
o
r
e
h
a
b
i
t
s
o
f
m
in
d
e
i
t
h
e
r
i
n
l
e
a
r
n
i
n
g
o
r
w
h
e
n
s
o
l
v
i
n
g
p
r
o
b
l
e
ms
th
an
m
ale
s
tu
d
en
ts
.
Fig
u
r
e
3
.
B
ar
d
ia
g
r
am
o
f
s
tu
d
e
n
ts
’
h
ab
its
o
f
m
in
d
q
u
esti
o
n
n
a
ir
e
b
ased
o
n
g
en
d
er
3
.
3
.
Student
s
’
inte
rv
iew
re
s
ults
I
n
ter
v
iews
wer
e
co
n
d
u
cted
with
r
ep
r
esen
tativ
es
o
f
ea
ch
ca
te
g
o
r
y
o
f
m
ath
em
atica
l
ex
p
l
o
r
at
io
n
ab
ilit
y
th
at
h
ad
b
ee
n
p
r
e
v
io
u
s
ly
s
elec
ted
to
r
ev
ea
l
in
d
ep
th
th
eir
m
a
th
em
atica
l
ex
p
lo
r
atio
n
ab
ilit
ie
s
th
at
m
ig
h
t
n
o
t
b
e
o
b
s
er
v
ed
.
I
n
th
e
h
ig
h
ca
teg
o
r
y
,
f
em
ale
s
tu
d
e
n
ts
g
av
e
m
o
r
e
p
r
ec
is
e
an
d
co
m
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lete
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r
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I
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3679
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ased
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,
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u
b
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-
1
k
n
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d
p
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at
m
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s
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ed
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b
ject
M
-
1
was
also
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le
to
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p
lain
th
e
p
atter
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f
o
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m
ed
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atter
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m
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la
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awin
g
.
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e
as
in
an
s
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n
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m
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er
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u
b
ject
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-
1
was
ab
le
to
an
s
wer
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r
r
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tly
a
n
d
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p
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ter
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e
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u
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ject
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1
.
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le
t
o
r
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ex
p
lai
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o
u
t
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er
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d
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f
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e
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u
b
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1
c
o
n
f
id
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tly
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x
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m
o
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d
etails
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d
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th
an
s
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b
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d
d
o
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ts
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o
r
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ased
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d
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p
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te
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ially
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M
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f
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ar
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Fig
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an
d
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f
o
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Fig
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v
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ased
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B
ased
o
n
th
e
th
eo
r
etica
l
r
elatio
n
b
etwe
en
m
ath
e
m
atica
l
ex
p
lo
r
atio
n
a
b
ilit
y
an
d
h
a
b
its
o
f
m
in
d
,
t
h
er
e
ar
e
also
a
r
elatio
n
b
etwe
en
ea
c
h
in
d
icato
r
o
f
th
em
.
T
h
u
s
,
in
T
ab
le
5
,
it
is
s
h
o
wn
h
o
w
th
e
h
ab
its
o
f
m
in
d
in
ea
ch
ca
teg
o
r
y
o
f
m
ath
em
atica
l
ex
p
l
o
r
atio
n
ab
ili
ty
b
etwe
en
f
e
m
ale
an
d
m
ale
s
tu
d
en
ts
.
B
ased
o
n
th
e
tab
le,
th
e
r
elatio
n
b
etwe
en
m
ath
em
atica
l e
x
p
l
o
r
a
tio
n
ab
ilit
y
an
d
s
tu
d
en
ts
'
h
ab
its
o
f
m
in
d
is
ex
p
lain
ed
in
d
ep
t
h
.
T
ab
le
5
.
Ach
iev
a
b
ilit
y
o
f
h
ab
i
ts
o
f
m
in
d
o
n
m
ath
e
m
atica
l e
x
p
lo
r
atio
n
a
b
ilit
y
I
n
d
i
c
a
t
o
r
s o
f
mat
h
e
m
a
t
i
c
a
l
e
x
p
l
o
r
a
t
i
o
n
a
b
i
l
i
t
y
I
n
d
i
c
a
t
o
r
s o
f
h
a
b
i
t
s
o
f
mi
n
d
A
c
h
i
e
v
a
b
i
l
i
t
y
o
f
h
a
b
i
t
s
o
f
m
i
n
d
H
i
g
h
M
o
d
e
r
a
t
e
Lo
w
M
-
1
F
-
1
M
-
2
F
-
2
M
-
3
F
-
3
U
n
d
e
r
st
a
n
d
i
n
g
t
h
e
p
r
o
b
l
e
m
Li
st
e
n
i
n
g
t
o
o
t
h
e
r
s
-
w
i
t
h
u
n
d
e
r
st
a
n
d
i
n
g
a
n
d
e
mp
a
t
h
y
√
−
√
√
−
−
G
a
t
h
e
r
i
n
g
d
a
t
a
t
h
r
o
u
g
h
a
l
l
se
n
ses
√
√
√
√
√
√
R
e
s
p
o
n
d
i
n
g
w
i
t
h
w
o
n
d
e
r
me
n
t
a
n
d
a
w
e
−
√
−
−
−
−
Ex
a
m
i
n
i
n
g
p
a
t
t
e
r
n
s
P
e
r
si
st
i
n
g
√
√
−
√
−
−
M
a
n
a
g
i
n
g
i
mp
u
l
s
i
v
i
t
y
√
√
√
√
−
√
Th
i
n
k
i
n
g
f
l
e
x
i
b
l
y
√
√
√
√
−
−
S
t
r
i
v
i
n
g
f
o
r
a
c
c
u
r
a
c
y
a
n
d
p
r
e
c
i
s
i
o
n
√
√
√
√
√
√
S
e
a
r
c
h
i
n
g
i
n
f
o
r
mal
l
y
Q
u
e
st
i
o
n
i
n
g
a
n
d
p
o
si
n
g
p
r
o
b
l
e
ms
√
√
−
√
−
−
F
i
n
d
i
n
g
h
u
m
o
r
√
−
√
−
√
√
Th
i
n
k
i
n
g
i
n
t
e
r
d
e
p
e
n
d
e
n
t
l
y
−
−
√
√
√
√
Le
a
r
n
i
n
g
c
o
n
t
i
n
u
o
u
s
l
y
√
√
−
√
−
−
C
l
a
r
i
f
y
i
n
g
t
h
e
so
l
u
t
i
o
n
e
f
f
o
r
t
Th
i
n
k
i
n
g
a
b
o
u
t
t
h
i
n
k
i
n
g
√
√
−
−
−
−
Th
i
n
k
i
n
g
a
n
d
c
o
mm
u
n
i
c
a
t
i
n
g
w
i
t
h
c
l
a
r
i
t
y
a
n
d
p
r
e
c
i
si
o
n
−
√
−
−
−
−
C
r
e
a
t
i
n
g
,
i
ma
g
i
n
i
n
g
,
a
n
d
i
n
n
o
v
a
t
i
n
g
√
√
−
−
−
−
S
y
mb
o
l
i
z
a
t
i
o
n
a
n
d
g
e
n
e
r
a
l
i
z
a
t
i
o
n
A
p
p
l
y
i
n
g
p
a
st
k
n
o
w
l
e
d
g
e
t
o
n
e
w
si
t
u
a
t
i
o
n
√
√
−
−
−
−
Ta
k
i
n
g
r
e
sp
o
n
si
b
l
e
r
i
s
k
s
√
√
−
−
−
−
3
.
4
.
1
.
H
ig
h c
a
t
e
g
o
ry
o
f
m
a
t
h
em
a
t
ica
l e
x
plo
ra
t
io
n a
bil
it
y
W
h
en
f
ac
ed
with
p
r
o
b
lem
s
r
elate
d
to
m
ath
em
atica
l
e
x
p
lo
r
ati
o
n
,
b
o
th
m
ale
a
n
d
f
em
ale
s
tu
d
en
ts
wer
e
ab
le
to
h
av
e
a
g
o
o
d
u
n
d
e
r
s
tan
d
in
g
o
f
th
e
p
r
o
b
lem
.
Ma
le
s
tu
d
en
ts
ten
d
to
s
h
o
w
th
e
h
a
b
it
o
f
lis
ten
in
g
to
o
th
e
r
s
em
p
ath
etica
lly
an
d
tr
y
in
g
to
u
n
d
er
s
tan
d
wh
at
th
ey
h
ea
r
.
I
n
c
o
n
tr
ast
to
f
em
ale
s
tu
d
en
ts
,
th
e
y
d
o
n
o
t
s
h
o
w
th
is
h
ab
it,
th
ey
ar
e
n
o
t
u
s
ed
to
lis
ten
in
g
to
o
th
er
s
b
y
in
v
o
lv
in
g
e
m
p
ath
y
.
Alth
o
u
g
h
th
er
e
ar
e
d
if
f
er
en
ce
s
r
elate
d
to
th
ese
h
ab
its
,
th
ey
ar
e
s
till
ab
le
to
u
n
d
e
r
s
tan
d
th
e
p
r
o
b
lem
s
g
iv
en
b
ec
a
u
s
e
th
ey
h
av
e
a
h
a
b
it
o
f
u
n
d
er
s
tan
d
in
g
s
o
m
eth
in
g
b
y
u
tili
zin
g
th
ei
r
s
en
s
es
in
o
b
s
er
v
in
g
an
d
lo
o
k
i
n
g
at
ev
er
y
d
etail
o
f
t
h
e
p
r
o
b
lem
to
b
e
s
o
lv
e
d
.
T
h
is
allo
ws
th
em
to
ab
s
o
r
b
as
m
u
c
h
in
f
o
r
m
atio
n
th
at
is
co
n
s
id
er
ed
ca
p
ab
le
o
f
b
ein
g
u
s
ed
i
n
p
r
o
b
lem
s
o
lv
in
g
[
2
9
]
.
I
n
th
e
n
ex
t
h
ab
it,
o
n
ly
f
em
ale
s
tu
d
en
ts
s
h
o
w
th
e
h
ab
it
to
ten
d
to
b
e
in
ter
ested
i
n
th
e
p
r
o
b
l
em
s
g
iv
en
a
n
d
h
av
e
g
r
ea
t c
u
r
io
s
ity
.
T
h
is
s
h
o
ws th
at
m
ale
s
tu
d
en
ts
ten
d
to
b
e
less
in
ter
ested
wh
en
g
iv
e
n
s
o
m
eth
i
n
g
n
ew.
Ma
th
em
atica
l
ex
p
lo
r
atio
n
ab
ili
ty
r
elate
d
to
th
e
a
b
ilit
y
to
e
x
am
in
e
p
atter
n
s
is
s
h
o
wn
th
r
o
u
g
h
t
h
eir
h
ab
it
o
f
wo
r
k
in
g
o
n
s
o
m
et
h
in
g
g
i
v
e
n
s
er
io
u
s
ly
u
n
til
co
m
p
letio
n
a
n
d
n
o
t
g
i
v
in
g
u
p
ea
s
ily
.
W
h
eth
er
th
e
s
tu
d
en
ts
ar
e
m
ale
o
r
f
e
m
ale,
th
ey
ar
e
a
b
le
t
o
ex
am
in
e
p
a
tter
n
s
b
ec
a
u
s
e
th
ey
tr
y
c
o
n
tin
u
o
u
s
ly
to
l
o
o
k
f
o
r
r
elatio
n
s
h
ip
s
in
th
e
in
f
o
r
m
atio
n
o
b
tain
ed
an
d
m
ak
e
co
n
jectu
r
es a
b
o
u
t p
atter
n
s
th
at
m
ig
h
t
b
e
a
p
p
r
o
p
r
iate
[
3
3
]
.
I
n
ad
d
itio
n
,
th
ey
ar
e
also
u
s
ed
to
th
in
k
in
g
b
ef
o
r
e
a
ctin
g
wh
ich
h
elp
s
i
n
u
n
d
er
s
tan
d
in
g
t
h
e
d
i
r
ec
tio
n
o
f
th
e
p
r
o
b
lem
an
d
co
n
s
id
er
in
g
alter
n
ativ
e
s
u
itab
le
p
atter
n
s
.
T
h
is
is
o
f
co
u
r
s
e
also
in
s
ep
ar
ab
le
f
r
o
m
th
e
h
ab
it
o
f
f
lex
ib
le
th
in
k
in
g
wh
er
e
th
ey
ar
e
ab
le
to
ch
a
n
g
e
th
ei
r
m
in
d
s
wh
en
r
ec
eiv
in
g
n
ew
d
ata
an
d
allo
w
th
em
to
id
en
tif
y
o
th
e
r
p
atter
n
r
elatio
n
s
h
ip
s
if
th
e
p
r
e
v
io
u
s
p
atter
n
is
co
n
s
id
er
ed
in
a
p
p
r
o
p
r
iate.
T
h
is
h
ab
i
t is also
b
ec
au
s
e
th
ey
alwa
y
s
s
tr
iv
e
f
o
r
ac
c
u
r
ac
y
.
I
n
th
e
ab
ilit
y
r
elate
d
to
s
ea
r
ch
in
g
in
f
o
r
m
ally
,
in
th
e
ab
ilit
y
r
e
lated
to
in
f
o
r
m
al
s
ea
r
ch
,
s
tu
d
e
n
ts
h
av
e
a
h
ab
it
o
f
ask
in
g
q
u
esti
o
n
s
an
d
p
r
o
p
o
s
in
g
p
r
o
b
lem
s
s
o
th
at
t
h
ey
ca
n
e
x
p
lo
r
e
h
o
w
to
s
o
lv
e
a
g
iv
en
p
r
o
b
lem
.
Ho
wev
er
,
in
th
e
h
a
b
its
o
f
m
a
le
an
d
f
em
ale
s
tu
d
en
ts
,
th
er
e
ar
e
d
if
f
er
en
ce
s
wh
en
it
co
m
es
to
h
u
m
o
r
.
Ma
le
s
tu
d
en
ts
ten
d
t
o
o
f
ten
d
o
h
u
m
o
r
wh
ic
h
ca
n
ce
r
tain
l
y
f
r
ee
u
p
cr
ea
tiv
ity
to
u
n
d
e
r
s
tan
d
t
h
e
s
itu
atio
n
f
r
o
m
a
d
if
f
er
en
t
p
er
s
p
ec
tiv
e
o
n
a
p
r
o
b
lem
,
in
co
n
tr
ast
to
f
em
al
e
s
t
u
d
en
ts
wh
o
ten
d
n
o
t
to
b
e
ac
cu
s
t
o
m
ed
to
d
o
in
g
h
u
m
o
r
.
Similar
r
esear
ch
in
to
g
en
d
er
a
n
d
h
u
m
o
r
ex
p
lain
ed
th
at
th
is
th
is
co
u
ld
b
e
d
u
e
to
a
lo
wer
u
n
d
er
s
tan
d
in
g
o
f
h
u
m
o
r
in
f
em
ale
s
tu
d
en
ts
co
m
p
ar
ed
to
m
ale
s
tu
d
en
ts
[
3
8
]
.
Ho
wev
er
,
th
ey
ar
e
u
s
ed
to
co
n
tin
u
o
u
s
lear
n
in
g
,
wh
ich
is
s
h
o
wn
th
r
o
u
g
h
th
eir
en
th
u
s
iasm
an
d
cu
r
io
s
ity
,
s
o
t
h
ey
will
alwa
y
s
d
is
co
v
er
n
ew
th
in
g
s
.
T
h
e
h
ab
it
o
f
th
in
k
in
g
will
ce
r
tain
ly
h
elp
s
tu
d
en
ts
in
ex
p
l
o
r
in
g
u
n
u
s
u
al
s
o
lu
tio
n
s
[
3
0
]
.
T
h
e
s
im
ilar
ity
b
etwe
en
m
ale
an
d
f
em
ale
s
tu
d
en
ts
ca
n
also
b
e
s
ee
n
f
r
o
m
th
e
h
ab
it
o
f
t
h
in
k
in
g
in
ter
d
ep
en
d
en
tly
o
n
o
th
e
r
s
th
at
is
n
o
t
s
h
o
wn
s
o
th
is
ca
n
s
h
o
w
th
at
b
o
th
m
ale
an
d
f
em
ale
s
tu
d
en
ts
ten
d
to
wo
r
k
in
d
iv
id
u
ally
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
2
5
2
-
8
8
2
2
I
n
t
J
E
v
al
&
R
es E
d
u
c
,
Vo
l
.
13
,
No
.
6
,
Dec
em
b
er
2
0
2
4
:
3
6
7
3
-
3
6
8
5
3682
Fu
r
th
er
m
o
r
e
,
s
tu
d
e
n
ts
wh
o
a
r
e
ab
le
to
clar
if
y
p
r
o
b
lem
s
o
lv
in
g
ef
f
o
r
ts
ar
e
s
h
o
wn
th
r
o
u
g
h
h
a
b
its
r
elate
d
to
m
etac
o
g
n
itio
n
.
Pre
v
io
u
s
s
tu
d
y
[
2
5
]
s
tated
th
at
th
e
h
ab
it
o
f
t
h
in
k
in
g
m
etac
o
g
n
itio
n
m
ak
es
s
tu
d
en
ts
u
n
d
er
s
tan
d
th
em
s
elv
es,
b
ec
au
s
e
s
tu
d
en
ts
ar
e
awa
r
e
o
f
wh
at
k
n
o
wled
g
e
t
h
ey
h
av
e
an
d
wh
at
p
lan
s
th
ey
will
d
o
.
T
h
is
s
h
o
ws
th
at
b
o
th
m
ale
an
d
f
e
m
ale
s
tu
d
en
ts
ten
d
to
r
o
u
tin
ely
ev
al
u
ate
th
e
s
o
lu
tio
n
s
th
ey
h
av
e
d
o
n
e
b
ec
au
s
e
th
ey
r
ea
lize
an
d
u
n
d
er
s
tan
d
wh
at
th
ey
k
n
o
w
an
d
wh
at
t
h
ey
d
o
n
o
t
k
n
o
w
[
3
1
]
,
[
3
2
]
.
Fem
ale
s
tu
d
en
ts
s
h
o
wed
a
h
a
b
it
o
f
th
in
k
in
g
an
d
co
m
m
u
n
icatin
g
c
lear
ly
co
m
p
ar
e
d
to
m
ale
s
tu
d
en
ts
.
T
h
is
is
a
ls
o
ev
id
en
t
f
r
o
m
th
e
wo
r
k
o
f
f
e
m
ale
s
tu
d
en
ts
wh
o
ten
d
to
b
e
clea
r
e
r
an
d
m
o
r
e
s
y
s
tem
atic
th
an
m
ale
s
tu
d
en
ts
wh
ich
in
d
icate
s
t
h
at
f
em
ale
s
tu
d
en
ts
ar
e
a
b
le
to
co
m
m
u
n
icate
th
ei
r
p
r
o
b
lem
s
o
lv
in
g
id
ea
s
well.
T
h
e
ab
ilit
y
to
clar
if
y
t
h
e
s
o
lu
tio
n
ef
f
o
r
t
is
also
in
s
ep
ar
ab
le
f
r
o
m
th
e
s
tu
d
en
ts
'
h
ab
it
o
f
wo
r
k
in
g
an
d
in
n
o
v
ati
n
g
b
y
tr
y
in
g
v
a
r
io
u
s
s
o
lu
tio
n
s
in
s
o
lv
in
g
p
r
o
b
lem
s
an
d
im
a
g
in
in
g
h
o
w
th
e
s
o
lu
tio
n
will
wo
r
k
.
B
o
th
m
ale
a
n
d
f
e
m
ale
s
tu
d
en
ts
tr
ied
to
ex
p
lo
r
e
an
d
in
n
o
v
ate
to
f
in
d
th
e
r
ig
h
t so
lu
tio
n
.
Stu
d
en
ts
wh
o
ar
e
t
h
en
also
ab
le
to
s
y
m
b
o
lize
a
n
d
g
en
er
aliz
e
ar
e
s
h
o
wn
b
y
th
eir
h
ab
it
o
f
co
n
n
ec
tin
g
th
e
g
iv
en
p
r
o
b
lem
with
th
eir
p
ast
ex
p
er
ien
ce
s
to
s
o
lv
e
th
e
p
r
o
b
lem
.
T
h
er
e
is
n
o
d
if
f
er
e
n
ce
in
h
ab
its
in
th
is
p
h
ase
f
o
r
m
ale
an
d
f
em
ale
s
tu
d
en
ts
.
T
h
ey
u
tili
ze
o
l
d
ex
p
e
r
ien
ce
s
an
d
m
ak
e
a
n
alo
g
ies
th
at
all
o
w
th
em
to
e
x
p
r
ess
th
eir
s
o
lu
tio
n
s
in
th
e
f
o
r
m
o
f
s
y
m
b
o
ls
th
at
ar
e
g
en
er
alize
d
[
3
9
]
,
[
4
0
]
.
I
n
a
d
d
itio
n
,
s
tu
d
en
ts
a
r
e
also
ac
cu
s
to
m
ed
to
tak
in
g
r
is
k
s
b
y
tak
in
g
o
p
p
o
r
tu
n
ities
th
at
ex
is
t a
n
d
tr
y
in
g
to
p
r
esen
t th
e
r
ig
h
t c
o
n
cl
u
s
io
n
s
.
3
.
4
.
2
.
M
o
dera
t
e
ca
t
e
g
o
ry
o
f
m
a
t
hema
t
ica
l e
x
plo
ra
t
io
n a
b
ility
I
n
th
e
f
ir
s
t
in
d
icato
r
o
f
m
ath
e
m
atica
l
ex
p
lo
r
atio
n
ab
ilit
y
,
s
t
u
d
en
ts
ar
e
a
b
le
to
id
e
n
tify
th
e
p
r
o
b
lem
b
ec
au
s
e
th
ey
h
a
v
e
a
h
ab
it
o
f
u
n
d
er
s
tan
d
in
g
s
o
m
eth
in
g
t
h
ey
h
ea
r
b
y
in
v
o
lv
in
g
em
p
ath
y
an
d
f
u
ll
u
n
d
e
r
s
tan
d
in
g
.
J
u
s
t
lik
e
s
tu
d
en
ts
in
th
e
h
ig
h
ca
teg
o
r
y
,
t
h
ey
ar
e
also
ac
cu
s
to
m
ed
to
u
tili
zin
g
t
h
eir
s
en
s
es
in
o
b
s
er
v
in
g
an
d
lo
o
k
in
g
at
ev
e
r
y
d
etail
o
f
t
h
e
p
r
o
b
lem
to
b
e
s
o
lv
ed
.
T
h
is
allo
ws
th
em
to
ab
s
o
r
b
s
o
m
e
in
f
o
r
m
atio
n
th
at
is
co
n
s
id
er
ed
ca
p
ab
le
o
f
b
ein
g
u
s
ed
in
p
r
o
b
lem
s
o
lv
in
g
.
Ho
wev
er
,
in
th
e
n
ex
t
h
ab
it,
b
o
th
m
a
le
an
d
f
em
ale
s
tu
d
en
ts
d
id
n
o
t
s
h
o
w
t
h
at
th
e
y
wer
e
in
ter
ested
in
th
e
p
r
o
b
lem
s
g
iv
en
a
n
d
h
ad
cu
r
io
s
ity
a
b
o
u
t
s
o
m
eth
in
g
n
ew.
Nev
er
th
eless
,
th
ey
wer
e
s
till
ab
le
to
u
n
d
er
s
tan
d
th
e
g
iv
en
p
r
o
b
lem
ap
p
r
o
p
r
iately
[
2
5
]
.
I
n
th
e
ab
ilit
y
to
e
x
am
in
e
p
atte
r
n
s
,
s
tu
d
en
ts
a
r
e
ab
le
to
f
in
d
p
atter
n
s
to
a
n
s
wer
th
e
q
u
esti
o
n
s
g
iv
en
.
I
n
f
em
ale
s
tu
d
en
ts
,
th
e
y
a
r
e
ac
cu
s
to
m
ed
to
w
o
r
k
in
g
o
n
s
o
m
et
h
in
g
g
i
v
en
s
er
io
u
s
ly
u
n
til
it
is
co
m
p
leted
an
d
d
o
n
o
t
g
iv
e
u
p
ea
s
ily
.
T
h
ey
u
n
d
er
s
tan
d
th
e
r
elat
io
n
s
h
ip
o
f
th
e
p
atter
n
f
o
r
m
ed
b
ec
a
u
s
e
th
ey
tr
y
co
n
t
in
u
o
u
s
ly
t
o
f
i
n
d
th
e
r
elatio
n
s
h
ip
in
th
e
in
f
o
r
m
atio
n
o
b
tain
ed
.
I
n
co
n
tr
ast to
m
ale
s
tu
d
en
ts
wh
er
e
th
ey
d
o
n
o
t sh
o
w
th
ese
h
ab
its
th
at
m
ak
e
th
e
m
ea
s
ily
g
iv
e
u
p
in
s
o
lv
in
g
e
x
is
tin
g
p
r
o
b
lem
s
.
T
h
is
is
alig
n
ed
w
ith
p
r
e
v
io
u
s
s
t
u
d
y
[
2
2
]
,
[
2
7
]
,
[
3
3
]
wh
ich
s
tates
th
at
p
eo
p
le
wh
o
h
av
e
p
er
s
is
tin
g
h
ab
its
will
tr
y
co
n
tin
u
o
u
s
ly
u
n
til
th
eir
p
r
o
b
lem
-
s
o
lv
in
g
s
tr
ateg
y
is
s
u
cc
ess
f
u
l.
Alth
o
u
g
h
d
if
f
er
e
n
t
in
ter
m
s
o
f
p
e
r
s
is
ten
ce
,
th
ey
all
s
h
o
w
th
e
s
am
e
h
ab
it
o
f
m
a
n
ag
in
g
im
p
u
ls
iv
ity
b
y
th
in
k
in
g
b
ef
o
r
e
ac
tin
g
wh
ic
h
h
elp
s
in
u
n
d
er
s
tan
d
in
g
th
e
d
i
r
ec
tio
n
o
f
th
e
p
r
o
b
lem
an
d
co
n
s
id
er
in
g
alter
n
ativ
e
s
u
itab
le
p
atter
n
s
.
T
h
is
is
o
f
co
u
r
s
e
also
d
u
e
to
t
h
e
h
ab
it
o
f
f
le
x
ib
le
th
in
k
in
g
wh
er
e
t
h
ey
ar
e
ab
le
to
ch
an
g
e
th
eir
m
in
d
s
wh
en
r
ec
eiv
i
n
g
n
ew
d
ata
wh
ich
allo
ws
th
em
to
id
e
n
tif
y
o
th
e
r
p
atter
n
r
elatio
n
s
h
ip
s
if
th
e
p
r
e
v
io
u
s
p
atter
n
is
n
o
t
co
n
s
id
er
e
d
ap
p
r
o
p
r
iate.
I
t
also
in
d
icate
s
th
at
s
tu
d
e
n
ts
in
th
is
ca
teg
o
r
y
a
r
e
also
u
s
ed
to
s
tr
iv
in
g
f
o
r
th
e
ac
cu
r
ac
y
o
f
th
eir
a
n
s
wer
s
.
Fu
r
th
er
m
o
r
e
,
o
n
t
h
e
in
d
icato
r
o
f
s
ea
r
ch
in
g
i
n
f
o
r
m
ally
,
s
o
m
e
d
if
f
er
en
ce
s
wer
e
s
ee
n
in
h
ab
i
ts
o
f
m
ale
an
d
f
em
ale
s
tu
d
en
ts
in
th
is
ca
t
eg
o
r
y
ev
e
n
th
o
u
g
h
th
ey
ar
e
ab
l
e
to
co
n
d
u
ct
in
f
o
r
m
al
s
ea
r
ch
es
to
s
o
lv
e
p
r
o
b
lem
s
.
Fem
ale
s
tu
d
en
ts
ten
d
t
o
s
h
o
w
th
e
h
a
b
it
o
f
ask
in
g
q
u
esti
o
n
s
an
d
p
r
o
p
o
s
in
g
p
r
o
b
le
m
s
s
o
th
at
th
ey
ca
n
ex
p
lo
r
e
h
o
w
to
s
o
lv
e
a
g
iv
e
n
p
r
o
b
lem
an
d
th
e
h
ab
it
o
f
co
n
tin
u
o
u
s
lear
n
in
g
f
r
o
m
p
r
e
v
io
u
s
ly
o
b
tain
ed
p
r
o
b
lem
s
,
wh
ile
m
ale
s
tu
d
en
ts
d
o
n
o
t
s
h
o
w
th
e
s
e
h
ab
its
.
Ho
wev
er
,
ju
s
t
lik
e
in
th
e
h
ig
h
ca
teg
o
r
y
,
o
n
ly
m
al
e
s
tu
d
en
ts
s
h
o
w
th
e
h
ab
it
o
f
d
o
in
g
h
u
m
o
r
t
h
at
ca
n
ex
er
cise
cr
ea
tiv
ity
s
o
th
at
th
ey
ca
n
u
n
d
er
s
tan
d
th
e
s
itu
at
io
n
f
r
o
m
d
if
f
er
e
n
t
p
er
s
p
ec
tiv
es
o
n
a
p
r
o
b
lem
.
Ne
v
er
th
eless
,
th
ey
all
s
h
o
wed
th
e
h
ab
it
o
f
in
ter
d
ep
en
d
en
t
t
h
in
k
in
g
wh
e
r
e
th
e
y
ar
e
u
s
ed
to
an
d
ab
le
t
o
wo
r
k
in
g
r
o
u
p
s
an
d
r
esp
ec
t o
th
e
r
p
eo
p
le'
s
id
ea
s
an
d
id
ea
s
in
d
ea
lin
g
with
a
p
r
o
b
lem
.
I
n
th
e
f
o
u
r
th
in
d
icato
r
,
s
tu
d
en
ts
in
th
e
m
o
d
er
ate
ca
te
g
o
r
y
w
er
e
u
n
ab
le
t
o
clar
if
y
t
h
e
s
o
lu
t
io
n
ef
f
o
r
t.
B
o
th
m
ale
an
d
f
em
ale
s
tu
d
en
t
s
,
th
ey
all
d
id
n
o
t
s
h
o
w
an
y
h
ab
its
o
f
m
in
d
r
elate
d
to
th
e
a
b
ilit
y
to
clar
if
y
th
e
s
o
lu
tio
n
ef
f
o
r
t.
T
h
is
s
h
o
ws
th
at
th
ey
ar
e
n
o
t
u
s
ed
to
ev
alu
atin
g
th
e
s
o
lu
tio
n
s
th
ey
h
av
e
d
o
n
e
b
ec
au
s
e
th
ey
h
a
v
e
n
o
t
r
ea
lized
a
n
d
u
n
d
e
r
s
to
o
d
wh
at
th
ey
k
n
o
w
a
n
d
wh
at
t
h
e
y
d
o
n
o
t
k
n
o
w.
T
h
ey
ar
e
also
n
o
t
u
s
ed
to
m
a
k
in
g
co
n
jectu
r
es
an
d
ex
p
r
ess
in
g
id
ea
s
.
Pre
v
io
u
s
r
esear
ch
r
elev
an
t
to
th
is
i
s
s
u
e
s
ta
tes
th
at
th
i
s
o
f
co
u
r
s
e
also
r
esu
lts
in
th
em
n
o
t b
ein
g
u
s
ed
to
wo
r
k
an
d
i
n
n
o
v
ate
to
tr
y
v
ar
io
u
s
s
o
lu
tio
n
s
in
s
o
lv
in
g
p
r
o
b
lem
s
[
2
9
]
.
J
u
s
t
lik
e
th
e
in
d
icato
r
o
f
clar
if
y
in
g
th
e
s
o
lu
tio
n
ef
f
o
r
t,
s
tu
d
e
n
ts
in
th
is
ca
teg
o
r
y
ar
e
u
n
a
b
le
to
s
y
m
b
o
lize
an
d
g
en
er
alize
.
T
h
is
is
also
s
h
o
wn
b
y
th
e
n
o
n
-
ex
is
tin
g
h
ab
it
s
o
f
m
in
d
th
at
em
er
g
e.
T
h
ey
d
o
n
o
t h
av
e
th
e
h
ab
it
o
f
co
n
n
ec
tin
g
th
e
g
iv
en
p
r
o
b
l
em
with
th
eir
p
ast
ex
p
er
ien
ce
s
to
s
o
lv
e
th
e
p
r
o
b
lem
.
T
h
ey
w
er
e
also
n
o
t
u
s
ed
to
tak
in
g
r
is
k
s
,
wh
ich
was sh
o
wn
b
y
th
eir
lac
k
o
f
co
n
f
i
d
en
ce
in
wr
itin
g
th
eir
co
n
clu
s
io
n
s
.
3
.
4
.
3
.
L
o
w
ca
t
eg
o
ry
o
f
ma
t
h
em
a
t
ica
l e
x
plo
ra
t
io
n a
bil
it
y
I
n
th
e
f
ir
s
t
in
d
icato
r
o
f
m
ath
em
atica
l
ex
p
lo
r
atio
n
a
b
ilit
y
,
d
if
f
er
en
t
f
r
o
m
s
tu
d
e
n
ts
in
th
e
h
ig
h
an
d
m
ed
iu
m
ca
teg
o
r
ies,
in
th
e
lo
w
ca
teg
o
r
y
,
m
ale
an
d
f
em
ale
s
tu
d
en
ts
o
n
ly
s
h
o
we
d
th
at
th
e
y
w
er
e
u
s
ed
to
u
tili
zin
g
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