I
nte
rna
t
io
na
l J
o
urna
l o
f
E
v
a
lua
t
io
n a
nd
Resea
rc
h in E
du
ca
t
io
n (
I
J
E
RE
)
Vo
l.
14
,
No
.
4
,
A
u
g
u
s
t
20
25
,
p
p
.
3
2
0
1
~
3
2
1
0
I
SS
N:
2
2
5
2
-
8
8
2
2
,
DOI
: 1
0
.
1
1
5
9
1
/
ijer
e
.
v
14
i
4
.
3
3
3
4
7
3201
J
o
ur
na
l ho
m
ep
a
g
e
:
h
ttp
:
//ij
ere.
ia
esco
r
e.
co
m
Im
pro
v
ing
alg
ebra
ic problem
-
so
lv
ing
achiev
ement o
f
seco
nda
ry
lev
el studen
ts by
using
gra
ph
-
ba
sed v
isua
liza
tion o
f
three
mo
dels
I
m
da
d Ali,
Sa
m
ira
n Da
s
D
e
p
a
r
t
me
n
t
o
f
M
a
t
h
e
m
a
t
i
c
s,
C
e
n
t
r
a
l
I
n
st
i
t
u
t
e
o
f
T
e
c
h
n
o
l
o
g
y
K
o
k
r
a
j
h
a
r
,
K
o
k
r
a
j
h
a
r
,
I
n
d
i
a
Art
icle
I
nfo
AB
S
T
RAC
T
A
r
ticle
his
to
r
y:
R
ec
eiv
ed
Oct
21
,
2
0
2
4
R
ev
is
ed
May
30
,
2
0
2
5
Acc
ep
ted
J
u
n
6
,
2
0
2
5
Th
e
stu
d
y
a
ims
to
i
n
v
e
sti
g
a
te
th
e
a
lg
e
b
ra
ic
p
r
o
b
lem
-
so
l
v
in
g
a
c
h
iev
e
m
e
n
t
(P
S
A)
o
f
se
c
o
n
d
a
r
y
sc
h
o
o
l
st
u
d
e
n
ts
.
Th
e
st
u
d
y
is
c
o
n
d
u
c
ted
to
imp
ro
v
e
stu
d
e
n
ts’
a
lg
e
b
ra
ic P
S
A b
y
u
sin
g
th
e
g
ra
p
h
ica
l
v
isu
a
li
z
a
ti
o
n
o
f
t
h
re
e
m
o
d
e
ls,
wh
ich
a
re
in
teg
ra
ted
t
o
a
n
a
l
y
z
e
c
e
rtain
fa
c
to
rs
th
a
t
c
o
n
tri
b
u
te
to
a
lg
e
b
ra
ic
P
S
A.
T
h
e
first
m
o
d
e
l
is
c
o
n
st
ru
c
ted
t
o
a
n
a
ly
z
e
stu
d
e
n
ts’
in
tr
in
sic
a
n
d
e
x
tri
n
sic
m
o
ti
v
a
ti
o
n
i
n
th
e
p
re
d
ict
io
n
o
f
a
lg
e
b
ra
ic
P
S
A
.
T
h
e
se
c
o
n
d
m
o
d
e
l
is
c
o
n
stru
c
ted
t
o
a
n
a
ly
z
e
with
th
e
h
e
lp
o
f
c
o
g
n
it
i
v
e
a
n
d
a
ffe
c
ti
v
e
se
l
f
-
c
o
n
c
e
p
t.
Also
,
th
e
th
ir
d
m
o
d
e
l
is
c
o
n
stru
c
t
e
d
to
a
n
a
ly
z
e
with
t
h
e
h
e
lp
o
f
f
o
u
r
fa
c
to
rs,
su
c
h
a
s
c
o
g
n
it
i
v
e
a
n
d
a
ffe
c
ti
v
e
se
lf
-
c
o
n
c
e
p
t,
a
n
d
i
n
tri
n
sic
a
n
d
e
x
tri
n
sic
m
o
ti
v
a
ti
o
n
.
I
n
th
is
stu
d
y
,
a
d
e
sc
rip
ti
v
e
su
r
v
e
y
d
e
sig
n
is
u
se
d
,
i
n
wh
ich
4
0
0
n
in
t
h
-
g
ra
d
e
stu
d
e
n
ts
a
re
in
v
o
l
v
e
d
in
th
e
M
o
ri
g
a
o
n
District
o
f
As
sa
m
,
In
d
ia.
Th
e
re
su
lt
s
o
f
re
g
re
ss
io
n
a
n
a
ly
sis
in
th
e
first
m
o
d
e
l
e
x
p
lain
2
9
.
8
%
v
a
rian
c
e
o
f
a
lg
e
b
ra
ic
P
S
A
,
th
e
se
c
o
n
d
m
o
d
e
l
d
e
m
o
n
stra
ted
t
h
a
t
6
4
.
9
%
v
a
rian
c
e
o
f
a
lg
e
b
ra
ic
a
n
d
t
h
e
th
ir
d
m
o
d
e
l
h
a
s
h
ig
h
e
st
6
6
.
1
%
v
a
rian
c
e
o
f
a
lg
e
b
ra
ic
P
S
A
c
a
n
b
e
a
c
c
o
u
n
ted
b
y
t
h
e
li
n
e
a
r
c
o
m
b
in
a
ti
o
n
o
f
t
h
e
fo
u
r
in
d
e
p
e
n
d
e
n
t
v
a
riab
les
.
Th
e
f
in
d
in
g
s
o
f
th
is
st
u
d
y
c
a
n
b
e
u
ti
l
ize
d
b
y
se
c
o
n
d
a
ry
e
d
u
c
a
ti
o
n
in
stit
u
ti
o
n
s to
imp
r
o
v
e
stu
d
e
n
ts’
m
a
th
e
m
a
ti
c
a
l
a
c
h
iev
e
m
e
n
t
a
s a
w
h
o
le.
K
ey
w
o
r
d
s
:
Alg
eb
r
a
Mo
tiv
atio
n
Pro
b
lem
-
s
o
lv
in
g
Seco
n
d
ar
y
s
ch
o
o
l
Self
-
co
n
ce
p
t
T
h
is i
s
a
n
o
p
e
n
a
c
c
e
ss
a
rticle
u
n
d
e
r th
e
CC B
Y
-
SA
li
c
e
n
se
.
C
o
r
r
e
s
p
o
nd
ing
A
uth
o
r
:
I
m
d
ad
Ali
Dep
ar
tm
en
t o
f
Ma
th
em
atics,
C
en
tr
al
I
n
s
titu
te
o
f
T
ec
h
n
o
lo
g
y
Ko
k
r
ajh
a
r
Ko
k
r
ajh
ar
,
I
n
d
ia
E
m
ail:
ali.
im
d
ad
9
9
@
g
m
ail.
co
m
1.
I
NT
RO
D
UCT
I
O
N
Ma
th
em
atics
is
wid
ely
r
eg
a
r
d
ed
as
a
n
ess
en
tial
s
k
ill
with
s
ig
n
if
ican
t
im
p
licatio
n
s
ac
r
o
s
s
v
ar
io
u
s
f
ac
ets
o
f
life
.
I
t
co
n
tr
i
b
u
tes
to
th
e
d
ev
elo
p
m
en
t
o
f
a
s
tu
d
en
t
’
s
cr
itical
th
in
k
in
g
an
d
r
ea
s
o
n
in
g
,
as
well
as
th
eir
f
u
n
d
am
e
n
tal
p
r
o
b
lem
-
s
o
lv
in
g
ab
ilit
ies,
wh
ich
ar
e
v
er
y
u
s
ef
u
l
in
ev
er
y
d
a
y
life
.
Ma
th
em
atic
s
is
a
f
u
n
d
am
en
tal
s
u
b
ject
tau
g
h
t
in
s
ch
o
o
ls
f
r
o
m
p
r
im
ar
y
t
o
s
ec
o
n
d
ar
y
s
ch
o
o
l
ar
o
u
n
d
th
e
g
lo
b
e.
T
h
e
r
o
le
o
f
m
ath
em
atics
is
in
d
is
p
en
s
ab
le
f
o
r
a
s
tu
d
en
t
at
th
e
s
ec
o
n
d
ar
y
s
tag
e
f
o
r
th
eir
a
ca
d
em
ic
ac
h
iev
em
e
n
t,
h
ig
h
er
s
tu
d
ies,
an
d
f
u
t
u
r
e
p
u
r
s
u
its
[
1
]
.
Desp
ite
th
e
im
p
o
r
tan
ce
o
f
m
ath
em
atics
at
th
i
s
s
tag
e,
s
tu
d
en
ts
’
ac
h
iev
em
en
t
is
n
o
t
s
atis
f
ac
to
r
y
[
2
]
.
I
n
I
n
d
ia,
it
h
as
b
ee
n
co
n
s
id
er
ed
o
n
e
o
f
th
e
m
o
s
t
ch
alle
n
g
in
g
s
u
b
jects
in
s
ch
o
o
l
an
d
ac
ts
as
th
e
p
r
im
ar
y
r
ea
s
o
n
f
o
r
s
tu
d
e
n
ts
’
f
ailu
r
e
at
th
e
s
ec
o
n
d
ar
y
lev
el.
Ma
n
y
s
ec
o
n
d
ar
y
s
ch
o
o
l
s
tu
d
en
ts
f
i
n
d
d
if
f
icu
lty
in
s
o
lv
in
g
a
m
ath
em
atics p
r
o
b
lem
[
3
]
.
Alg
eb
r
a
is
a
m
a
jo
r
ar
ea
o
f
m
at
h
e
m
a
tics
d
ea
l
in
g
w
it
h
s
y
m
b
o
ls
a
n
d
n
u
m
b
er
s
a
r
e
m
o
s
t
ly
u
s
e
d
i
n
p
r
o
b
l
em
-
s
o
l
v
i
n
g
.
I
t
o
c
c
u
p
ies
a
m
aj
o
r
p
o
r
ti
o
n
o
f
th
e
m
id
d
l
e
a
n
d
s
ec
o
n
d
a
r
y
s
c
h
o
o
l
m
at
h
em
a
tics
c
u
r
r
i
cu
lu
m
.
Alg
eb
r
aic
k
n
o
wl
ed
g
e
i
m
p
r
o
v
e
s
s
tu
d
e
n
ts
’
cr
iti
ca
l
th
in
k
i
n
g
an
d
p
r
o
b
l
em
-
s
o
l
v
i
n
g
s
k
ills
.
T
h
e
r
o
le
o
f
v
a
r
i
a
b
les
a
n
d
co
n
s
t
an
ts
m
u
s
t
b
e
u
n
d
e
r
s
t
o
o
d
in
o
r
d
e
r
t
o
b
u
il
d
m
at
h
e
m
at
ica
l
r
e
as
o
n
in
g
a
n
d
c
o
m
p
r
e
h
en
s
io
n
.
H
o
w
ev
er
,
th
e
r
e
is
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
.
14
,
No
.
4
,
Au
g
u
s
t
20
25
:
320
1
-
3
2
1
0
3202
a
m
aj
o
r
c
h
a
lle
n
g
e
t
h
at
s
t
u
d
e
n
t
s
f
a
ce
w
h
il
e
l
ea
r
n
i
n
g
a
b
o
u
t
v
a
r
ia
b
l
es
an
d
c
o
n
s
ta
n
ts
a
n
d
i
n
te
r
ac
ti
n
g
w
it
h
al
g
e
b
r
a
in
g
e
n
e
r
al
,
as
th
e
y
d
o
n
o
t
k
n
o
w
h
o
w
q
u
a
n
ti
ties
a
n
d
v
ar
i
a
b
les
r
ela
te
t
o
o
n
e
a
n
o
t
h
e
r
in
a
lg
e
b
r
aic
e
x
p
r
ess
i
o
n
s
[
4
]
.
Alg
eb
r
aic
p
r
o
b
lem
-
s
o
lv
in
g
is
a
m
ajo
r
f
ac
et
o
f
m
ath
em
atics.
I
n
r
ec
en
t
y
ea
r
s
,
r
esear
ch
h
as
f
o
cu
s
ed
o
n
alg
eb
r
aic
p
r
o
b
lem
-
s
o
lv
in
g
at
s
ec
o
n
d
ar
y
s
ch
o
o
l
[
5
]
–
[
7
]
.
Ma
n
y
f
ac
to
r
s
ass
o
ciate
d
with
u
n
d
er
s
tan
d
in
g
o
f
alg
eb
r
aic
ex
p
r
ess
io
n
s
an
d
ac
h
iev
em
en
t
in
p
r
o
b
lem
-
s
o
lv
in
g
s
u
ch
as
s
tu
d
en
ts
’
s
elf
-
co
n
ce
p
t,
s
elf
-
ef
f
icac
y
,
m
o
tiv
atio
n
,
e
x
p
er
ien
ce
in
s
ch
o
o
l,
an
d
attitu
d
es
[
8
]
–
[
11
]
.
M
o
tiv
atio
n
an
d
s
elf
-
co
n
ce
p
t
ar
e
th
e
m
o
s
t
in
f
lu
e
n
tial
f
ac
to
r
s
th
at
h
av
e
p
o
s
itiv
e
co
n
n
ec
tio
n
with
ac
ad
em
ic
ac
h
iev
em
en
t
at
s
ec
o
n
d
ar
y
lev
el
ar
e
to
b
e
in
v
esti
g
ated
[
12
]
,
[
13
]
.
Acc
o
r
d
in
g
to
Ham
m
o
u
d
i
[
14
]
,
s
tu
d
e
n
ts
’
co
g
n
itiv
e
an
d
af
f
ec
tiv
e
s
elf
-
co
n
ce
p
t
an
d
ex
tr
in
s
ic
m
o
tiv
atio
n
ar
e
th
e
s
ig
n
if
ican
t
f
ac
to
r
s
o
f
m
at
h
em
atica
l
ac
h
iev
em
en
t
at
u
n
d
er
g
r
a
d
u
ate
lev
el.
Als
o
,
Yild
ir
im
[
15
]
d
em
o
n
s
tr
ated
t
h
at
wh
ile
b
o
th
in
tr
in
s
ic
an
d
ex
tr
in
s
ic
m
o
tiv
atio
n
s
h
a
v
e
d
if
f
er
en
t
m
ed
iatio
n
ef
f
ec
ts
,
th
ey
ca
n
b
o
th
p
ar
tially
m
o
d
er
ate
th
e
in
f
lu
e
n
ce
o
f
teac
h
er
s
u
p
p
o
r
t o
n
m
ath
em
atics l
ea
r
n
in
g
.
Acc
o
r
d
in
g
to
L
u
m
s
d
en
[
16
]
,
t
h
e
willin
g
n
ess
o
f
s
tu
d
e
n
ts
to
e
n
g
ag
e
i
n
th
e
lear
n
in
g
p
r
o
ce
s
s
as
well
as
th
e
m
o
tiv
es
o
r
g
o
als
th
at
e
n
co
u
r
ag
e
s
tu
d
en
ts
to
tak
e
p
ar
t
o
r
n
o
t
in
ac
a
d
em
ic
ac
tiv
itie
s
ar
e
r
ef
e
r
r
ed
to
as
s
tu
d
en
t
m
o
tiv
atio
n
.
M
o
tiv
atio
n
is
o
n
e
o
f
th
e
m
o
s
t
c
r
u
c
ial
co
m
p
o
n
en
ts
f
o
r
th
e
lear
n
in
g
a
n
d
ac
ad
em
ic
ac
h
iev
em
en
t
o
f
s
tu
d
en
ts
[
17
]
.
Sin
ce
m
o
tiv
atio
n
af
f
ec
ts
t
h
e
in
s
tig
atio
n
,
way
a
n
d
in
te
n
s
ity
o
f
co
g
n
itiv
e
p
r
o
ce
s
s
es,
it
is
b
eliev
ed
to
b
e
h
elp
f
u
l
to
p
r
o
b
lem
-
s
o
lv
i
n
g
[
18
]
.
Acc
o
r
d
in
g
to
B
r
o
p
h
y
[
19
]
,
m
o
tiv
atio
n
is
a
co
g
n
itiv
e
p
r
o
ce
s
s
in
th
e
m
a
th
em
atica
l
d
o
m
ain
t
h
at
in
v
o
lv
es
g
r
asp
in
g
m
at
h
em
atica
l
task
s
,
r
elatin
g
th
e
in
f
o
r
m
atio
n
to
p
r
io
r
k
n
o
wled
g
e,
an
d
b
e
co
m
in
g
p
r
o
f
icien
t
in
th
e
m
ath
em
atica
l
ab
ilit
ies
it
s
u
p
p
o
r
ts
.
Stu
d
en
ts
'
m
o
tiv
atio
n
is
t
h
e
m
o
s
t
im
p
o
r
tan
t
f
ac
to
r
am
o
n
g
m
an
y
th
at
d
e
ter
m
in
es
h
o
w
well
th
ey
s
u
cc
e
ed
in
a
m
at
h
em
atics
co
u
r
s
e
[
20
]
.
A
s
u
b
s
tan
tial
b
o
d
y
o
f
r
ec
en
t
ev
id
e
n
ce
h
as
h
ig
h
lig
h
ted
th
e
ass
o
ciatio
n
b
etwe
en
m
o
tiv
atio
n
an
d
m
ath
em
atics
ac
h
iev
em
en
t
[
21
]
–
[
23
].
Mo
r
eo
v
e
r
,
J
in
g
et
a
l.
[
24
]
estab
lis
h
ed
th
e
s
ig
n
if
ican
t
ef
f
ec
t
o
f
m
o
tiv
atio
n
o
n
lear
n
in
g
alg
eb
r
a
an
d
alg
eb
r
aic
ac
h
iev
em
e
n
t
a
m
o
n
g
Ma
lay
s
ian
s
tu
d
e
n
ts
.
Als
o
,
ac
co
r
d
in
g
to
t
h
e
s
elf
-
d
eter
m
in
atio
n
th
e
o
r
y
by
R
y
an
an
d
Dec
i
[
25
]
,
m
o
tiv
ati
o
n
ca
n
b
e
ca
teg
o
r
ized
as
ex
tr
in
s
ic
an
d
in
tr
in
s
ic.
T
h
e
ter
m
“
in
tr
in
s
ic
m
o
ti
v
atio
n
”
d
escr
ib
es
th
e
ac
t
o
f
d
o
in
g
s
o
m
eth
in
g
b
ec
au
s
e
it
is
ess
en
tially
f
u
n
o
r
in
ter
esti
n
g
.
I
n
co
n
t
r
ast,
“e
x
tr
i
n
s
ic
m
o
tiv
atio
n
”
m
o
tiv
ates
s
tu
d
en
ts
to
e
n
g
ag
e
in
ac
a
d
em
i
c
ac
tiv
ities
d
u
e
to
ex
ter
n
al
r
ewa
r
d
s
lik
e
p
r
aise
,
awa
r
d
s
,
o
r
p
u
n
is
h
m
en
ts
.
T
h
e
s
ig
n
if
ican
t
r
elatio
n
s
h
ip
b
etwe
en
in
tr
in
s
ic
m
o
tiv
atio
n
a
n
d
m
at
h
em
atics
ac
h
iev
em
en
t
h
as
b
ee
n
h
ig
h
li
g
h
ted
b
y
s
ev
er
al
s
tu
d
ies
[
26
]
–
[
28
]
.
I
n
a
r
ec
en
t
s
tu
d
y
,
T
r
an
an
d
Ng
u
y
en
[
29
]
also
estab
lis
h
ed
th
e
p
o
s
itiv
e
r
elatio
n
s
h
ip
b
etwe
en
in
tr
in
s
ic
m
o
tiv
atio
n
an
d
m
ath
em
atics a
ch
iev
em
en
t.
On
th
e
o
th
er
h
an
d
,
s
ev
er
al
o
th
er
s
ig
n
if
ies th
e
im
p
o
r
ta
n
ce
o
f
e
x
tr
in
s
ic
m
o
tiv
atio
n
in
m
ath
em
atica
l a
ch
iev
em
en
t
[
9
]
,
[
30
].
R
esear
ch
h
as
id
en
tifie
d
s
tu
d
e
n
ts
’
m
ath
em
atics
s
elf
-
co
n
ce
p
t
as
o
n
e
o
f
th
e
m
o
s
t
in
f
lu
en
tial
f
ac
to
r
s
o
f
m
ath
em
atics
ac
h
iev
em
en
t.
I
n
g
en
er
al,
r
esear
ch
h
as
u
s
ed
t
h
e
co
n
s
tr
u
ct
o
f
s
elf
-
c
o
n
ce
p
t
as
a
m
ea
s
u
r
e
o
f
an
in
d
iv
id
u
al'
s
co
g
n
itiv
e
an
d
af
f
ec
tiv
e
ab
ilit
ies
[
12
]
,
[
31
].
T
h
e
ter
m
“
m
ath
em
atics
s
elf
-
co
n
c
ep
t
”
d
escr
ib
es
h
o
w
p
u
p
ils
v
iew
th
eir
o
wn
co
m
p
eten
cy
an
d
ca
p
ac
ity
t
o
ac
q
u
ir
e
m
ath
em
atics
an
d
d
o
wel
l
o
n
m
ath
em
atica
l
p
r
o
b
lem
s
[
32
]
.
Acc
o
r
d
in
g
to
T
an
n
er
an
d
J
o
n
es
[
33
]
,
s
tu
d
e
n
ts
’
co
g
n
itiv
e
s
elf
-
co
n
ce
p
t
is
awa
r
e
o
f
th
eir
o
wn
m
ath
em
atica
l
k
n
o
wled
g
e,
in
clu
d
in
g
th
eir
ca
p
ac
ity
to
r
e
co
g
n
ize
th
eir
o
wn
m
ath
e
m
a
tical
s
tr
en
g
th
s
an
d
lim
itatio
n
s
,
lin
k
th
e
v
ar
io
u
s
to
p
ics
in
th
e
s
u
b
ject,
an
d
u
s
e
th
eir
ab
s
tr
ac
tio
n
p
r
o
ce
s
s
es.
T
h
e
in
n
er
s
et
o
f
b
elief
s
th
at
s
tu
d
en
ts
n
ee
d
to
s
u
cc
ee
d
in
m
ath
em
atics
is
k
n
o
wn
as
th
eir
af
f
ec
tiv
e
s
elf
-
c
o
n
ce
p
t.
T
h
ese
b
elief
s
in
clu
d
e
th
eir
co
n
f
i
d
en
ce
in
t
h
e
n
atu
r
e
o
f
m
ath
em
atica
l
k
n
o
wled
g
e
,
t
h
eir
f
ee
lin
g
s
o
f
s
elf
-
esteem
as
m
ath
lear
n
er
s
,
a
n
d
th
eir
ca
p
ac
ity
to
lear
n
an
d
ac
h
iev
e
in
th
e
s
u
b
ject.
Nu
m
e
r
o
u
s
s
tu
d
ies
h
av
e
estab
lis
h
ed
th
e
p
o
s
itiv
e
r
elatio
n
s
h
ip
b
etwe
en
m
ath
em
atics
s
elf
-
co
n
ce
p
t
an
d
m
ath
em
atics
ac
h
iev
e
m
en
t
[
5
]
,
[
34
]
.
Fo
r
i
n
s
tan
ce
,
Ab
d
u
llah
et
a
l.
[
5
]
estab
lis
h
ed
th
e
r
elatio
n
s
h
ip
b
etwe
en
s
elf
-
co
n
ce
p
t,
em
o
tio
n
al
in
tellig
en
ce
,
an
d
p
r
o
b
lem
-
s
o
lv
in
g
s
k
ills
an
d
attitu
d
es
to
war
d
s
s
o
lv
in
g
alg
e
b
r
aic
p
r
o
b
lem
s
a
m
o
n
g
Nig
er
i
an
s
ec
o
n
d
a
r
y
s
ch
o
o
l
s
tu
d
en
ts
.
Acc
o
r
d
in
g
to
th
ei
r
f
in
d
in
g
s
s
tu
d
en
ts
’
p
r
o
b
lem
-
s
o
lv
in
g
s
k
ills
ca
n
b
e
d
ev
elo
p
ed
b
y
tak
in
g
in
to
ac
co
u
n
t
o
f
th
e
ir
s
elf
-
co
n
ce
p
t
an
d
em
o
tio
n
al
in
tellig
en
ce
.
I
n
an
o
th
er
s
tu
d
y
,
Ham
m
o
u
d
i
a
n
d
Gr
ir
a
[
9
]
estab
lis
h
ed
th
at
s
tu
d
en
ts
’
co
g
n
itiv
e
an
d
af
f
ec
tiv
e
s
elf
-
co
n
ce
p
t h
av
e
a
s
ig
n
if
ican
t in
f
lu
e
n
ce
o
n
m
o
tiv
a
tio
n
f
o
r
th
eir
m
ath
em
atica
l a
ch
iev
em
en
t.
T
h
e
s
u
b
s
tan
tial
b
o
d
y
o
f
th
e
d
is
cu
s
s
ed
liter
atu
r
e
in
d
icate
s
s
tu
d
en
ts
’
m
o
tiv
atio
n
an
d
s
elf
-
c
o
n
ce
p
t
a
r
e
cr
u
cial
f
ac
to
r
s
in
th
eir
s
u
cc
ess
in
m
ath
em
atics.
T
h
o
u
g
h
a
l
o
t
o
f
r
esear
ch
h
as
b
ee
n
d
o
n
e
co
n
ce
r
n
in
g
to
t
h
e
ef
f
ec
t
o
f
s
tu
d
en
ts
’
m
o
tiv
atio
n
an
d
s
elf
-
co
n
ce
p
t
o
n
m
ath
em
atics
ac
h
iev
em
en
t,
litt
le
s
tu
d
y
h
as
ex
p
lo
r
e
d
alg
eb
r
aic
p
r
o
b
lem
-
s
o
lv
in
g
.
Mo
r
eo
v
er
,
th
e
r
elatio
n
s
h
ip
b
etwe
en
in
tr
in
s
ic
m
o
tiv
atio
n
,
ex
tr
in
s
ic
m
o
tiv
atio
n
,
co
g
n
itiv
e
s
elf
-
co
n
ce
p
t,
af
f
ec
ti
v
e
s
elf
-
co
n
ce
p
t,
an
d
alg
eb
r
aic
p
r
o
b
lem
-
s
o
lv
i
n
g
h
as
n
o
t
b
ee
n
ex
p
lo
r
ed
in
d
etail.
T
h
is
s
tu
d
y
ex
am
in
es
th
e
ef
f
ec
t
o
f
s
elf
-
co
n
ce
p
t
an
d
m
o
ti
v
atio
n
al
asp
ec
ts
o
n
alg
eb
r
ai
c
p
r
o
b
lem
-
s
o
l
v
in
g
ac
h
iev
em
en
t
(
PS
A)
.
B
ased
o
n
th
e
o
b
jectiv
es,
th
e
f
o
llo
win
g
th
r
ee
m
ath
em
atica
l
m
o
d
els
ar
e
co
n
s
tr
u
cted
to
im
p
r
o
v
e
s
tu
d
en
ts
’
alg
e
b
r
aic
P
SA
b
y
u
s
in
g
g
r
ap
h
-
b
ased
v
is
u
aliza
tio
n
o
f
th
e
th
r
ee
m
o
d
els.
T
h
ese
m
o
d
els
ar
e
co
m
b
in
ed
to
an
aly
ze
s
tu
d
en
ts
’
alg
eb
r
aic
PS
A
with
th
e
h
elp
o
f
th
eir
s
elf
-
co
n
ce
p
t a
n
d
m
o
tiv
a
tio
n
al
asp
ec
ts
.
I
n
th
e
f
ir
s
t
m
o
d
el,
it
is
ca
taly
z
ed
to
ex
am
in
e
th
e
s
ig
n
if
ican
t a
s
s
o
ciatio
n
o
f
al
g
eb
r
aic
PS
A
an
d
th
e
two
asp
ec
ts
o
f
m
o
tiv
atio
n
.
I
n
ed
u
ca
tio
n
al
r
esear
ch
,
s
tu
d
en
ts
’
m
o
tiv
atio
n
is
th
e
im
p
o
r
ta
n
t
f
ac
to
r
s
[
20
]
,
[
21
]
an
d
m
ath
em
atica
l
ac
h
iev
em
en
t
ca
n
b
e
ass
o
ciate
d
to
b
o
th
in
tr
in
s
ic
an
d
ex
ten
s
iv
e
m
o
tiv
atio
n
.
Acc
o
r
d
in
g
to
s
tu
d
y
by
R
y
an
an
d
Dec
i
[
35
]
,
in
tr
i
n
s
ic
m
o
tiv
atio
n
is
n
o
t
s
u
f
f
ici
en
t
to
e
n
g
ag
e
in
o
r
en
jo
y
a
s
p
ec
if
ic
task
with
o
u
t
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
I
mp
r
o
vin
g
a
lg
e
b
r
a
ic
p
r
o
b
lem
-
s
o
lvin
g
a
ch
ieve
men
t o
f seco
n
d
a
r
y
leve
l stu
d
en
ts
b
y
u
s
in
g
…
(
I
md
a
d
A
li
)
3203
ex
tr
in
s
ic
m
o
tiv
atio
n
.
T
h
u
s
,
i
t
m
ay
b
e
h
y
p
o
th
esized
th
at
s
tu
d
en
ts
lear
n
in
g
m
at
h
em
at
ics
ar
e
m
o
tiv
ated
in
tr
in
s
ically
f
o
r
th
eir
i
n
ter
ests
an
d
ex
tr
in
s
ically
in
o
r
d
e
r
to
ta
k
e
ad
v
a
n
tag
e
o
f
f
u
tu
r
e
p
r
o
s
p
e
cts.
Sin
ce
s
o
lv
in
g
alg
eb
r
aic
p
r
o
b
l
em
s
is
a
n
ec
es
s
ar
y
asp
ec
t
o
f
m
ath
em
atics
lear
n
in
g
,
it
is
h
y
p
o
th
esized
th
at
s
tu
d
en
ts
’
in
tr
in
s
ic
a
n
d
e
x
tr
in
s
ic
m
o
tiv
atio
n
m
ay
h
av
e
an
im
p
ac
t
o
n
th
eir
ab
ilit
y
to
s
o
lv
e
alg
eb
r
ai
c
p
r
o
b
lem
s
.
Fo
r
th
is
p
u
r
p
o
s
e,
th
e
f
o
llo
win
g
two
h
y
p
o
th
eses
ar
e
cr
ea
ted
to
ex
a
m
in
e
s
tu
d
e
n
ts
’
alg
eb
r
aic
PS
A.
T
h
e
f
o
llo
win
g
two
h
y
p
o
th
eses
ar
e
ex
am
in
e
d
b
y
u
s
in
g
th
e
f
ir
s
t
m
o
d
el
’
s
p
er
f
o
r
m
an
ce
:
-
0
1
:
t
h
er
e
is
n
o
r
elatio
n
s
h
i
p
b
etw
ee
n
s
tu
d
en
ts
’
in
tr
in
s
ic
m
o
tiv
at
io
n
an
d
alg
eb
r
aic
PS
A.
-
0
2
:
t
h
er
e
is
n
o
r
elatio
n
s
h
i
p
b
etw
ee
n
s
tu
d
en
ts
’
ex
tr
in
s
ic
m
o
tiv
at
io
n
an
d
alg
eb
r
aic
PS
A.
T
h
e
s
ec
o
n
d
m
o
d
el
aim
s
to
ex
am
in
e
th
e
asp
ec
ts
o
f
s
elf
-
co
n
ce
p
t,
in
r
elatio
n
t
o
alg
eb
r
aic
PS
A.
Acc
o
r
d
in
g
t
o
Ar
en
s
et
a
l.
[
36
]
,
c
o
g
n
itiv
e
s
elf
-
c
o
n
ce
p
t
i
s
o
n
e’
s
p
er
ce
p
tio
n
an
d
b
elie
f
s
ab
o
u
t
th
eir
o
wn
ab
ilit
ies,
s
k
ill
s
,
an
d
k
n
o
wled
g
e
in
m
ath
em
atics.
I
t
in
v
o
lv
es
b
elief
s
ab
o
u
t
o
n
e’
s
p
r
o
b
le
m
-
s
o
lv
in
g
ab
ilit
ies,
u
n
d
er
s
tan
d
i
n
g
o
f
m
ath
em
atic
al
co
n
ce
p
ts
an
d
o
v
er
all
co
m
p
eten
ce
in
m
ath
em
atics.
I
n
e
d
u
ca
tio
n
al
s
tu
d
ies,
co
g
n
itiv
e
s
elf
-
co
n
ce
p
t
is
th
e
m
o
s
t
im
p
o
r
tan
t
f
ac
to
r
in
m
at
h
em
atica
l
ac
h
iev
em
en
t.
Als
o
,
th
e
af
f
ec
tiv
e
s
elf
-
co
n
ce
p
t
in
m
ath
em
atics
p
er
tai
n
s
to
em
o
tio
n
al
p
er
ce
p
tio
n
s
a
n
d
attitu
d
es
to
war
d
s
m
at
h
em
a
tics
[
37
]
.
Stu
d
e
n
ts
’
p
o
s
itiv
e
af
f
ec
tiv
e
s
elf
-
co
n
ce
p
t
ca
n
en
h
a
n
ce
lear
n
in
g
an
d
ac
h
iev
em
en
t
in
m
ath
em
atic
s
[
10
]
.
T
h
u
s
,
b
o
th
co
g
n
itiv
e
an
d
a
f
f
ec
tiv
e
s
elf
-
c
o
n
ce
p
ts
m
ig
h
t
co
n
s
id
er
a
b
ly
i
n
f
lu
en
ce
o
n
s
tu
d
en
t’
s
ap
p
r
o
a
ch
an
d
s
u
cc
ess
in
m
ath
em
atica
l ta
s
k
s
.
I
n
lig
h
t
o
f
th
is
,
it
is
p
o
s
s
ib
le
t
o
h
y
p
o
th
esize
th
at
s
tu
d
en
t’
af
f
ec
tiv
e
an
d
co
g
n
itiv
e
s
elf
-
co
n
ce
p
t
m
ig
h
t
b
e
lin
k
ed
to
m
ath
em
atica
l
PS
A.
Fo
r
th
is
p
u
r
p
o
s
e,
th
e
f
o
l
lo
win
g
f
o
u
r
h
y
p
o
th
eses
ar
e
cr
ea
ted
to
ex
am
in
e
s
tu
d
en
ts
’
alg
eb
r
aic
PS
A.
T
h
e
f
o
llo
win
g
two
h
y
p
o
th
eses
ar
e
e
x
am
in
ed
b
y
th
e
s
ec
o
n
d
m
o
d
el
’
s
p
er
f
o
r
m
a
n
ce
:
-
0
3
:
t
h
er
e
is
n
o
r
elatio
n
s
h
i
p
b
etw
ee
n
s
tu
d
en
ts
’
co
g
n
itiv
e
s
elf
-
co
n
ce
p
t a
n
d
alg
eb
r
aic
PS
A.
-
0
4
:
t
h
er
e
is
n
o
r
elatio
n
s
h
i
p
b
etw
ee
n
s
tu
d
en
ts
’
af
f
ec
tiv
e
s
elf
-
c
o
n
ce
p
t a
n
d
alg
eb
r
aic
PS
A.
A
cc
o
r
d
in
g
to
Kap
la
n
[
27
]
,
s
tr
ateg
ies
s
u
ch
as
b
o
o
s
tin
g
ex
tr
in
s
ic
an
d
in
tr
in
s
ic
m
o
tiv
atio
n
an
d
d
ev
elo
p
in
g
a
g
o
o
d
s
elf
-
co
n
ce
p
t
th
r
o
u
g
h
im
p
r
o
v
in
g
in
t
r
in
s
ic
m
o
tiv
atio
n
s
h
o
u
ld
b
e
g
i
v
en
p
r
io
r
ity
in
o
r
d
er
to
im
p
r
o
v
e
m
ath
em
atica
l
ac
h
iev
em
en
t.
B
y
e
m
p
h
asizin
g
th
ese
ap
p
r
o
ac
h
es,
s
tu
d
en
ts
ar
e
en
c
o
u
r
ag
e
d
to
ex
p
lo
r
e
m
ath
,
s
et
g
o
als,
an
d
b
u
ild
c
o
n
f
id
en
ce
,
u
ltima
tely
lead
in
g
to
g
r
ea
ter
s
u
cc
ess
in
th
e
s
u
b
ject.
Fu
r
th
er
m
o
r
e,
Ham
m
o
u
d
i
[
14
]
p
o
in
ts
o
u
t
th
at
th
e
m
o
s
t
s
ig
n
if
ica
n
t
v
ar
iab
les
f
o
r
m
ath
e
m
atics
ac
h
iev
em
en
t
at
th
e
u
n
d
er
g
r
ad
u
ate
lev
el
ar
e
s
tu
d
en
ts
’
s
elf
-
co
n
ce
p
ts
an
d
th
eir
m
o
tiv
atio
n
al
asp
ec
ts
.
Hen
ce
,
th
e
th
ir
d
m
o
d
el
is
cr
ea
ted
to
in
v
esti
g
ate
t
h
e
in
f
lu
en
ce
o
n
alg
eb
r
aic
PS
A
o
f
th
e
f
o
u
r
f
ac
to
r
s
,
s
u
c
h
as
s
tu
d
en
ts
’
co
g
n
itiv
e
s
elf
-
co
n
ce
p
t,
a
f
f
ec
tiv
e
s
elf
-
c
o
n
ce
p
t,
i
n
tr
in
s
ic
m
o
tiv
atio
n
an
d
ex
t
r
in
s
ic
m
o
tiv
atio
n
.
Fo
r
th
is
p
u
r
p
o
s
e,
t
h
e
f
o
llo
win
g
f
o
u
r
h
y
p
o
th
eses
ar
e
cr
ea
ted
to
p
r
ed
ict
s
tu
d
en
ts
’
alg
eb
r
aic
PS
A.
T
h
e
f
o
llo
win
g
h
y
p
o
th
eses
ar
e
ex
am
in
ed
b
y
th
e
t
h
ir
d
m
o
d
el
’
s
p
er
f
o
r
m
an
ce
:
-
0
5
:
t
h
er
e
is
n
o
r
elatio
n
s
h
i
p
b
etw
ee
n
s
tu
d
en
ts
’
in
tr
in
s
ic
m
o
tiv
at
io
n
an
d
alg
eb
r
aic
PS
A.
-
0
6
:
t
h
er
e
is
n
o
r
elatio
n
s
h
i
p
b
etw
ee
n
s
tu
d
en
ts
’
ex
tr
in
s
ic
m
o
tiv
at
io
n
an
d
alg
eb
r
aic
PS
A.
-
0
7
:
t
h
er
e
is
n
o
r
elatio
n
s
h
i
p
b
etw
ee
n
s
tu
d
en
ts
’
co
g
n
itiv
e
s
elf
-
co
n
ce
p
t a
n
d
alg
eb
r
aic
PS
A.
-
0
8
:
t
h
er
e
is
n
o
r
elatio
n
s
h
i
p
b
etw
ee
n
s
tu
d
en
ts
’
af
f
ec
tiv
e
s
elf
-
c
o
n
ce
p
t a
n
d
alg
eb
r
aic
PS
A.
2.
RE
S
E
ARCH
M
E
T
H
O
D
A
q
u
an
titativ
e
r
esear
c
h
d
esig
n
is
u
s
ed
f
o
r
t
h
is
p
r
esen
t
s
tu
d
y
.
I
n
th
is
s
tu
d
y
,
a
v
ar
iety
o
f
r
eg
r
ess
io
n
test
s
h
av
e
b
ee
n
u
s
ed
to
ex
a
m
in
e
th
e
r
elatio
n
s
h
ip
b
etwe
e
n
m
u
ltip
le
in
d
ep
en
d
en
t
v
a
r
ia
b
les
an
d
a
s
in
g
le
d
ep
en
d
e
n
t
v
ar
iab
le
f
o
r
th
e
e
f
f
ec
tiv
en
ess
o
f
ea
ch
o
f
t
h
e
m
o
d
els.
Fo
r
th
e
s
tu
d
y
,
a
s
am
p
le
o
f
4
0
0
n
in
th
-
g
r
a
d
e
p
u
p
ils
is
r
a
n
d
o
m
l
y
s
elec
ted
f
r
o
m
t
h
e
Mo
r
ig
ao
n
D
is
tr
ict
o
f
Ass
am
.
T
h
is
s
am
p
le
s
ize
is
d
eter
m
in
e
d
u
s
in
g
C
o
ch
r
an
’
s
s
am
p
le
s
ize
f
o
r
m
u
l
a
(
f
o
r
f
in
ite
p
o
p
u
latio
n
)
.
T
h
r
ee
in
s
tr
u
m
en
ts
ar
e
u
s
ed
to
co
llect
th
e
d
ata.
T
h
e
in
s
tr
u
m
en
ts
ar
e
th
e
m
o
tiv
atio
n
s
ca
le,
th
e
s
elf
-
co
n
ce
p
t
s
ca
le
,
an
d
alg
eb
r
aic
PS
A
test
.
T
h
e
m
o
tiv
atio
n
s
ca
le
is
ad
o
p
ted
an
d
m
o
d
if
ied
f
r
o
m
Ham
m
o
u
d
i
[
14
]
f
o
r
th
e
s
tu
d
y
.
T
h
e
s
ca
le
co
n
s
is
ts
o
f
10
item
s
:
f
iv
e
f
o
r
in
tr
in
s
ic
an
d
f
iv
e
f
o
r
e
x
tr
in
s
ic
m
o
tiv
atio
n
s
.
On
e
ex
a
m
p
le
o
f
i
n
tr
in
s
ic
m
o
tiv
atio
n
is
“
m
ath
e
m
atics
is
m
ain
ly
ab
o
u
t
f
ac
ts
a
n
d
o
p
e
r
atio
n
s
,
”
an
d
o
n
e
ex
a
m
p
le
o
f
e
x
tr
in
s
ic
m
o
t
iv
atio
n
is
“
l
ea
r
n
in
g
m
at
h
em
at
ics
is
u
s
ef
u
l
f
o
r
m
e
b
ec
au
s
e
i
t
will
im
p
r
o
v
e
m
y
ca
r
ee
r
p
r
o
s
p
ec
ts
.
”
T
h
e
L
ik
er
t
5
-
p
o
in
t
s
ca
le
is
u
s
ed
f
o
r
t
h
e
s
tu
d
y
.
I
t
r
an
g
es
f
r
o
m
s
tr
o
n
g
ly
d
is
ag
r
ee
=
1
to
s
tr
o
n
g
ly
ag
r
ee
=
5
.
T
h
e
C
r
o
n
b
a
ch
alp
h
a
r
eliab
ilit
y
o
f
th
e
m
o
ti
v
atio
n
al
s
ca
le
is
0
.
8
1
5
.
T
h
e
s
elf
-
co
n
ce
p
t
s
ca
le
is
ad
o
p
ted
f
r
o
m
Ham
m
o
u
d
i
[
14
].
T
h
e
s
elf
-
co
n
ce
p
t
s
ca
le
co
n
s
is
ts
o
f
10
item
s
:
s
ix
f
o
r
“c
o
g
n
itiv
e
s
elf
-
co
n
ce
p
t”
an
d
f
o
u
r
f
o
r
“a
f
f
ec
ti
v
e
s
elf
-
co
n
ce
p
t”
o
n
th
e
L
ik
e
r
t
f
i
v
e
-
p
o
in
t
s
ca
le.
On
e
ex
am
p
le
o
f
c
o
g
n
itiv
e
s
elf
-
co
n
ce
p
t
is
“I
b
eliev
e
I
h
av
e
a
g
o
o
d
u
n
d
er
s
tan
d
i
n
g
o
f
m
ath
em
a
tical
co
n
ce
p
ts
,
”
an
d
o
n
e
ex
am
p
le
o
f
af
f
ec
tiv
e
s
elf
-
co
n
ce
p
t
is
“I
f
ee
l
co
n
f
id
en
t
i
n
m
y
m
ath
em
atica
l
p
r
o
b
lem
-
s
o
lv
in
g
s
k
ills
,
ev
en
wh
en
th
e
p
r
o
b
lem
s
ar
e
d
if
f
ic
u
lt.
”
T
h
e
L
i
k
er
t
5
-
p
o
i
n
t
s
ca
le
is
u
s
ed
f
o
r
th
e
s
tu
d
y
.
I
t
r
a
n
g
es
f
r
o
m
s
tr
o
n
g
ly
d
is
ag
r
ee
=
1
to
s
tr
o
n
g
ly
a
g
r
ee
=
5
.
T
h
e
C
r
o
n
b
ac
h
al
p
h
a
r
eliab
il
ity
o
f
th
e
m
o
tiv
atio
n
al
s
ca
le
is
0
.
9
0
1
.
Stu
d
en
ts
’
alg
eb
r
aic
PS
A
is
m
ea
s
u
r
ed
b
y
a
test
.
I
t
co
n
s
is
ts
o
f
2
0
m
u
ltip
le
ch
o
ice
item
s
b
a
s
ed
o
n
th
e
class
I
X
s
y
llab
u
s
.
T
h
e
r
eliab
il
ity
o
f
th
e
item
s
is
ca
lcu
lated
b
y
u
s
in
g
C
r
o
n
b
ac
h
alp
h
a,
wh
i
ch
is
0
.
8
9
a
n
d
is
in
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
.
14
,
No
.
4
,
Au
g
u
s
t
20
25
:
320
1
-
3
2
1
0
3204
an
ac
ce
p
tab
le
r
an
g
e.
An
ex
a
m
p
le
o
f
th
e
test
item
i
s
“
t
h
e
co
-
ef
f
icien
t
o
f
x
2
in
th
e
ex
p
an
s
io
n
o
f
(
2
x
-
3
)
(x
2
-
3
x
+
1
)
is
”
:
(
i)
9
,
(
ii)
-
9
,
(
iii)
1
,
an
d
(
iv
)
-
1
.
T
h
e
co
llected
d
ata
ar
e
a
n
aly
z
ed
b
y
u
s
in
g
SP
SS
2
6
.
0
v
er
s
io
n
.
R
eg
r
ess
io
n
an
aly
s
is
is
d
o
n
e
to
test
th
e
p
er
f
o
r
m
an
ce
o
f
ea
c
h
o
f
th
e
th
r
ee
m
o
d
els.
Stu
d
en
ts
’
alg
eb
r
a
ic
PS
A
i
s
th
e
d
ep
en
d
en
t
v
ar
ia
b
le
in
all
th
e
th
r
ee
m
o
d
els.
Als
o
,
d
is
tin
ct
in
d
ep
en
d
en
t
v
ar
iab
les
ar
e
u
s
ed
f
o
r
ea
ch
o
f
th
e
th
r
ee
m
o
d
els.
T
h
e
ac
cu
r
ac
y
o
f
ea
ch
m
o
d
el
is
p
r
ed
icted
u
s
in
g
th
e
co
ef
f
icien
t
o
f
d
eter
m
in
a
n
t
(
R
2
)
.
I
t
d
eter
m
i
n
es
h
o
w
well
th
e
d
ata
p
o
in
ts
f
it
th
e
r
eg
r
ess
io
n
lin
e.
3.
RE
SU
L
T
S
T
h
e
r
esu
lts
p
r
esen
ted
in
T
ab
le
s
1
-
3
h
av
e
b
ee
n
u
s
ed
t
o
ex
a
m
i
n
e
th
e
co
r
r
elatio
n
o
f
alg
e
b
r
aic
PS
A
with
th
e
two
asp
ec
ts
o
f
m
o
tiv
atio
n
.
Fro
m
T
ab
le
3
,
it
is
s
ee
n
th
at
ex
tr
in
s
ic
m
o
tiv
atio
n
(
b
=1
.
4
9
0
,
t=3
.
2
6
2
,
p
<
0
.
0
0
1
)
an
d
in
tr
i
n
s
ic
m
o
tiv
atio
n
(
b
=
3
.
3
9
3
,
t=6
.
6
4
0
,
p
<0
.
0
0
0
)
h
as
s
i
g
n
if
ican
t
r
elatio
n
s
h
ip
with
al
g
eb
r
aic
PS
A.
Hen
ce
,
th
e
h
y
p
o
th
eses
0
1
an
d
0
2
h
av
e
b
ee
n
r
ejec
ted
.
Als
o
,
f
r
o
m
T
a
b
les
1
an
d
2
,
it
i
s
f
o
u
n
d
th
at
alg
e
b
r
aic
PS
A
ca
n
b
e
e
x
p
lain
ed
b
y
b
o
th
th
e
in
t
r
in
s
ic
an
d
ex
tr
in
s
ic
m
o
tiv
atio
n
s
,
as
F
(
2
,
3
9
7
)
=
8
5
.
3
2
8
,
p
<0
.
0
0
0
.
T
h
e
v
alu
e
o
f
th
e
c
o
ef
f
icien
t
o
f
d
eter
m
in
atio
n
(R
2
=0
.
2
9
8
)
s
h
o
ws
th
at
co
g
n
itiv
e
an
d
af
f
ec
tiv
e
s
elf
-
c
o
n
ce
p
t
ex
p
lain
s
2
9
.
8
%
o
f
t
h
e
v
a
r
iatio
n
o
f
alg
eb
r
aic
PS
A.
Als
o
,
th
e
m
u
ltip
le
co
r
r
elatio
n
co
ef
f
icien
t
(
R
=0
.
5
4
6
)
in
d
icate
s
th
at
s
tu
d
en
ts
’
in
tr
in
s
ic
an
d
ex
tr
in
s
ic
m
o
tiv
atio
n
ar
e
s
ig
n
if
ican
tly
ass
o
ciate
d
to
alg
eb
r
aic
PS
A.
T
ab
le
1
.
Mo
d
el
I
s
u
m
m
a
r
y
M
o
d
e
l
R
R
2
A
d
j
u
st
e
d
R
2
S
t
a
n
d
a
r
d
e
r
r
o
r
o
f
t
h
e
e
s
t
i
m
a
t
e
1
0
.
5
4
6
a
0
.
2
9
8
0
.
2
9
5
0
.
4
5
2
2
4
3
a
P
r
e
d
i
c
t
o
r
s
:
(
c
o
n
st
a
n
t
)
,
e
x
t
r
i
n
si
c
m
o
t
i
v
a
t
i
o
n
,
i
n
t
r
i
n
si
c
m
o
t
i
v
a
t
i
o
n
T
ab
le
2
.
Mo
d
el
I
ANOVA
a
M
o
d
e
I
S
u
m
o
f
s
q
u
a
r
e
df
M
e
a
n
sq
u
a
r
e
F
S
i
g
.
1
R
e
g
r
e
ssi
o
n
R
e
si
d
u
a
l
To
t
a
l
3
4
4
9
.
4
0
0
8
1
1
9
.
5
7
8
1
1
5
6
8
.
9
7
7
2
3
9
7
3
9
9
1
7
2
4
.
7
0
0
2
0
.
4
5
2
8
5
.
3
2
8
0
.
0
0
0
b
a
D
e
p
e
n
d
e
n
t
v
a
r
i
a
b
l
e
:
P
S
A
b
P
r
e
d
i
c
t
o
r
s:
(
c
o
n
s
t
a
n
t
)
,
e
x
t
r
i
n
s
i
c
mo
t
i
v
a
t
i
o
n
a
n
d
i
n
t
r
i
n
s
i
c
mo
t
i
v
a
t
i
o
n
T
ab
le
3
.
Mo
d
el
I
co
e
f
f
icien
ts
a
M
o
d
e
l
U
n
st
a
n
d
a
r
d
i
z
e
d
c
o
e
f
f
i
c
i
e
n
t
s
S
t
a
n
d
a
r
d
i
z
e
d
c
o
e
f
f
i
c
i
e
n
t
s
t
S
i
g
.
B
S
t
a
n
d
a
r
d
e
r
r
o
r
B
e
t
a
(
β)
1
(
c
o
n
st
a
n
t
)
Ex
t
r
i
n
s
i
c
mo
t
i
v
a
t
i
o
n
I
n
t
r
i
n
si
c
m
o
t
i
v
a
t
i
o
n
-
4
.
6
3
6
1
.
4
9
0
3
.
3
9
3
2
.
2
7
5
0
.
4
5
7
0
.
5
1
1
0
.
1
9
3
0
.
3
9
3
-
3
.
6
3
7
3
.
2
6
2
6
.
6
4
0
0
.
0
0
0
0
.
0
0
1
0
.
0
0
0
a
D
e
p
e
n
d
e
n
t
v
a
r
i
a
b
l
e
:
P
S
A
Fig
u
r
e
1
in
d
icate
s
th
at
s
tu
d
e
n
ts
’
in
tr
in
s
ic
m
o
tiv
atio
n
(
β=0
.
3
9
3
)
in
m
ath
em
atics
h
as
a
g
r
e
ater
ef
f
ec
t
o
n
alg
e
b
r
aic
PS
A
th
a
n
ex
t
r
in
s
ic
m
o
tiv
atio
n
(
β=0
.
1
9
3
)
a
n
d
b
o
th
th
e
i
n
d
ep
e
n
d
en
t
v
a
r
iab
le
h
a
v
e
2
9
.
8
%
co
m
b
in
ed
v
a
r
iatio
n
e
x
p
lan
ati
o
n
o
n
alg
eb
r
aic
PS
A.
T
h
e
f
ir
s
t
an
aly
tical
m
o
d
el
is
p
r
esen
ted
in
Fig
u
r
e
1
.
T
h
e
lin
ea
r
r
eg
r
ess
io
n
m
o
d
el
is
co
n
s
tr
u
cted
as in
(
1
)
.
=
−
4
.
636
+
1
.
490
1
+
3
.
393
2
+
(
1
)
W
h
er
e,
Y=
alg
eb
r
aic
PS
A,
X
1
=e
x
tr
in
s
ic
m
o
tiv
atio
n
,
X
2
=in
tr
in
s
ic
m
o
tiv
atio
n
,
an
d
ε=
m
o
d
el
’
s
er
r
o
r
.
T
h
e
r
esu
lts
p
r
esen
ted
in
T
a
b
les
4
-
6
h
as
b
ee
n
u
s
ed
to
ex
a
m
in
e
th
e
c
o
r
r
elatio
n
o
f
alg
e
b
r
aic
PS
A
with
th
e
two
asp
ec
ts
o
f
s
elf
-
co
n
ce
p
t.
Fro
m
T
ab
le
6
,
it
is
s
ee
n
th
at
co
g
n
itiv
e
s
elf
-
co
n
ce
p
t
(
b
=5
.
5
4
8
,
t=1
2
.
2
2
5
,
p
<0
.
0
0
0
)
an
d
af
f
ec
tiv
e
s
elf
-
co
n
ce
p
t
(
b
=2
.
0
8
3
,
t=5
.
1
1
9
,
p
<
0
.
0
0
0
)
h
as
s
ig
n
if
ican
t
r
elatio
n
s
h
ip
with
alg
eb
r
aic
PS
A.
Hen
ce
,
th
e
h
y
p
o
th
eses
0
3
an
d
0
4
h
av
e
b
ee
n
r
ejec
ted
.
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
I
mp
r
o
vin
g
a
lg
e
b
r
a
ic
p
r
o
b
lem
-
s
o
lvin
g
a
ch
ieve
men
t o
f seco
n
d
a
r
y
leve
l stu
d
en
ts
b
y
u
s
in
g
…
(
I
md
a
d
A
li
)
3205
Fig
u
r
e
1
.
Gr
a
p
h
-
b
ased
v
is
u
aliza
tio
n
m
o
d
el
I
T
ab
le
4
.
Mo
d
el
I
I
s
u
m
m
ar
y
M
o
d
e
l
R
R
2
A
d
j
u
st
e
d
R
2
S
t
a
n
d
a
r
d
e
r
r
o
r
o
f
t
h
e
e
s
t
i
m
a
t
e
2
0
.
8
0
6
a
0
.
6
4
9
0
.
6
4
7
0
.
3
1
9
8
3
1
a
P
r
e
d
i
c
t
o
r
s
:
(
c
o
n
st
a
n
t
)
,
a
f
f
e
c
t
i
v
e
s
e
l
f
-
c
o
n
c
e
p
t
,
c
o
g
n
i
t
i
v
e
s
e
l
f
-
c
o
n
c
e
p
t
T
ab
le
5
.
Mo
d
el
I
I
ANOV
A
a
M
o
d
e
l
S
u
m
o
f
s
q
u
a
r
e
df
M
e
a
n
sq
u
a
r
e
F
S
i
g
.
2
R
e
g
r
e
ssi
o
n
R
e
si
d
u
a
l
To
t
a
l
7
5
0
7
.
9
9
1
4
0
6
0
.
9
8
7
1
1
5
6
8
.
9
7
7
2
3
9
7
3
9
9
3
7
5
3
.
9
9
7
1
0
.
2
2
9
3
6
6
.
9
8
9
0
.
0
0
0
b
a
D
e
p
e
n
d
e
n
t
v
a
r
i
a
b
l
e
:
P
S
A
b
P
r
e
d
i
c
t
o
r
s:
(
C
o
n
s
t
a
n
t
)
,
a
f
f
e
c
t
i
v
e
se
l
f
-
c
o
n
c
e
p
t
,
c
o
g
n
i
t
i
v
e
se
l
f
-
c
o
n
c
e
p
t
T
ab
le
6
.
Mo
d
el
I
I
c
o
ef
f
icien
ts
a
M
o
d
e
l
U
n
st
a
n
d
a
r
d
i
z
e
d
c
o
e
f
f
i
c
i
e
n
t
s
S
t
a
n
d
a
r
d
i
z
e
d
c
o
e
f
f
i
c
i
e
n
t
s
t
S
i
g
.
B
S
t
a
n
d
a
r
d
e
r
r
o
r
B
e
t
a
(
β)
2
(
c
o
n
st
a
n
t
)
C
o
g
n
i
t
i
v
e
s
e
l
f
-
c
o
n
c
e
p
t
A
f
f
e
c
t
i
v
e
se
l
f
-
c
o
n
c
e
p
t
-
1
1
.
1
1
2
4
.
5
4
8
2
.
0
8
3
0
.
8
7
8
0
.
3
7
2
0
.
4
0
7
0
.
5
9
4
0
.
2
4
9
-
1
2
.
6
5
8
1
2
.
2
2
5
5
.
1
1
7
0
.
0
0
0
0
.
0
0
0
0
.
0
0
0
a
D
e
p
e
n
d
e
n
t
v
a
r
i
a
b
l
e
:
P
S
A
Als
o
,
f
r
o
m
T
ab
les
4
an
d
5
,
it
is
f
o
u
n
d
th
at
alg
eb
r
aic
PS
A
ca
n
b
e
ex
p
lain
ed
b
y
b
o
t
h
th
e
p
r
ed
icto
r
v
ar
iab
le
co
g
n
itiv
e
an
d
af
f
ec
ti
v
e
s
elf
-
co
n
ce
p
t
as
F
(
2
,
3
9
7
)
=
3
6
6
.
9
8
9
,
p
<0
.
0
0
0
.
Acc
o
r
d
in
g
to
th
e
co
ef
f
icien
t
o
f
d
eter
m
in
atio
n
(
R
2
=0
.
2
9
8
)
,
th
e
p
r
ed
icto
r
co
g
n
itiv
e
an
d
af
f
ec
tiv
e
s
elf
-
co
n
ce
p
t
e
x
p
lain
s
2
9
.
8
%
v
a
r
iatio
n
o
f
alg
eb
r
aic
PS
A.
Als
o
,
th
e
m
u
ltip
le
co
r
r
elatio
n
c
o
ef
f
icien
t
(
R
=0
.
8
0
6
)
s
h
o
ws
th
at
s
tu
d
e
n
ts
’
co
g
n
itiv
e
an
d
af
f
ec
tiv
e
s
elf
-
co
n
ce
p
t
ar
e
s
ig
n
if
ican
tly
ass
o
ciate
d
to
alg
eb
r
ai
c
PS
A.
Fig
u
r
e
2
in
d
icate
d
th
at
s
tu
d
e
n
ts
’
co
g
n
itiv
e
s
elf
-
co
n
ce
p
t
(
β=5
.
5
9
4
)
in
m
ath
em
atics
h
as
a
g
r
ea
ter
ef
f
ec
t
o
n
alg
eb
r
aic
PS
A
th
an
af
f
ec
tiv
e
s
elf
-
co
n
ce
p
t
(
β=2
.
2
4
9
)
an
d
b
o
th
th
e
in
d
e
p
en
d
e
n
t
v
ar
iab
le
h
av
e
6
4
.
9
%
co
m
b
in
ed
v
ar
iatio
n
ex
p
lan
atio
n
o
n
alg
eb
r
aic
PS
A.
B
ased
o
n
th
e
an
aly
tical
m
o
d
el
in
Fig
u
r
e
2
,
th
e
r
eg
r
ess
io
n
eq
u
atio
n
is
p
r
esen
ted
in
th
e
(
2
)
.
=
−
11
.
112
+
5
.
548
1
+
2
.
083
2
+
(
2
)
W
h
er
e,
Y=
alg
eb
r
aic
PS
A,
X
1
=c
o
g
n
itiv
e
s
elf
-
co
n
ce
p
t,
X
2
=a
f
f
ec
tiv
e
s
elf
-
co
n
ce
p
t,
an
d
ε=
m
o
d
el’
s
er
r
o
r
.
Fig
u
r
e
2
.
Gr
a
p
h
-
b
ased
v
is
u
aliza
tio
n
m
o
d
el
I
I
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
.
14
,
No
.
4
,
Au
g
u
s
t
20
25
:
320
1
-
3
2
1
0
3206
T
ab
les
7
-
9
p
r
o
v
id
e
th
e
esti
m
ate
o
f
th
e
p
r
o
p
o
s
ed
r
e
g
r
e
s
s
io
n
m
o
d
el,
wh
ich
d
em
o
n
s
tr
ates
th
e
r
elatio
n
s
h
ip
b
etwe
en
alg
eb
r
ai
c
PS
A
an
d
s
tu
d
en
ts
’
c
o
g
n
itiv
e
s
elf
-
co
n
ce
p
t,
af
f
ec
tiv
e
s
elf
-
co
n
ce
p
t,
in
tr
in
s
ic
a
n
d
ex
tr
in
s
ic
m
o
tiv
atio
n
.
Fro
m
T
ab
le
9
,
it
is
s
ee
n
th
at
s
tu
d
en
t’
s
co
g
n
itiv
e
s
elf
-
co
n
ce
p
t
(
b
=4
.
0
2
3
,
t=1
0
.
2
2
0
,
P<0
.
0
0
0
)
,
af
f
ec
tiv
e
s
elf
-
c
o
n
c
ep
t
(
b
=
2
.
0
2
5
,
t=5
.
0
3
3
,
P<0
.
0
0
0
)
an
d
in
tr
in
s
ic
m
o
tiv
atio
n
(
b
=0
.
9
8
6
,
t=2
.
6
1
9
,
P<0
.
0
0
9
)
h
as
s
ig
n
if
ican
t
ef
f
ec
t
o
n
alg
eb
r
aic
PS
A.
T
h
u
s
,
th
e
h
y
p
o
th
eses
0
5
,
0
6
,
an
d
0
7
ar
e
r
ejec
ted
,
b
u
t
t
h
e
h
y
p
o
th
esis
0
8
is
ac
ce
p
ted
.
T
ab
le
7
.
Mo
d
el
I
I
I
s
u
m
m
ar
y
M
o
d
e
l
R
R
2
A
d
j
u
st
e
d
R
2
S
t
a
n
d
a
r
d
e
r
r
o
r
o
f
t
h
e
e
s
t
i
m
a
t
e
3
0
.
8
1
3
a
0
.
6
6
1
0
.
6
5
7
0
.
3
1
5
2
5
7
a
P
r
e
d
i
c
t
o
r
s
:
(
c
o
n
st
a
n
t
)
,
i
n
t
r
i
n
si
c
m
o
t
i
v
a
t
i
o
n
,
e
x
t
r
i
n
si
c
m
o
t
i
v
a
t
i
o
n
,
c
o
g
n
i
t
i
v
e
sel
f
-
c
o
n
c
e
p
t
a
n
d
a
f
f
e
c
t
i
v
e
se
l
f
-
c
o
n
c
e
p
t
T
ab
le
8
.
Mo
d
el
I
I
I
ANOV
A
a
M
o
d
e
l
S
u
m
o
f
s
q
u
a
r
e
df
M
e
a
n
sq
u
a
r
e
F
S
i
g
.
3
R
e
g
r
e
ssi
o
n
R
e
si
d
u
a
l
To
t
a
l
7
6
4
3
.
1
8
4
3
9
2
5
.
7
9
4
1
1
5
6
8
.
9
7
7
4
1
9
1
0
.
7
9
6
1
9
2
.
2
5
8
0
.
0
0
0
b
a
D
e
p
e
n
d
e
n
t
v
a
r
i
a
b
l
e
:
P
S
A
b
P
r
e
d
i
c
t
o
r
s:
(
c
o
n
s
t
a
n
t
)
,
i
n
t
r
i
n
s
i
c
mo
t
i
v
a
t
i
o
n
,
e
x
t
r
i
n
si
c
m
o
t
i
v
a
t
i
o
n
,
c
o
g
n
i
t
i
v
e
sel
f
-
c
o
n
c
e
p
t
a
n
d
a
f
f
e
c
t
i
v
e
se
l
f
-
c
o
n
c
e
p
t
T
ab
le
9
.
Mo
d
el
I
I
I
co
ef
f
icien
t
s
a
M
o
d
e
l
U
n
st
a
n
d
a
r
d
i
z
e
d
c
o
e
f
f
i
c
i
e
n
t
s
S
t
a
n
d
a
r
d
i
z
e
d
c
o
e
f
f
i
c
i
e
n
t
s
t
S
i
g
.
B
S
t
a
n
d
a
r
d
e
r
r
o
r
B
e
t
a
(
β)
3
(
c
o
n
st
a
n
t
)
C
o
g
n
i
t
i
v
e
s
e
l
f
-
c
o
n
c
e
p
t
A
f
f
e
c
t
i
v
e
se
l
f
-
c
o
n
c
e
p
t
I
n
t
r
i
n
si
c
m
o
t
i
v
a
t
i
o
n
Ex
t
r
i
n
s
i
c
mo
t
i
v
a
t
i
o
n
-
1
2
.
9
4
7
4
.
0
2
3
2
.
0
2
5
0
.
9
8
6
0
.
1
7
6
1
.
0
0
3
0
.
3
9
4
0
.
4
0
2
0
.
3
7
7
0
.
3
2
5
0
.
5
2
6
0
.
2
4
2
0
.
1
1
4
0
.
0
2
3
-
1
2
.
9
0
2
1
0
.
2
2
0
5
.
0
3
3
2
.
6
1
9
0
.
5
4
1
0
.
0
0
0
0
.
0
0
0
0
.
0
0
0
0
.
0
0
9
0
.
5
8
9
a
D
e
p
e
n
d
e
n
t
v
a
r
i
a
b
l
e
:
P
S
A
T
h
e
th
ir
d
m
o
d
el
is
s
h
o
wn
in
th
e
g
r
a
p
h
ical
v
is
u
aliza
tio
n
in
Fig
u
r
e
3
.
Fro
m
Fig
u
r
e
3
,
it
is
s
ee
n
th
at
co
g
n
itiv
e
s
elf
-
co
n
ce
p
t
h
as
th
e
h
ig
h
est
co
n
tr
ib
u
tio
n
to
alg
eb
r
aic
PS
A
with
β=0
.
5
2
6
.
T
h
e
s
ec
o
n
d
lar
g
est
p
r
ed
icto
r
is
af
f
ec
tiv
e
s
elf
-
co
n
ce
p
t
with
β=0
.
2
4
2
,
f
o
llo
wed
b
y
in
tr
in
s
ic
m
o
tiv
atio
n
with
β=0
.
1
1
4
a
n
d
ex
tr
in
s
ic
m
o
tiv
atio
n
β=0
.
0
2
3
.
T
h
e
v
ar
i
atio
n
o
f
alg
e
b
r
aic
PS
A
is
ex
p
lain
in
g
b
y
t
h
ese
f
o
u
r
v
ar
iab
les
is
6
6
.
1
%.
T
h
e
3
rd
an
aly
tical
m
o
d
el
is
p
r
esen
ted
in
Fig
u
r
e
3
.
T
h
e
r
eg
r
ess
io
n
m
o
d
el
is
co
n
s
tr
u
cted
as in
(
3
)
.
=
−
12
.
947
+
4
.
023
1
+
2
.
025
2
+
0
.
986
3
+
0
.
0176
4
+
(
3
)
W
h
er
e,
Y=
Alg
eb
r
aic
PS
A,
X
1
=c
o
g
n
itiv
e
s
elf
-
co
n
ce
p
t,
X
2
=
af
f
ec
tiv
e
s
elf
-
co
n
ce
p
t,
X
3
=in
t
r
in
s
ic
m
o
tiv
atio
n
,
X
4
=e
x
tr
in
s
ic
m
o
tiv
atio
n
,
an
d
ε
=m
o
d
el’
s
er
r
o
r
.
Fig
u
r
e
3
.
Gr
a
p
h
-
b
ased
v
is
u
aliza
tio
n
m
o
d
el
I
I
I
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
I
mp
r
o
vin
g
a
lg
e
b
r
a
ic
p
r
o
b
lem
-
s
o
lvin
g
a
ch
ieve
men
t o
f seco
n
d
a
r
y
leve
l stu
d
en
ts
b
y
u
s
in
g
…
(
I
md
a
d
A
li
)
3207
4.
DIS
CU
SS
I
O
N
I
n
o
r
d
er
to
c
o
n
d
u
ct
th
is
s
tu
d
y
,
th
r
ee
m
ath
em
atica
l
m
o
d
els
a
r
e
co
n
s
tr
u
cte
d
,
an
d
th
e
y
ar
e
a
n
ticip
ated
to
im
p
r
o
v
e
s
tu
d
en
ts
’
alg
e
b
r
ai
c
PS
A.
T
h
e
m
o
s
t
im
p
o
r
tan
t
f
in
d
in
g
is
th
at
th
e
g
r
ap
h
ical
r
e
p
r
esen
tatio
n
o
f
th
e
th
r
ee
co
m
b
in
e
d
m
o
d
els
s
h
o
wn
in
Fig
u
r
e
4
.
T
h
e
co
m
b
i
n
ed
m
o
d
els
in
d
icate
th
at
m
o
d
el
I
I
I
h
as
h
ig
h
e
r
p
er
f
o
r
m
an
ce
in
ex
p
lain
i
n
g
s
tu
d
en
ts
’
alg
eb
r
aic
PS
A.
T
h
e
in
d
ep
en
d
e
n
t
f
ac
to
r
s
in
m
o
d
el
I
I
I
ex
p
lain
6
6
.
1
%
o
f
th
e
v
ar
ian
ce
o
f
th
e
alg
eb
r
aic
PS
A,
as seen
in
Fig
u
r
e
4
.
Ho
wev
er
,
in
m
o
d
els I
an
d
I
I
,
th
e
in
d
ep
en
d
en
t
v
ar
iab
les
ex
p
lain
6
4
.
9
% a
n
d
2
9
.
8
%,
r
esp
ec
tiv
ely
,
o
f
th
e
v
a
r
iatio
n
o
f
a
lg
eb
r
aic
PS
A.
Fig
u
r
e
4
s
h
o
ws
th
at
2
9
.
8
%
o
f
th
e
v
ar
iatio
n
in
al
g
eb
r
aic
PS
A
ca
n
b
e
e
x
p
lain
e
d
b
y
th
e
tw
o
f
ac
to
r
s
,
in
tr
in
s
ic
an
d
ex
tr
in
s
ic
m
o
tiv
atio
n
.
I
t
in
d
icate
s
th
at
s
tu
d
e
n
ts
’
in
tr
in
s
ic
an
d
e
x
tr
in
s
ic
m
o
tiv
atio
n
h
av
e
a
s
ig
n
if
ican
t r
elatio
n
s
h
ip
with
alg
eb
r
aic
PS
A.
T
h
is
f
in
d
in
g
is
co
n
s
is
ten
t w
ith
th
e
f
ac
t th
at
s
tu
d
en
ts
’
in
tr
in
s
ic
an
d
ex
tr
in
s
ic
m
o
tiv
atio
n
s
ar
e
th
e
s
ig
n
if
ican
t f
ac
to
r
s
o
f
m
ath
e
m
atica
l a
ch
iev
em
en
t [
26
]
,
[
29
].
As
s
h
o
wn
in
Fig
u
r
e
4
,
t
h
e
two
s
elf
-
co
n
ce
p
t
asp
ec
ts
o
f
m
ath
e
m
atics
ex
p
lain
6
4
.
9
% o
f
t
h
e
v
ar
iatio
n
o
f
alg
eb
r
aic
PS
A.
T
h
u
s
,
f
o
s
ter
in
g
co
g
n
itiv
e
an
d
af
f
ec
tiv
e
s
elf
-
co
n
ce
p
t
in
m
ath
e
m
atics
ca
n
im
p
r
o
v
e
alg
e
b
r
aic
PS
A.
Pre
v
io
u
s
r
esear
ch
s
u
p
p
o
r
ts
th
at
co
n
s
id
er
in
g
s
tu
d
en
ts
’
s
elf
-
co
n
ce
p
t
in
th
e
m
ath
em
atica
l
r
ea
lm
h
elp
s
th
em
s
o
lv
e
alg
eb
r
aic
p
r
o
b
lem
s
m
o
r
e
s
u
cc
ess
f
u
lly
an
d
m
o
tiv
ated
to
lear
n
[
5
]
,
[
11
]
.
Als
o
,
J
in
g
et
a
l.
[
24
]
p
o
in
t
o
u
t
th
at
th
e
s
tu
d
en
ts
’
m
o
tiv
atio
n
f
o
r
lear
n
i
n
g
alg
eb
r
a
an
d
alg
eb
r
aic
ac
h
iev
em
e
n
t
is
m
o
s
t
s
ig
n
if
ican
t.
Hen
ce
,
it
s
u
g
g
ests
th
at
s
tu
d
en
ts
’
p
o
s
itiv
e
s
elf
-
co
n
ce
p
t a
b
o
u
t th
eir
ab
ili
ty
to
s
o
lv
e
a
p
r
o
b
lem
m
ay
lead
th
em
to
en
g
ag
e
in
m
ath
em
atica
l
task
s
,
wh
ich
m
ay
in
cr
ea
s
e
t
h
eir
co
m
p
eten
cy
an
d
ca
p
ac
ity
to
p
er
f
o
r
m
we
ll
in
m
ath
em
atica
l
p
r
o
b
lem
-
s
o
lv
in
g
[
33
]
.
T
h
e
y
ar
e
also
m
o
tiv
ated
to
d
o
well
an
d
p
u
t g
r
ea
ter
ef
f
o
r
t in
m
ath
em
atica
l ta
s
k
s
[
19
].
Fig
u
r
e
4
.
Gr
a
p
h
-
b
ased
v
is
u
aliza
tio
n
o
f
th
r
ee
in
teg
r
ated
m
o
d
els
I
n
Fig
u
r
e
4
,
th
e
f
o
u
r
p
r
ed
icto
r
s
(
co
g
n
itiv
e
an
d
af
f
ec
tiv
e
s
elf
-
co
n
ce
p
t a
s
well
as
in
tr
in
s
ic
an
d
ex
tr
in
s
ic
m
o
tiv
atio
n
)
ex
p
lain
6
6
.
1
%
o
f
th
e
v
ar
iatio
n
in
al
g
eb
r
aic
PS
A.
T
h
e
f
o
u
r
in
d
e
p
en
d
e
n
t
v
a
r
iab
les
all
to
g
eth
e
r
h
av
e
a
s
ig
n
if
ican
t
r
elatio
n
s
h
i
p
with
alg
eb
r
aic
PS
A.
T
h
e
f
in
d
in
g
s
in
m
o
d
el
I
I
I
,
ev
id
e
n
c
e
th
at
th
e
co
g
n
itiv
e
s
elf
-
co
n
ce
p
t
is
th
e
b
est
p
r
ed
ic
to
r
o
f
alg
e
b
r
aic
PS
A.
Als
o
,
af
f
ec
tiv
e
s
elf
-
co
n
ce
p
t
h
as
a
h
ig
h
er
co
n
tr
i
b
u
tio
n
t
o
alg
eb
r
aic
PS
A
th
an
in
tr
in
s
ic
an
d
ex
tr
in
s
ic
m
o
tiv
atio
n
.
On
th
e
o
th
er
h
a
n
d
,
e
x
tr
in
s
ic
m
o
tiv
atio
n
h
as
f
o
u
n
d
a
wea
k
ass
o
ciatio
n
with
alg
e
b
r
a
ic
PS
A.
T
h
is
f
in
d
in
g
is
co
n
s
is
ten
t
with
Ham
m
o
u
d
i
a
n
d
Gr
ir
a
[
9
]
th
at
s
tu
d
e
n
ts
’
co
g
n
itiv
e
an
d
a
f
f
ec
tiv
e
s
elf
-
c
o
n
ce
p
t
h
av
e
a
s
tr
o
n
g
er
ef
f
ec
t
o
n
th
eir
m
ath
em
atica
l
ac
h
iev
e
m
en
t
th
an
ex
tr
in
s
ic
m
o
tiv
atio
n
.
Fu
r
th
er
m
o
r
e,
Ham
m
o
u
d
i
an
d
Gr
ir
a
[
10
]
em
p
h
asized
th
e
s
ig
n
if
ican
ce
o
f
b
o
th
ex
tr
in
s
ic
a
n
d
in
tr
in
s
ic
m
o
tiv
atio
n
f
o
r
ac
h
iev
em
en
t
in
m
ath
em
atics.
B
u
t,
ac
co
r
d
in
g
to
o
u
r
f
in
d
in
g
s
,
o
n
ly
i
n
tr
in
s
ic
m
o
tiv
atio
n
h
as
a
s
ig
n
if
ican
t
co
n
tr
ib
u
tio
n
to
alg
eb
r
aic
PS
A,
wh
ich
s
u
p
p
o
r
ts
s
ev
er
al
s
tu
d
ies
[
26
]
–
[
28
]
an
d
co
n
tr
a
d
icts
th
e
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
.
14
,
No
.
4
,
Au
g
u
s
t
20
25
:
320
1
-
3
2
1
0
3208
f
in
d
in
g
s
o
f
s
ev
er
al
o
th
er
s
[
9
]
,
[
30
].
T
h
e
r
ef
o
r
e
,
a
p
o
s
itiv
e
co
g
n
itiv
e
an
d
af
f
ec
tiv
e
s
elf
-
co
n
c
ep
t
co
m
b
in
ed
with
in
tr
in
s
ic
m
o
tiv
atio
n
ca
n
g
r
ea
tl
y
af
f
ec
t
s
tu
d
en
ts
’
c
o
g
n
itiv
e
a
n
d
af
f
ec
tiv
e
b
eh
av
io
r
to
im
p
r
o
v
e
an
d
s
u
cc
ee
d
in
alg
eb
r
aic
p
r
o
b
lem
-
s
o
lv
i
n
g
.
5.
CO
NCLU
SI
O
N
T
h
e
s
tu
d
y
is
d
esig
n
e
d
to
im
p
r
o
v
e
alg
e
b
r
aic
PS
A
am
o
n
g
s
ec
o
n
d
ar
y
s
ch
o
o
l
s
tu
d
e
n
ts
.
T
h
r
ee
m
o
d
els
ar
e
u
s
ed
a
n
d
in
te
g
r
ated
t
o
an
a
ly
ze
th
e
co
n
tr
ib
u
tio
n
o
f
ce
r
tai
n
f
ac
to
r
s
th
at
h
elp
s
in
im
p
r
o
v
in
g
alg
eb
r
aic
PS
A.
T
h
e
r
esu
lts
o
f
m
o
d
el
I
s
h
o
w
th
at
2
9
.
8
%
o
f
th
e
v
ar
ian
ce
o
f
alg
e
b
r
aic
PS
A
ca
n
b
e
e
x
p
lain
ed
b
y
th
e
lin
ea
r
co
m
b
in
atio
n
o
f
s
tu
d
en
ts
’
in
tr
i
n
s
ic
an
d
e
x
tr
in
s
ic
m
o
tiv
ati
o
n
.
T
h
e
r
esu
lts
o
f
m
o
d
el
I
I
d
e
m
o
n
s
tr
ated
th
at
6
4
.
9
%
v
ar
ian
ce
o
f
alg
e
b
r
aic
PS
A
ca
n
b
e
ac
co
u
n
ted
b
y
th
e
lin
ea
r
co
m
b
in
atio
n
o
f
s
tu
d
en
ts
’
co
g
n
itiv
e
an
d
af
f
ec
tiv
e
s
elf
-
co
n
ce
p
t.
Als
o
,
in
m
o
d
el
I
I
I
,
it
is
s
ee
n
th
at
th
e
h
ig
h
est
6
6
.
1
%
v
ar
ian
ce
o
f
alg
e
b
r
aic
PS
A
ca
n
b
e
ex
p
lain
ed
b
y
th
e
lin
ea
r
co
m
b
i
n
atio
n
o
f
t
h
e
f
o
u
r
in
d
e
p
en
d
e
n
t v
ar
iab
les.
B
ased
o
n
th
e
f
in
d
in
g
s
o
f
th
e
m
o
d
el
I
I
I
,
it
s
h
o
w
s
th
at
s
tu
d
en
ts
’
co
g
n
itiv
e
an
d
af
f
ec
tiv
e
s
elf
-
co
n
ce
p
t
h
av
e
h
ig
h
er
co
n
tr
ib
u
tio
n
o
n
al
g
eb
r
aic
PS
A
f
o
llo
wed
b
y
in
tr
i
n
s
ic
an
d
ex
tr
in
s
ic
m
o
tiv
atio
n
.
T
h
er
ef
o
r
e,
in
lig
h
t
o
f
th
e
r
esu
lts
,
th
e
f
o
llo
win
g
s
u
g
g
esti
o
n
s
ar
e
p
r
o
v
id
ed
t
o
ed
u
ca
to
r
s
,
p
o
licy
m
a
k
er
s
,
an
d
s
tak
eh
o
ld
er
s
to
im
p
r
o
v
e
s
tu
d
e
n
ts
’
alg
eb
r
aic
p
r
o
b
lem
-
s
o
lv
in
g
at
th
e
s
ec
o
n
d
ar
y
lev
el:
i)
en
h
an
cin
g
s
tu
d
en
ts
’
co
g
n
itiv
e
s
elf
-
co
n
ce
p
t
in
m
ath
em
atics
b
y
ass
ess
in
g
th
eir
b
elief
in
m
at
h
em
atics,
willin
g
n
ess
to
en
g
a
g
e
in
m
ath
em
atica
l
task
s
,
co
n
s
is
ten
t
p
r
ac
tice
to
b
u
ild
s
elf
-
co
n
f
id
e
n
ce
an
d
f
o
cu
s
o
n
alg
eb
r
aic
p
r
o
b
lem
-
s
o
lv
in
g
ac
tiv
ities
;
ii)
b
o
o
s
tin
g
s
tu
d
en
ts
’
af
f
ec
tiv
e
s
elf
-
co
n
ce
p
t
in
m
at
h
em
atics
b
y
b
u
ild
i
n
g
p
o
s
itiv
e
f
ee
lin
g
s
,
attitu
d
es
an
d
b
elief
s
to
war
d
s
m
ath
em
atica
l
p
r
o
b
le
m
s
;
iii)
en
co
u
r
a
g
in
g
an
d
f
o
s
te
r
in
g
in
tr
i
n
s
ic
m
o
tiv
atio
n
is
ess
en
tial
f
o
r
c
u
ltiv
atin
g
a
g
en
u
i
n
e
p
ass
io
n
f
o
r
m
ath
em
atics
lear
n
in
g
a
n
d
f
o
s
ter
in
g
lo
n
g
-
ter
m
s
u
cc
ess
in
p
r
o
b
lem
s
o
lv
in
g
;
an
d
iv
)
ex
t
r
in
s
ic
m
o
tiv
atio
n
ca
n
s
ig
n
if
ican
tly
im
p
ac
t
m
ath
e
m
atics
p
r
o
b
lem
-
s
o
lv
in
g
ab
ili
ties
b
y
in
f
lu
en
cin
g
s
tu
d
en
ts
’
en
g
a
g
em
en
t,
ef
f
o
r
t,
an
d
p
er
s
is
ten
ce
.
W
h
en
s
tu
d
e
n
ts
ar
e
ex
tr
in
s
ically
m
o
tiv
ated
,
th
ey
m
ay
b
e
m
o
r
e
lik
ely
to
tack
le
c
h
allen
g
in
g
m
ath
em
atica
l
p
r
o
b
lem
s
;
s
tay
f
o
cu
s
ed
o
n
task
s
f
o
r
lo
n
g
er
p
er
i
o
d
s
an
d
s
ee
k
s
o
u
t
ad
d
itio
n
al
r
eso
u
r
ce
s
o
r
s
u
p
p
o
r
t to
s
o
lv
e
p
r
o
b
lem
s
.
ACK
NO
WL
E
DG
M
E
N
T
S
W
e
ar
e
g
r
atef
u
l
to
all
th
e
h
ea
d
s
o
f
th
e
in
s
titu
tio
n
s
an
d
t
ea
ch
er
s
f
o
r
th
eir
v
alu
ab
le
s
u
p
p
o
r
t
an
d
p
er
m
is
s
io
n
.
Fin
ally
,
th
an
k
s
to
all
th
e
s
tu
d
en
ts
wh
o
h
a
v
e
willin
g
ly
p
a
r
ticip
ated
in
th
is
s
tu
d
y
.
F
UNDING
I
NF
O
R
M
A
T
I
O
N
Au
th
o
r
s
s
tate
th
er
e
is
n
o
f
u
n
d
i
n
g
in
v
o
lv
ed
.
AUTHO
R
CO
NT
RI
B
UT
I
O
NS ST
A
T
E
M
E
N
T
T
h
is
jo
u
r
n
al
u
s
es
th
e
C
o
n
t
r
ib
u
to
r
R
o
les
T
a
x
o
n
o
m
y
(
C
R
ed
iT)
to
r
ec
o
g
n
ize
in
d
iv
i
d
u
al
au
th
o
r
co
n
tr
ib
u
tio
n
s
,
r
ed
u
ce
au
th
o
r
s
h
ip
d
is
p
u
tes,
an
d
f
ac
ilit
ate
co
llab
o
r
atio
n
.
Na
m
e
o
f
Aut
ho
r
C
M
So
Va
Fo
I
R
D
O
E
Vi
Su
P
Fu
I
m
d
ad
Ali
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
Sam
ir
an
Das
✓
✓
✓
✓
✓
✓
✓
C
:
C
o
n
c
e
p
t
u
a
l
i
z
a
t
i
o
n
M
:
M
e
t
h
o
d
o
l
o
g
y
So
:
So
f
t
w
a
r
e
Va
:
Va
l
i
d
a
t
i
o
n
Fo
:
Fo
r
mal
a
n
a
l
y
s
i
s
I
:
I
n
v
e
s
t
i
g
a
t
i
o
n
R
:
R
e
so
u
r
c
e
s
D
:
D
a
t
a
C
u
r
a
t
i
o
n
O
:
W
r
i
t
i
n
g
-
O
r
i
g
i
n
a
l
D
r
a
f
t
E
:
W
r
i
t
i
n
g
-
R
e
v
i
e
w
&
E
d
i
t
i
n
g
Vi
:
Vi
su
a
l
i
z
a
t
i
o
n
Su
:
Su
p
e
r
v
i
s
i
o
n
P
:
P
r
o
j
e
c
t
a
d
mi
n
i
st
r
a
t
i
o
n
Fu
:
Fu
n
d
i
n
g
a
c
q
u
i
si
t
i
o
n
CO
NF
L
I
C
T
O
F
I
N
T
E
R
E
S
T
ST
A
T
E
M
E
NT
Au
th
o
r
s
s
tate
n
o
co
n
f
lict o
f
in
t
er
est.
DATA AV
AI
L
AB
I
L
I
T
Y
T
h
e
d
ata
th
at
s
u
p
p
o
r
t
th
e
f
i
n
d
in
g
s
o
f
th
is
s
tu
d
y
ar
e
av
aila
b
le
o
n
r
eq
u
est
f
r
o
m
th
e
co
r
r
esp
o
n
d
in
g
au
th
o
r
[
I
A]
.
T
h
e
d
ata,
w
h
ich
c
o
n
tain
in
f
o
r
m
atio
n
th
at
co
u
ld
co
m
p
r
o
m
is
e
th
e
p
r
iv
ac
y
o
f
r
es
ea
r
ch
p
ar
ticip
an
ts
,
ar
e
n
o
t p
u
b
licly
av
aila
b
le
d
u
e
to
ce
r
tain
r
estrictio
n
s
.
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
I
mp
r
o
vin
g
a
lg
e
b
r
a
ic
p
r
o
b
lem
-
s
o
lvin
g
a
ch
ieve
men
t o
f seco
n
d
a
r
y
leve
l stu
d
en
ts
b
y
u
s
in
g
…
(
I
md
a
d
A
li
)
3209
RE
F
E
R
E
NC
E
S
[
1
]
X
.
W
a
n
g
,
“
W
h
y
st
u
d
e
n
t
s
c
h
o
o
s
e
S
TEM
ma
j
o
r
s:
M
o
t
i
v
a
t
i
o
n
,
h
i
g
h
sc
h
o
o
l
l
e
a
r
n
i
n
g
,
a
n
d
p
o
s
t
se
c
o
n
d
a
r
y
c
o
n
t
e
x
t
o
f
s
u
p
p
o
r
t
,
”
Am
e
ri
c
a
n
E
d
u
c
a
t
i
o
n
a
l
Re
s
e
a
r
c
h
J
o
u
r
n
a
l
,
v
o
l
.
5
0
,
n
o
.
5
,
p
p
.
1
0
8
1
–
1
1
2
1
,
O
c
t
.
2
0
1
3
,
d
o
i
:
1
0
.
3
1
0
2
/
0
0
0
2
8
3
1
2
1
3
4
8
8
6
2
2
.
[
2
]
D
.
K
e
l
l
y
,
H
.
X
i
e
,
C
.
W
.
N
o
r
d
,
F
.
Je
n
k
i
n
s
,
J.
Y
.
C
h
a
n
,
a
n
d
D
.
K
a
st
b
e
r
g
,
Pe
r
f
o
rm
a
n
c
e
o
f
U
S
1
5
-
Y
e
a
r
-
O
l
d
S
t
u
d
e
n
t
s i
n
M
a
t
h
e
m
a
t
i
c
s,
S
c
i
e
n
c
e
,
a
n
d
Re
a
d
i
n
g
L
i
t
e
r
a
c
y
i
n
a
n
I
n
t
e
rn
a
t
i
o
n
a
l
C
o
n
t
e
x
t
.
F
i
rst
L
o
o
k
a
t
PI
S
A
2
0
1
2
.
N
C
E
S
2
0
1
4
-
0
2
4
.
W
a
sh
i
n
g
t
o
n
,
D
C
:
N
a
t
i
o
n
a
l
C
e
n
t
e
r
f
o
r
E
d
u
c
a
t
i
o
n
S
t
a
t
i
s
t
i
c
s
,
2
0
1
3
.
[
3
]
N
a
t
i
o
n
a
l
C
e
n
t
r
e
f
o
r
E
d
u
c
a
t
i
o
n
a
l
R
e
sea
r
c
h
a
n
d
Tr
a
i
n
i
n
g
,
“
P
o
s
i
t
i
o
n
p
a
p
e
r
o
f
t
h
e
N
a
t
i
o
n
a
l
F
o
c
u
s
G
r
o
u
p
o
n
t
h
e
T
e
a
c
h
i
n
g
o
f
M
a
t
h
e
ma
t
i
c
s,”
2
0
0
6
.
[
O
n
l
i
n
e
]
.
A
v
a
i
l
a
b
l
e
:
h
t
t
p
:
/
/
w
w
w
.
n
c
e
r
t
.
n
i
c
.
i
n
/
h
t
ml
/
p
d
f
/
sch
o
o
l
c
u
r
r
i
c
u
l
u
m
/
p
o
s
i
t
i
o
n
_
p
a
p
e
r
s/
m
a
t
h
.
p
d
f
[
4
]
A
.
El
G
u
e
n
y
a
r
i
,
M
.
C
h
e
r
g
u
i
,
a
n
d
B
.
El
W
a
h
b
i
,
“
A
s
t
u
d
y
o
n
t
h
e
e
f
f
e
c
t
i
v
e
n
e
ss
o
f
so
m
e
c
o
g
n
i
t
i
v
e
a
c
t
i
v
i
t
i
e
s
i
n
t
e
a
c
h
i
n
g
i
n
t
e
g
r
a
l
s
i
n
sec
o
n
d
a
r
y
s
c
h
o
o
l
,
”
Ma
t
h
e
m
a
t
i
c
s Te
a
c
h
i
n
g
-
Re
se
a
r
c
h
J
o
u
r
n
a
l
,
v
o
l
.
1
4
,
n
o
.
5
,
p
p
.
6
6
–
8
3
,
2
0
2
2
.
[
5
]
A
.
H
.
A
b
d
u
l
l
a
h
e
t
a
l
.
,
“
R
e
l
a
t
i
o
n
sh
i
p
b
e
t
w
e
e
n
S
e
l
f
-
C
o
n
c
e
p
t
,
Em
o
t
i
o
n
a
l
I
n
t
e
l
l
i
g
e
n
c
e
a
n
d
P
r
o
b
l
e
m
-
S
o
l
v
i
n
g
S
k
i
l
l
s
o
n
S
e
c
o
n
d
a
r
y
S
c
h
o
o
l
S
t
u
d
e
n
t
s’
A
t
t
i
t
u
d
e
t
o
w
a
r
d
s
S
o
l
v
i
n
g
A
l
g
e
b
r
a
i
c
P
r
o
b
l
e
ms,
”
S
u
s
t
a
i
n
a
b
i
l
i
t
y
,
v
o
l
.
1
4
,
n
o
.
2
1
,
p
.
1
4
4
0
2
,
N
o
v
.
2
0
2
2
,
d
o
i
:
1
0
.
3
3
9
0
/
s
u
1
4
2
1
1
4
4
0
2
.
[
6
]
L.
R
a
d
f
o
r
d
,
C
.
B
a
r
d
i
n
i
,
a
n
d
C
.
S
a
b
e
n
a
,
“
P
e
r
c
e
i
v
i
n
g
t
h
e
g
e
n
e
r
a
l
:
T
h
e
m
u
l
t
i
se
mi
o
t
i
c
d
i
m
e
n
s
i
o
n
o
f
st
u
d
e
n
t
s’
a
l
g
e
b
r
a
i
c
a
c
t
i
v
i
t
y
,
”
J
o
u
rn
a
l
f
o
r
Re
s
e
a
r
c
h
i
n
Ma
t
h
e
m
a
t
i
c
s
Ed
u
c
a
t
i
o
n
,
v
o
l
.
3
8
,
n
o
.
5
,
p
p
.
5
0
7
–
5
3
0
,
2
0
0
7
,
d
o
i
:
1
0
.
2
3
0
7
/
3
0
0
3
4
9
6
3
.
[
7
]
I
.
A
l
i
a
n
d
S
.
D
a
s,
“
D
i
r
e
c
t
a
n
d
I
n
d
i
r
e
c
t
Ef
f
e
c
t
o
f
S
e
l
f
-
Ef
f
i
c
a
c
y
,
A
n
x
i
e
t
y
a
n
d
I
n
t
e
r
e
st
o
n
A
l
g
e
b
r
a
i
c
P
r
o
b
l
e
m
-
S
o
l
v
i
n
g
A
c
h
i
e
v
e
me
n
t
,
”
Ma
t
h
e
m
a
t
i
c
s Te
a
c
h
i
n
g
-
Re
s
e
a
r
c
h
J
o
u
r
n
a
l
,
v
o
l
.
1
6
,
n
o
.
2
,
p
p
.
1
0
2
–
1
1
8
,
2
0
2
4
.
[
8
]
R.
-
N
.
A
w
a
n
,
N
.
G
h
a
z
a
l
a
,
a
n
d
N
.
A
n
j
u
m,
“
A
S
t
u
d
y
o
f
R
e
l
a
t
i
o
n
sh
i
p
b
e
t
w
e
e
n
A
c
h
i
e
v
e
me
n
t
M
o
t
i
v
a
t
i
o
n
,
A
c
a
d
e
mi
c
S
e
l
f
C
o
n
c
e
p
t
a
n
d
A
c
h
i
e
v
e
me
n
t
i
n
E
n
g
l
i
s
h
a
n
d
M
a
t
h
e
mat
i
c
s
a
t
S
e
c
o
n
d
a
r
y
Le
v
e
l
,
”
I
n
t
e
r
n
a
t
i
o
n
a
l
E
d
u
c
a
t
i
o
n
S
t
u
d
i
e
s
,
v
o
l
.
4
,
n
o
.
3
,
p
p
.
7
2
–
7
8
,
Ju
l
.
2
0
1
1
,
d
o
i
:
1
0
.
5
5
3
9
/
i
e
s.
v
4
n
3
p
7
2
.
[
9
]
M
.
M
.
H
a
mm
o
u
d
i
a
n
d
S
.
G
r
i
r
a
,
“
O
p
t
i
mi
z
a
t
i
o
n
o
f
t
h
e
m
o
t
i
v
a
t
i
o
n
f
o
r
su
c
c
e
ss
i
n
m
a
t
h
e
mat
i
c
s
a
t
h
i
g
h
e
r
e
d
u
c
a
t
i
o
n
:
Ev
i
d
e
n
c
e
f
r
o
m
n
o
v
e
l
g
r
a
p
h
-
b
a
se
d
v
i
su
a
l
i
z
a
t
i
o
n
o
f
t
h
r
e
e
m
o
d
e
l
s,”
E
u
r
a
si
a
J
o
u
r
n
a
l
o
f
M
a
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
E
d
u
c
a
t
i
o
n
,
v
o
l
.
1
9
,
n
o
.
1
1
,
p
.
e
m
2
3
5
6
,
N
o
v
.
2
0
2
3
,
d
o
i
:
1
0
.
2
9
3
3
3
/
e
j
ms
t
e
/
1
3
8
0
8
.
[
10
]
M
.
M
.
H
a
mm
o
u
d
i
a
n
d
S
.
G
r
i
r
a
,
“
I
mp
r
o
v
i
n
g
s
t
u
d
e
n
t
s’
m
o
t
i
v
a
t
i
o
n
i
n
c
a
l
c
u
l
u
s
c
o
u
r
ses
a
t
i
n
s
t
i
t
u
t
i
o
n
s
o
f
h
i
g
h
e
r
e
d
u
c
a
t
i
o
n
:
E
v
i
d
e
n
c
e
f
r
o
m
g
r
a
p
h
-
b
a
s
e
d
v
i
su
a
l
i
z
a
t
i
o
n
o
f
t
w
o
mo
d
e
l
s,
”
E
u
ra
s
i
a
J
o
u
r
n
a
l
o
f
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
Ed
u
c
a
t
i
o
n
,
v
o
l
.
1
9
,
n
o
.
1
,
p
.
e
m
2
2
0
9
,
Ja
n
.
2
0
2
3
,
d
o
i
:
1
0
.
2
9
3
3
3
/
e
j
ms
t
e
/
1
2
7
7
1
.
[
11
]
L.
S
u
g
i
a
r
t
i
a
n
d
H
.
R
e
t
n
a
w
a
t
i
,
“
A
n
a
l
y
si
s
o
f
st
u
d
e
n
t
d
i
f
f
i
c
u
l
t
i
e
s
o
n
a
l
g
e
b
r
a
p
r
o
b
l
e
m
s
o
l
v
i
n
g
i
n
j
u
n
i
o
r
h
i
g
h
s
c
h
o
o
l
,
”
J
o
u
r
n
a
l
o
f
Ph
y
s
i
c
s:
C
o
n
f
e
r
e
n
c
e
S
e
ri
e
s
,
v
o
l
.
1
3
2
0
,
n
o
.
1
,
p
.
0
1
2
1
0
3
,
O
c
t
.
2
0
1
9
,
d
o
i
:
1
0
.
1
0
8
8
/
1
7
4
2
-
6
5
9
6
/
1
3
2
0
/
1
/
0
1
2
1
0
3
.
[
12
]
H
.
W
.
M
a
r
s
h
a
n
d
R
.
G
.
C
r
a
v
e
n
,
“
A
c
a
d
e
mi
c
se
l
f
-
c
o
n
c
e
p
t
:
B
e
y
o
n
d
t
h
e
d
u
st
b
o
w
l
,
”
i
n
H
a
n
d
b
o
o
k
o
f
C
l
a
ssr
o
o
m
Assessm
e
n
t
L
e
a
r
n
i
n
g
,
A
c
h
i
e
v
e
m
e
n
t
,
a
n
d
A
d
j
u
st
m
e
n
t
,
G
.
D
.
P
h
y
e
,
E
d
.
O
r
l
a
n
d
o
,
F
L
:
A
c
a
d
e
m
i
c
P
r
e
ss,
1
9
9
7
,
p
p
.
1
3
1
–
1
9
8
.
[
13
]
A
.
Te
l
l
a
,
“
T
h
e
I
mp
a
c
t
o
f
M
o
t
i
v
a
t
i
o
n
o
n
S
t
u
d
e
n
t
’
s
A
c
a
d
e
mi
c
A
c
h
i
e
v
e
m
e
n
t
a
n
d
L
e
a
r
n
i
n
g
O
u
t
c
o
mes
i
n
M
a
t
h
e
m
a
t
i
c
s
a
m
o
n
g
S
e
c
o
n
d
a
r
y
S
c
h
o
o
l
S
t
u
d
e
n
t
s
i
n
N
i
g
e
r
i
a
,
”
EU
RA
S
I
A
J
o
u
rn
a
l
o
f
M
a
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
Ed
u
c
a
t
i
o
n
,
v
o
l
.
3
,
n
o
.
2
,
p
p
.
1
4
9
–
1
5
6
,
J
u
n
.
2
0
0
7
,
d
o
i
:
1
0
.
1
2
9
7
3
/
e
j
mste
/
7
5
3
9
0
.
[
14
]
M
.
M
.
H
a
m
mo
u
d
i
,
“
P
r
e
d
i
c
t
i
v
e
f
a
c
t
o
r
s
o
f
s
t
u
d
e
n
t
s’
m
o
t
i
v
a
t
i
o
n
t
o
s
u
c
c
e
e
d
i
n
i
n
t
r
o
d
u
c
t
o
r
y
ma
t
h
e
ma
t
i
c
s
c
o
u
r
s
e
s:
e
v
i
d
e
n
c
e
f
r
o
m
h
i
g
h
e
r
e
d
u
c
a
t
i
o
n
i
n
t
h
e
U
A
E,
”
I
n
t
e
rn
a
t
i
o
n
a
l
J
o
u
rn
a
l
o
f
M
a
t
h
e
m
a
t
i
c
a
l
E
d
u
c
a
t
i
o
n
i
n
S
c
i
e
n
c
e
a
n
d
T
e
c
h
n
o
l
o
g
y
,
v
o
l
.
5
0
,
n
o
.
5
,
p
p
.
6
4
7
–
6
6
4
,
J
u
l
.
2
0
1
9
,
d
o
i
:
1
0
.
1
0
8
0
/
0
0
2
0
7
3
9
X
.
2
0
1
8
.
1
5
2
9
3
3
9
.
[
15
]
S
.
Y
ı
l
d
ı
r
ı
m
,
“
Te
a
c
h
e
r
S
u
p
p
o
r
t
,
M
o
t
i
v
a
t
i
o
n
,
Le
a
r
n
i
n
g
S
t
r
a
t
e
g
y
U
se,
a
n
d
A
c
h
i
e
v
e
me
n
t
:
A
M
u
l
t
i
l
e
v
e
l
M
e
d
i
a
t
i
o
n
M
o
d
e
l
,
”
T
h
e
J
o
u
rn
a
l
o
f
Ex
p
e
r
i
m
e
n
t
a
l
Ed
u
c
a
t
i
o
n
,
v
o
l
.
8
0
,
n
o
.
2
,
p
p
.
1
5
0
–
1
7
2
,
J
a
n
.
2
0
1
2
,
d
o
i
:
1
0
.
1
0
8
0
/
0
0
2
2
0
9
7
3
.
2
0
1
1
.
5
9
6
8
5
5
.
[
16
]
L.
S
.
L
u
ms
d
e
n
,
“
S
t
u
d
e
n
t
m
o
t
i
v
a
t
i
o
n
t
o
l
e
a
r
n
.
ER
I
C
D
i
g
e
st
,
”
ERIC
D
i
g
e
s
t
,
n
o
.
9
2
,
p
p
.
1
–
7
,
1
9
9
4
,
[
O
n
l
i
n
e
]
.
A
v
a
i
l
a
b
l
e
:
h
t
t
p
:
/
/
w
w
w
.
k
i
d
s
o
u
r
c
e
.
c
o
m/
k
i
d
s
o
u
r
c
e
/
c
o
n
t
e
n
t
2
/
S
t
u
d
e
n
t
_
M
o
t
i
v
a
t
a
t
i
o
n
.
h
t
ml
#
c
r
e
d
i
t
s
[
17
]
A
.
J.
El
l
i
o
t
a
n
d
C
.
S
.
D
w
e
c
k
,
H
a
n
d
b
o
o
k
o
f
C
o
m
p
e
t
e
n
c
e
a
n
d
M
o
t
i
v
a
t
i
o
n
.
N
e
w
Y
o
r
k
:
G
u
i
l
f
o
r
d
P
r
e
ss,
2
0
0
5
.
[
18
]
M
.
B
a
a
r
s,
L
.
W
i
j
n
i
a
,
a
n
d
F
.
P
a
a
s
,
“
Th
e
A
sso
c
i
a
t
i
o
n
b
e
t
w
e
e
n
M
o
t
i
v
a
t
i
o
n
,
A
f
f
e
c
t
,
a
n
d
S
e
l
f
-
r
e
g
u
l
a
t
e
d
Le
a
r
n
i
n
g
W
h
e
n
S
o
l
v
i
n
g
P
r
o
b
l
e
ms,
”
F
ro
n
t
i
e
rs
i
n
Ps
y
c
h
o
l
o
g
y
,
v
o
l
.
8
,
p
.
1
3
4
6
,
A
u
g
.
2
0
1
7
,
d
o
i
:
1
0
.
3
3
8
9
/
f
p
s
y
g
.
2
0
1
7
.
0
1
3
4
6
.
[
19
]
J.
E.
B
r
o
p
h
y
,
Mo
t
i
v
a
t
i
n
g
s
t
u
d
e
n
t
s t
o
l
e
a
r
n
,
3
r
d
e
d
.
L
o
n
d
o
n
:
R
o
u
t
l
e
d
g
e
,
2
0
1
0
.
[
20
]
M
.
Li
s
h
c
h
y
n
s
k
a
,
C
.
P
a
l
mer,
S
.
La
c
e
y
,
a
n
d
D
.
O
’
C
o
n
n
o
r
,
“
I
s
mo
t
i
v
a
t
i
o
n
t
h
e
k
e
y
?
F
a
c
t
o
r
s
i
mp
a
c
t
i
n
g
p
e
r
f
o
r
ma
n
c
e
i
n
f
i
r
st
y
e
a
r
serv
i
c
e
ma
t
h
e
ma
t
i
c
s
m
o
d
u
l
e
s,
”
E
u
r
o
p
e
a
n
J
o
u
rn
a
l
o
f
S
c
i
e
n
c
e
a
n
d
Ma
t
h
e
m
a
t
i
c
s
E
d
u
c
a
t
i
o
n
,
v
o
l
.
1
1
,
n
o
.
1
,
p
p
.
1
4
6
–
1
6
6
,
J
a
n
.
2
0
2
3
,
d
o
i
:
1
0
.
3
0
9
3
5
/
s
c
i
m
a
t
h
/
1
2
5
2
9
.
[
21
]
S
.
S
c
h
u
k
a
j
l
o
w
,
J.
B
l
o
mb
e
r
g
,
J
.
R
e
l
l
e
n
sma
n
n
,
a
n
d
C
.
L
e
o
p
o
l
d
,
“
Th
e
r
o
l
e
o
f
st
r
a
t
e
g
y
-
b
a
s
e
d
m
o
t
i
v
a
t
i
o
n
i
n
m
a
t
h
e
mat
i
c
a
l
p
r
o
b
l
e
m
so
l
v
i
n
g
:
Th
e
c
a
se
o
f
l
e
a
r
n
e
r
-
g
e
n
e
r
a
t
e
d
d
r
a
w
i
n
g
s,”
L
e
a
rn
i
n
g
a
n
d
I
n
st
r
u
c
t
i
o
n
,
v
o
l
.
8
0
,
p
.
1
0
1
5
6
1
,
A
u
g
.
2
0
2
2
,
d
o
i
:
1
0
.
1
0
1
6
/
j
.
l
e
a
r
n
i
n
s
t
r
u
c
.
2
0
2
1
.
1
0
1
5
6
1
.
[
22
]
A
.
A
b
í
n
,
J.
C
.
N
ú
ñ
e
z
,
C
.
R
o
d
r
í
g
u
e
z
,
M
.
C
u
e
l
i
,
T
.
G
a
r
c
í
a
,
a
n
d
P
.
R
o
sár
i
o
,
“
P
r
e
d
i
c
t
i
n
g
M
a
t
h
e
ma
t
i
c
s
A
c
h
i
e
v
e
m
e
n
t
i
n
S
e
c
o
n
d
a
r
y
Ed
u
c
a
t
i
o
n
:
T
h
e
R
o
l
e
o
f
C
o
g
n
i
t
i
v
e
,
M
o
t
i
v
a
t
i
o
n
a
l
,
a
n
d
Em
o
t
i
o
n
a
l
V
a
r
i
a
b
l
e
s
,
”
Fro
n
t
i
e
rs
i
n
Psy
c
h
o
l
o
g
y
,
v
o
l
.
1
1
,
p
.
8
7
6
,
M
a
y
2
0
2
0
,
d
o
i
:
1
0
.
3
3
8
9
/
f
p
s
y
g
.
2
0
2
0
.
0
0
8
7
6
.
[
23
]
T.
G
a
r
c
í
a
,
C
.
R
o
d
r
í
g
u
e
z
,
L.
B
e
t
t
s,
D
.
A
r
e
c
e
s
,
a
n
d
P
.
G
o
n
z
á
l
e
z
-
C
a
s
t
r
o
,
“
H
o
w
a
f
f
e
c
t
i
v
e
-
mo
t
i
v
a
t
i
o
n
a
l
v
a
r
i
a
b
l
e
s
a
n
d
a
p
p
r
o
a
c
h
e
s
t
o
l
e
a
r
n
i
n
g
p
r
e
d
i
c
t
m
a
t
h
e
m
a
t
i
c
s
a
c
h
i
e
v
e
men
t
i
n
u
p
p
e
r
e
l
e
m
e
n
t
a
r
y
l
e
v
e
l
s
,
”
L
e
a
rn
i
n
g
a
n
d
I
n
d
i
v
i
d
u
a
l
D
i
f
f
e
re
n
c
e
s
,
v
o
l
.
4
9
,
p
p
.
2
5
–
3
1
,
Ju
l
.
2
0
1
6
,
d
o
i
:
1
0
.
1
0
1
6
/
j
.
l
i
n
d
i
f
.
2
0
1
6
.
0
5
.
0
2
1
.
[
24
]
T.
J.
Ji
n
g
,
R
.
A
.
Ta
r
m
i
z
i
,
K
.
A
.
B
a
k
a
r
,
a
n
d
D
.
A
r
a
l
a
s,
“
Th
e
A
d
o
p
t
i
o
n
o
f
V
a
r
i
a
t
i
o
n
Th
e
o
r
y
i
n
t
h
e
C
l
a
ssr
o
o
m:
Ef
f
e
c
t
o
n
S
t
u
d
e
n
t
s
’
A
l
g
e
b
r
a
i
c
A
c
h
i
e
v
e
m
e
n
t
a
n
d
M
o
t
i
v
a
t
i
o
n
t
o
Le
a
r
n
,
”
E
l
e
c
t
r
o
n
i
c
J
o
u
r
n
a
l
o
f
Re
se
a
rc
h
i
n
Ed
u
c
a
t
i
o
n
a
l
Psy
c
h
o
l
o
g
y
,
v
o
l
.
1
5
,
n
o
.
2
,
p
p
.
3
0
7
–
3
2
5
,
S
e
p
.
2
0
1
7
,
d
o
i
:
1
0
.
1
4
2
0
4
/
e
j
r
e
p
.
4
2
.
1
6
0
7
0
.
[
25
]
R
.
M
.
R
y
a
n
a
n
d
E
.
L.
D
e
c
i
,
“
I
n
t
r
i
n
si
c
a
n
d
Ex
t
r
i
n
si
c
M
o
t
i
v
a
t
i
o
n
s
:
C
l
a
ssi
c
D
e
f
i
n
i
t
i
o
n
s
a
n
d
N
e
w
D
i
r
e
c
t
i
o
n
s,”
C
o
n
t
e
m
p
o
ra
r
y
Ed
u
c
a
t
i
o
n
a
l
Psy
c
h
o
l
o
g
y
,
v
o
l
.
2
5
,
n
o
.
1
,
p
p
.
5
4
–
6
7
,
J
a
n
.
2
0
0
0
,
d
o
i
:
1
0
.
1
0
0
6
/
c
e
p
s.
1
9
9
9
.
1
0
2
0
.
[
26
]
S
.
A
r
e
e
p
a
t
t
a
m
a
n
n
i
l
,
J.
G
.
F
r
e
e
ma
n
,
a
n
d
D
.
A
.
K
l
i
n
g
e
r
,
“
I
n
t
r
i
n
s
i
c
mo
t
i
v
a
t
i
o
n
,
e
x
t
r
i
n
s
i
c
m
o
t
i
v
a
t
i
o
n
,
a
n
d
a
c
a
d
e
mi
c
a
c
h
i
e
v
e
me
n
t
a
mo
n
g
I
n
d
i
a
n
a
d
o
l
e
sc
e
n
t
s
i
n
C
a
n
a
d
a
a
n
d
I
n
d
i
a
,
”
S
o
c
i
a
l
Ps
y
c
h
o
l
o
g
y
o
f
Ed
u
c
a
t
i
o
n
,
v
o
l
.
1
4
,
n
o
.
3
,
p
p
.
4
2
7
–
4
3
9
,
S
e
p
.
2
0
1
1
,
d
o
i
:
1
0
.
1
0
0
7
/
s
1
1
2
1
8
-
011
-
9
1
5
5
-
1.
[
27
]
O
.
K
a
p
l
a
n
,
“
A
n
a
l
y
z
i
n
g
t
h
e
Ef
f
e
c
t
s
o
f
M
o
t
i
v
a
t
i
o
n
,
E
x
p
e
c
t
a
t
i
o
n
s
a
n
d
S
e
l
f
-
C
o
n
c
e
p
t
o
n
Ei
g
h
t
h
-
G
r
a
d
e
S
t
u
d
e
n
t
s’
M
a
t
h
e
m
a
t
i
c
s
A
c
h
i
e
v
e
me
n
t
,
”
P
h
.
D
.
d
i
ss
e
r
t
a
t
i
o
n
,
D
e
p
t
.
E
d
u
c
.
,
U
n
i
v
e
r
si
t
y
o
f
F
l
o
r
i
d
a
,
G
a
i
n
e
sv
i
l
l
e
,
F
L,
U
S
A
,
2
0
1
8
.
[
28
]
H
.
Le
e
a
n
d
Y
.
K
i
m,
“
K
o
r
e
a
n
a
d
o
l
e
s
c
e
n
t
s’
l
o
n
g
i
t
u
d
i
n
a
l
c
h
a
n
g
e
o
f
i
n
t
r
i
n
si
c
mo
t
i
v
a
t
i
o
n
i
n
l
e
a
r
n
i
n
g
E
n
g
l
i
s
h
a
n
d
m
a
t
h
e
m
a
t
i
c
s
d
u
r
i
n
g
sec
o
n
d
a
r
y
s
c
h
o
o
l
y
e
a
r
s:
F
o
c
u
si
n
g
o
n
g
e
n
d
e
r
d
i
f
f
e
r
e
n
c
e
a
n
d
sc
h
o
o
l
c
h
a
r
a
c
t
e
r
i
st
i
c
s,”
L
e
a
r
n
i
n
g
a
n
d
I
n
d
i
v
i
d
u
a
l
D
i
f
f
e
re
n
c
e
s
,
v
o
l
.
3
6
,
p
p
.
1
3
1
–
1
3
9
,
D
e
c
.
2
0
1
4
,
d
o
i
:
1
0
.
1
0
1
6
/
j
.
l
i
n
d
i
f
.
2
0
1
4
.
0
7
.
0
1
8
.
[
29
]
L.
T.
Tr
a
n
a
n
d
T
.
S
.
S
.
N
g
u
y
e
n
,
“
M
o
t
i
v
a
t
i
o
n
a
n
d
M
a
t
h
e
m
a
t
i
c
s A
c
h
i
e
v
e
m
e
n
t
:
A
V
i
e
t
n
a
m
e
se
C
a
s
e
S
t
u
d
y
,
”
J
o
u
r
n
a
l
o
n
M
a
t
h
e
m
a
t
i
c
s
Ed
u
c
a
t
i
o
n
,
v
o
l
.
1
2
,
n
o
.
3
,
p
p
.
4
4
9
–
4
6
8
,
S
e
p
.
2
0
2
1
,
d
o
i
:
1
0
.
2
2
3
4
2
/
j
me
.
1
2
.
3
.
1
4
2
7
4
.
4
4
9
-
4
6
8
.
[
30
]
Y
.
Li
u
a
n
d
S
.
H
o
u
,
“
P
o
t
e
n
t
i
a
l
r
e
c
i
p
r
o
c
a
l
r
e
l
a
t
i
o
n
s
h
i
p
b
e
t
w
e
e
n
m
o
t
i
v
a
t
i
o
n
a
n
d
a
c
h
i
e
v
e
m
e
n
t
:
A
l
o
n
g
i
t
u
d
i
n
a
l
st
u
d
y
,
”
S
c
h
o
o
l
Psy
c
h
o
l
o
g
y
I
n
t
e
r
n
a
t
i
o
n
a
l
,
v
o
l
.
3
9
,
n
o
.
1
,
p
p
.
3
8
–
5
5
,
F
e
b
.
2
0
1
8
,
d
o
i
:
1
0
.
1
1
7
7
/
0
1
4
3
0
3
4
3
1
7
7
1
0
5
7
4
.
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
.
14
,
No
.
4
,
Au
g
u
s
t
20
25
:
320
1
-
3
2
1
0
3210
[
31
]
U
.
T.
J
a
n
k
v
i
s
t
,
“
C
h
a
n
g
i
n
g
s
t
u
d
e
n
t
s’
i
ma
g
e
s
o
f
‘
ma
t
h
e
ma
t
i
c
s
a
s
a
d
i
s
c
i
p
l
i
n
e
,
’
”
T
h
e
J
o
u
rn
a
l
o
f
M
a
t
h
e
m
a
t
i
c
a
l
B
e
h
a
v
i
o
r
,
v
o
l
.
3
8
,
p
p
.
4
1
–
5
6
,
J
u
n
.
2
0
1
5
,
d
o
i
:
1
0
.
1
0
1
6
/
j
.
j
mat
h
b
.
2
0
1
5
.
0
2
.
0
0
2
.
[
32
]
L.
H
.
R
e
y
e
s
,
“
A
f
f
e
c
t
i
v
e
V
a
r
i
a
b
l
e
s
a
n
d
M
a
t
h
e
ma
t
i
c
s E
d
u
c
a
t
i
o
n
,
”
T
h
e
E
l
e
m
e
n
t
a
ry
S
c
h
o
o
l
J
o
u
r
n
a
l
,
v
o
l
.
8
4
,
n
o
.
5
,
p
p
.
5
5
8
–
5
8
1
,
M
a
y
1
9
8
4
,
d
o
i
:
1
0
.
1
0
8
6
/
4
6
1
3
8
4
.
[
33
]
H
.
T
a
n
n
e
r
a
n
d
S
.
Jo
n
e
s
,
“
S
c
a
f
f
o
l
d
i
n
g
f
o
r
s
u
c
c
e
ss
:
R
e
f
l
e
c
t
i
v
e
d
i
s
c
o
u
r
se
a
n
d
t
h
e
e
f
f
e
c
t
i
v
e
t
e
a
c
h
i
n
g
o
f
ma
t
h
e
ma
t
i
c
a
l
t
h
i
n
k
i
n
g
sk
i
l
l
s,”
Re
se
a
rc
h
i
n
M
a
t
h
e
m
a
t
i
c
s
E
d
u
c
a
t
i
o
n
,
v
o
l
.
2
,
n
o
.
1
,
p
p
.
1
9
–
3
2
,
A
p
r
.
2
0
0
0
,
d
o
i
:
1
0
.
1
0
8
0
/
1
4
7
9
4
8
0
0
0
0
8
5
2
0
0
6
5
.
[
34
]
T.
G
o
e
t
z
,
H
.
C
r
o
n
j
a
e
g
e
r
,
A
.
C
.
F
r
e
n
z
e
l
,
O
.
Lü
d
t
k
e
,
a
n
d
N
.
C
.
H
a
l
l
,
“
A
c
a
d
e
mi
c
se
l
f
-
c
o
n
c
e
p
t
a
n
d
e
m
o
t
i
o
n
r
e
l
a
t
i
o
n
s
:
D
o
mai
n
sp
e
c
i
f
i
c
i
t
y
a
n
d
a
g
e
e
f
f
e
c
t
s,
”
C
o
n
t
e
m
p
o
r
a
r
y
Ed
u
c
a
t
i
o
n
a
l
Ps
y
c
h
o
l
o
g
y
,
v
o
l
.
3
5
,
n
o
.
1
,
p
p
.
4
4
–
5
8
,
J
a
n
.
2
0
1
0
,
d
o
i
:
1
0
.
1
0
1
6
/
j
.
c
e
d
p
s
y
c
h
.
2
0
0
9
.
1
0
.
0
0
1
.
[
35
]
R
.
M
.
R
y
a
n
a
n
d
E.
L.
D
e
c
i
,
“
I
n
t
r
i
n
si
c
a
n
d
e
x
t
r
i
n
s
i
c
m
o
t
i
v
a
t
i
o
n
f
r
o
m
a
se
l
f
-
d
e
t
e
r
mi
n
a
t
i
o
n
t
h
e
o
r
y
p
e
r
sp
e
c
t
i
v
e
:
D
e
f
i
n
i
t
i
o
n
s,
t
h
e
o
r
y
,
p
r
a
c
t
i
c
e
s,
a
n
d
f
u
t
u
r
e
d
i
r
e
c
t
i
o
n
s,
”
C
o
n
t
e
m
p
o
r
a
ry
E
d
u
c
a
t
i
o
n
a
l
P
syc
h
o
l
o
g
y
,
v
o
l
.
6
1
,
p
.
1
0
1
8
6
0
,
A
p
r
.
2
0
2
0
,
d
o
i
:
1
0
.
1
0
1
6
/
j
.
c
e
d
p
s
y
c
h
.
2
0
2
0
.
1
0
1
8
6
0
.
[
36
]
A
.
K
.
A
r
e
n
s
,
H
.
W
.
M
a
r
s
h
,
R
.
G
.
C
r
a
v
e
n
,
A
.
S
.
Y
e
u
n
g
,
E.
R
a
n
d
h
a
w
a
,
a
n
d
M
.
H
a
sse
l
h
o
r
n
,
“
M
a
t
h
se
l
f
-
c
o
n
c
e
p
t
i
n
p
r
e
s
c
h
o
o
l
c
h
i
l
d
r
e
n
:
S
t
r
u
c
t
u
r
e
,
a
c
h
i
e
v
e
me
n
t
r
e
l
a
t
i
o
n
s
,
a
n
d
g
e
n
e
r
a
l
i
z
a
b
i
l
i
t
y
a
c
r
o
ss
g
e
n
d
e
r
,
”
E
a
rl
y
C
h
i
l
d
h
o
o
d
Re
s
e
a
r
c
h
Q
u
a
r
t
e
r
l
y
,
v
o
l
.
3
6
,
p
p
.
3
9
1
–
4
0
3
,
2
0
1
6
,
d
o
i
:
1
0
.
1
0
1
6
/
j
.
e
c
r
e
sq
.
2
0
1
5
.
1
2
.
0
2
4
.
[
37
]
M
.
S
.
K
a
d
i
r
,
A
.
S
.
Y
e
u
n
g
,
a
n
d
T.
M
.
O
.
D
i
a
l
l
o
,
“
S
i
m
u
l
t
a
n
e
o
u
s
t
e
s
t
i
n
g
o
f
f
o
u
r
d
e
c
a
d
e
s
o
f
a
c
a
d
e
mi
c
s
e
l
f
-
c
o
n
c
e
p
t
m
o
d
e
l
s
,
”
C
o
n
t
e
m
p
o
r
a
ry
Ed
u
c
a
t
i
o
n
a
l
Ps
y
c
h
o
l
o
g
y
,
v
o
l
.
5
1
,
p
p
.
4
2
9
–
4
4
6
,
O
c
t
.
2
0
1
7
,
d
o
i
:
1
0
.
1
0
1
6
/
j
.
c
e
d
p
sy
c
h
.
2
0
1
7
.
0
9
.
0
0
8
.
B
I
O
G
RAP
H
I
E
S O
F
AUTH
O
RS
Im
d
a
d
Ali
is
a
P
h
.
D.
c
a
n
d
id
a
t
e
,
De
p
a
rtme
n
t
o
f
M
a
t
h
e
m
a
ti
c
s,
Ce
n
tral
In
sti
tu
te
o
f
Tec
h
n
o
l
o
g
y
Ko
k
ra
jh
a
r,
De
p
a
rtme
n
t
o
f
M
a
th
e
m
a
ti
c
s,
In
d
ia.
He
h
a
s
p
u
b
l
ish
e
d
m
a
n
y
re
se
a
rc
h
p
a
p
e
rs
in
m
a
th
e
m
a
ti
c
s
e
d
u
c
a
ti
o
n
in
re
p
u
ted
jo
u
rn
a
ls
a
n
d
is
a
n
a
c
ti
v
e
re
se
a
rc
h
e
r
in
m
a
th
e
m
a
ti
c
s
e
d
u
c
a
ti
o
n
.
He
c
a
n
b
e
c
o
n
tac
ted
a
t
e
m
a
il
:
a
li
.
imd
a
d
9
9
@g
m
a
il
.
c
o
m
.
S
a
m
ir
a
n
Da
s
is
a
n
a
ss
o
c
iate
p
ro
fe
ss
o
r
,
De
p
a
rtme
n
t
o
f
M
a
t
h
e
m
a
ti
c
s,
Ce
n
tra
l
In
stit
u
te
o
f
Tec
h
n
o
lo
g
y
K
o
k
ra
j
h
a
r,
In
d
ia.
Be
sid
e
s
m
a
th
e
m
a
ti
c
s
e
d
u
c
a
ti
o
n
,
h
e
is
a
n
a
c
ti
v
e
re
se
a
rc
h
e
r
in
p
las
m
a
p
h
y
sic
s
a
n
d
p
u
b
li
sh
e
d
m
a
n
y
re
se
a
rc
h
p
a
p
e
rs
in
m
a
n
y
re
p
u
ted
jo
u
rn
a
ls
su
c
h
a
s
Am
e
rica
n
I
n
stit
u
te
o
f
P
h
y
sic
s
(AIP)
,
S
p
rin
g
e
r
Na
tu
re
,
IE
EE
,
Ca
m
b
ri
d
g
e
,
a
n
d
I
n
stit
u
te
o
f
P
h
y
sic
s (IOP
).
He
c
a
n
b
e
c
o
n
tac
ted
a
t
e
m
a
il
:
s.d
a
s@
c
it
.
a
c
.
in
.
Evaluation Warning : The document was created with Spire.PDF for Python.