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f
E
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rica
l a
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m
pu
t
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ineering
(
I
J
E
CE
)
Vo
l.
14
,
No
.
6
,
Dec
em
b
er
20
24
,
p
p
.
6
4
1
1
~
6
4
2
2
I
SS
N:
2088
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8
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DOI
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v
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.
pp
6
4
1
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6411
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ftwa
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K
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d
s
:
Help
in
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s
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d
en
t
L
ea
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n
in
g
Ma
th
em
atics
Me
ta
an
aly
s
is
Vir
tu
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r
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tatio
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T
h
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s
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CC B
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C
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s
p
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A
uth
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:
Yu
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izk
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Me
lia
Sar
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Dep
ar
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Ma
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atics E
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ca
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Neg
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Ketin
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3
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ac
.
id
1.
I
NT
RO
D
UCT
I
O
N
Ma
th
em
atics,
as
a
co
m
p
lex
d
is
cip
lin
e
b
y
n
atu
r
e,
in
v
o
lv
es
ess
en
tial
s
k
ill
s
s
u
ch
as
lin
k
in
g
,
ab
s
tr
ac
tio
n
,
an
d
g
en
er
aliza
tio
n
[
1
]
.
I
n
r
ec
e
n
t
y
ea
r
s
,
r
esear
ch
in
m
ath
em
atics
ed
u
ca
tio
n
h
as
h
ig
h
lig
h
ted
t
h
e
co
m
p
lex
ity
o
f
teac
h
in
g
ch
alle
n
g
es
an
d
s
tr
ateg
ies
[
2
]
.
Ma
th
e
m
atics
teac
h
in
g
g
en
er
ally
u
n
d
er
g
o
es
a
tr
an
s
itio
n
f
r
o
m
co
n
c
r
ete
to
ab
s
tr
ac
t
[
3
]
,
[
4
]
.
T
h
is
in
d
icate
s
an
u
n
d
er
s
t
an
d
in
g
th
at
co
n
cr
ete
m
ath
em
a
tical
th
in
k
in
g
s
k
ills
ar
e
tau
g
h
t
in
th
e
ea
r
ly
y
ea
r
s
o
f
ed
u
ca
tio
n
at
th
e
p
r
im
ar
y
lev
el,
wh
ile
m
ath
em
atica
l
th
in
k
in
g
s
k
ills
th
at
r
eq
u
ir
e
f
u
r
th
er
in
f
er
e
n
ce
an
d
ju
s
tific
atio
n
ar
e
p
r
o
v
id
ed
i
n
m
o
r
e
ad
v
a
n
ce
d
ed
u
ca
tio
n
al
p
r
o
ce
s
s
es.
Sev
er
al
s
tu
d
ies
h
av
e
b
ee
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c
o
n
d
u
cted
o
n
th
e
co
m
p
lex
ity
o
f
m
at
h
em
atics
wh
ich
is
n
o
t
o
n
l
y
v
iewe
d
as
a
s
er
ies
o
f
co
n
ce
p
ts
,
b
u
t
als
o
as
a
s
er
ies
o
f
in
ter
r
elate
d
c
o
g
n
itiv
e
s
k
ills
[
2
]
.
R
esear
ch
er
s
a
s
s
er
t
th
at
ef
f
ec
tiv
e
m
ath
em
ati
cs
teac
h
in
g
r
eq
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ir
es
a
d
ee
p
u
n
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e
r
s
tan
d
in
g
o
f
h
o
w
s
tu
d
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b
u
ild
r
elatio
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h
ip
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etwe
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p
ts
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p
er
f
o
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m
ab
s
tr
ac
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s
,
an
d
m
ak
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g
en
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aliza
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s
.
Stu
d
en
ts
o
f
ten
ex
p
e
r
ien
ce
d
i
f
f
icu
lties
in
lear
n
in
g
m
ath
e
m
atics
b
ec
au
s
e
th
ey
f
ac
e
d
if
f
icu
lties
in
u
n
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er
s
tan
d
i
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co
n
ce
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ts
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p
er
f
o
r
m
in
g
ab
s
tr
ac
tio
n
s
,
an
d
g
e
n
er
alizin
g
[
5
]
,
[
6
]
.
T
h
is
is
b
ec
au
s
e
s
o
m
e
s
tu
d
en
ts
d
o
Evaluation Warning : The document was created with Spire.PDF for Python.
I
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I
n
t J E
lec
&
C
o
m
p
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g
,
Vo
l.
14
,
No
.
6
,
Dec
em
b
e
r
20
24
:
6
4
1
1
-
6
4
2
2
6412
n
o
t
u
n
d
er
s
tan
d
m
ath
em
atica
l
to
p
ics
th
o
r
o
u
g
h
ly
s
o
th
e
y
f
a
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if
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icu
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estab
lis
h
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co
n
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ec
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atica
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'
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tan
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atic
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k
n
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wled
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e
ca
n
h
in
d
er
t
h
e
u
n
d
er
s
tan
d
in
g
o
f
m
o
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e
co
m
p
lex
m
ath
em
atica
l
o
p
er
atio
n
co
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ce
p
ts
[
7
]
.
Ab
s
tr
ac
tio
n
ab
ilit
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is
an
o
b
s
tacle
f
o
r
s
tu
d
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ts
,
b
ec
a
u
s
e
th
ey
h
av
e
d
if
f
icu
lty
lin
k
in
g
m
ath
em
atica
l
co
n
ce
p
ts
to
r
ea
l
-
wo
r
ld
s
itu
atio
n
s
o
r
f
o
r
m
u
latin
g
p
r
o
b
lem
s
ab
s
tr
ac
tly
[
8
]
.
I
n
ad
d
itio
n
,
d
if
f
icu
lti
es
in
g
en
er
aliza
tio
n
also
o
f
t
en
o
cc
u
r
,
b
ec
au
s
e
s
tu
d
en
ts
h
av
e
d
if
f
icu
lty
ap
p
l
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in
g
th
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co
n
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p
ts
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lear
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to
d
if
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er
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t
co
n
tex
ts
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r
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n
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tan
d
in
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ca
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b
e
a
p
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lied
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v
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r
o
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[
9
]
.
T
h
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r
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elatio
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s
h
i
p
s
b
etwe
en
co
n
ce
p
ts
,
p
er
f
o
r
m
ab
s
tr
ac
tio
n
,
an
d
g
en
e
r
alize
is
k
ey
in
o
v
er
c
o
m
in
g
m
ath
em
atics lea
r
n
in
g
d
i
f
f
icu
lt
ies.
I
n
lear
n
in
g
m
ath
em
atics,
f
r
o
m
th
e
co
n
cr
ete
t
o
th
e
ab
s
tr
ac
t
lev
el,
o
n
e
o
f
th
e
s
tr
ateg
ies
th
at
teac
h
er
s
ca
n
u
s
e
to
im
p
r
o
v
e
u
n
d
er
s
tan
d
in
g
o
f
m
ath
e
m
atica
l
co
n
ce
p
ts
is
to
u
s
e
v
is
u
al
r
ep
r
esen
tati
o
n
s
o
r
r
ea
l
o
b
jects
[
1
0
]
.
At
th
e
c
o
n
cr
ete
s
tag
e,
f
o
r
ex
am
p
le
o
n
th
e
t
o
p
ic
o
f
n
u
m
b
er
s
,
teac
h
er
s
teac
h
b
asic
m
ath
o
p
er
atio
n
s
with
n
u
m
b
er
b
lo
c
k
s
,
wh
ich
h
elp
s
s
tu
d
en
ts
u
n
d
er
s
tan
d
th
e
c
o
n
ce
p
ts
b
etter
[
1
1
]
–
[
1
3
]
.
I
n
th
e
r
ep
r
esen
tatio
n
al
s
tag
e,
s
tu
d
en
ts
ar
e
in
tr
o
d
u
ce
d
to
r
ea
l
o
b
jects
s
u
ch
as
n
u
m
b
er
b
lo
c
k
s
.
I
n
th
e
last
s
tag
e,
th
e
a
b
s
tr
ac
t
s
tag
e,
s
tu
d
en
ts
lear
n
m
ath
th
r
o
u
g
h
m
ath
em
ati
ca
l
s
y
m
b
o
ls
in
a
m
o
r
e
g
en
er
al
way
af
ter
g
ain
i
n
g
co
n
cr
ete
a
n
d
r
ep
r
esen
tatio
n
al
f
o
u
n
d
atio
n
s
.
I
f
s
tu
d
e
n
ts
f
ac
e
d
if
f
icu
lties
,
it
is
im
p
o
r
tan
t
to
r
etu
r
n
to
th
e
c
o
n
cr
ete
lev
el
[
1
4
]
.
Ma
th
e
m
atica
l
m
an
ip
u
latio
n
a
n
d
v
is
u
al
r
ep
r
e
s
en
tatio
n
ar
e
ess
en
tial f
o
r
s
o
lv
in
g
s
tu
d
en
ts
'
m
ath
lear
n
in
g
d
if
f
icu
lties
.
T
ec
h
n
o
lo
g
ical
ap
p
r
o
ac
h
es
to
cr
ea
tin
g
v
is
u
al
p
r
esen
tat
io
n
s
h
av
e
tr
an
s
f
o
r
m
ed
th
e
lear
n
in
g
en
v
ir
o
n
m
en
t
b
y
p
r
o
v
id
in
g
a
v
ar
iety
o
f
way
s
to
p
r
esen
t
i
n
f
o
r
m
atio
n
in
d
y
n
am
ic
a
n
d
en
g
a
g
in
g
way
s
[
1
5
]
.
A
v
ar
iety
o
f
d
ig
ital
to
o
ls
an
d
s
o
f
twar
e
en
ab
le
teac
h
er
s
an
d
s
tu
d
en
ts
to
cr
ea
te
e
n
g
ag
in
g
a
n
d
i
n
f
o
r
m
ativ
e
im
ag
es
[
1
6
]
,
[
1
7
]
.
Fro
m
g
r
a
p
h
ic
an
im
atio
n
s
to
in
ter
ac
tiv
e
s
im
u
latio
n
s
,
tech
n
o
lo
g
y
h
elp
s
s
tu
d
en
ts
u
n
d
er
s
tan
d
c
o
m
p
le
x
co
n
ce
p
ts
b
y
cr
ea
tin
g
v
is
u
al
p
r
esen
tatio
n
s
th
at
en
r
ich
th
e
lea
r
n
in
g
ex
p
er
ie
n
ce
[
1
6
]
,
[
1
8
]
.
T
h
ese
b
en
ef
its
cr
ea
te
a
m
o
r
e
d
y
n
a
m
ic
an
d
ad
ap
tiv
e
lear
n
in
g
en
v
ir
o
n
m
e
n
t
th
at
s
u
p
p
o
r
ts
m
u
ltip
le
way
s
to
m
e
et
s
tu
d
en
ts
'
d
iv
er
s
e
n
ee
d
s
in
u
n
d
er
s
tan
d
in
g
th
e
m
a
ter
ial.
T
ec
h
n
o
lo
g
ical
d
ev
elo
p
m
en
ts
h
av
e
led
to
th
e
cr
ea
tio
n
o
f
v
ar
io
u
s
ap
p
licatio
n
s
an
d
to
o
ls
f
o
r
v
ir
t
u
al
r
ep
r
esen
tatio
n
in
m
ath
em
atics
lear
n
in
g
.
So
m
e
tech
n
o
lo
g
ical
ap
p
r
o
ac
h
es
i
n
m
at
h
em
atics
lear
n
in
g
in
clu
d
e
th
e
u
s
e
o
f
v
ar
i
o
u
s
to
o
ls
s
u
ch
as
Geo
Geb
r
a,
3
D
m
o
d
els
an
d
Go
o
g
le
Sit
es
ap
p
licatio
n
s
.
Geo
Geb
r
a
ca
n
c
r
ea
te
v
is
u
aliza
tio
n
s
to
ex
p
lain
m
ath
em
atica
l
co
n
ce
p
ts
an
d
3
D
m
o
d
els
p
r
o
v
i
d
e
a
v
is
u
al
an
d
in
te
r
ac
tiv
e
ex
p
e
r
ien
ce
,
m
ak
in
g
lear
n
in
g
m
o
r
e
f
u
n
an
d
im
m
er
s
iv
e
[
1
9
]
.
G
o
o
g
le
Sit
es
ap
p
licatio
n
is
a
u
s
ef
u
l
to
o
l
f
o
r
p
r
ac
ticin
g
v
is
u
al
r
ep
r
esen
tatio
n
o
f
m
ath
e
m
atica
l
m
o
d
els
[
2
0
]
.
T
h
is
tech
n
o
lo
g
y
also
f
ac
ilit
ates
v
ar
io
u
s
v
is
u
a
lizatio
n
s
,
in
clu
d
in
g
d
r
awin
g
s
ce
n
es,
c
r
ea
tin
g
g
r
a
p
h
s
,
an
d
u
s
in
g
g
en
e
r
al
v
is
u
a
lizatio
n
to
o
ls
.
B
y
u
s
in
g
th
is
tech
n
o
lo
g
y
,
m
ath
lear
n
in
g
b
ec
o
m
es
m
o
r
e
in
ter
a
ctiv
e
an
d
f
u
n
,
s
u
p
p
o
r
tin
g
s
tu
d
en
ts
'
co
n
ce
p
t
u
n
d
er
s
tan
d
in
g
th
r
o
u
g
h
b
etter
v
is
u
al
r
ep
r
esen
tatio
n
.
Ma
th
em
atics
lear
n
in
g
in
te
g
r
a
ted
with
elec
tr
o
n
ic
d
ev
ices,
s
u
ch
as
tab
lets
o
r
lap
to
p
s
,
h
as
b
ee
n
t
h
e
s
u
b
ject
o
f
s
ig
n
if
ican
t
r
esear
c
h
o
v
er
th
e
p
ast
1
0
y
e
ar
s
[
2
1
]
.
T
h
e
d
ev
elo
p
m
en
t
o
f
tec
h
n
o
lo
g
y
-
in
teg
r
ated
m
ath
em
atics
lear
n
in
g
to
o
ls
,
s
u
ch
as
e
-
wo
r
k
s
h
ee
ts
an
d
i
n
ter
ac
tiv
e
e
-
b
o
o
k
s
,
h
as
s
u
cc
ess
f
u
lly
in
cr
ea
s
ed
th
e
ef
f
ec
tiv
en
ess
o
f
m
ath
em
atic
s
lear
n
in
g
[
2
2
]
,
[
2
3
]
.
T
h
is
s
h
o
ws
th
at
th
e
in
teg
r
atio
n
o
f
tech
n
o
lo
g
y
in
m
ath
em
atics
ed
u
ca
tio
n
h
as
s
h
o
wn
a
p
o
s
itiv
e
im
p
ac
t
o
n
s
tu
d
en
ts
'
ab
ilit
ie
s
.
Stu
d
ies
s
h
o
w
th
at
th
e
k
ey
s
u
cc
ess
f
ac
to
r
s
o
f
ad
v
a
n
ce
d
tech
n
o
l
o
g
y
in
m
ath
em
atics
ed
u
ca
tio
n
in
v
o
lv
e
th
e
d
esig
n
o
f
co
m
p
u
ter
-
b
ased
to
o
ls
an
d
th
e
way
th
ey
ar
e
im
p
lem
e
n
ted
[
2
4
]
.
T
h
e
u
s
e
o
f
m
o
b
ile
d
e
v
ices
h
as
also
b
ee
n
s
h
o
wn
to
h
av
e
a
p
o
s
itiv
e
ef
f
ec
t
o
n
m
ath
lear
n
in
g
,
wh
e
r
e
th
e
i
n
teg
r
atio
n
o
f
tech
n
o
lo
g
y
a
n
d
c
o
m
p
u
ter
s
ca
n
h
elp
s
tu
d
e
n
ts
b
et
ter
u
n
d
e
r
s
tan
d
m
ath
co
n
ce
p
ts
[
2
5
]
.
I
n
ad
d
itio
n
,
t
h
e
ap
p
licatio
n
o
f
a
r
ea
lis
tic
m
ath
em
atics
ed
u
ca
tio
n
ap
p
r
o
ac
h
with
tech
n
o
lo
g
ical
s
u
p
p
o
r
t
h
as
s
u
cc
ee
d
ed
in
im
p
r
o
v
in
g
s
tu
d
en
ts
'
ab
ilit
y
to
r
el
ate
m
ath
em
atica
l
co
n
ce
p
ts
to
ev
er
y
d
a
y
life
[
2
6
]
.
T
h
u
s
,
th
e
p
o
r
t
r
ait
o
f
lear
n
in
g
s
u
cc
ess
u
s
in
g
elec
tr
o
n
ic
d
ev
ices
in
clu
d
es
in
cr
ea
s
in
g
lear
n
in
g
ef
f
ec
tiv
e
n
ess
,
d
ev
elo
p
in
g
s
tu
d
e
n
t
ab
ilit
ies,
a
n
d
in
te
g
r
atin
g
tech
n
o
lo
g
y
to
e
n
r
ich
th
e
m
ath
e
m
atics
lear
n
in
g
ex
p
er
ien
ce
.
T
h
er
e
ar
e
s
ev
er
al
o
p
in
i
o
n
s
ag
ain
s
t d
i
g
ital le
ar
n
in
g
o
r
u
s
in
g
elec
tr
o
n
ic
d
ev
ices
[
2
7
]
,
[
2
8
]
.
First,
th
e
u
s
e
o
f
elec
tr
o
n
ic
s
cr
ee
n
s
ca
n
lead
to
ey
estra
in
an
d
d
ec
r
ea
s
ed
f
o
cu
s
,
wh
ich
ca
n
h
in
d
er
lear
n
er
s
'
co
m
p
r
eh
en
s
io
n
o
f
th
e
m
ater
ial
[
2
8
]
.
Oth
er
s
tu
d
ies
h
av
e
also
s
h
o
wn
t
h
at
r
ea
d
i
n
g
co
m
p
r
eh
en
s
io
n
o
n
p
ap
er
b
o
o
k
s
is
b
etter
th
an
o
n
elec
tr
o
n
ic
b
o
o
k
s
[
2
9
]
.
T
h
is
s
u
g
g
ests
th
at
elec
tr
o
n
ic
d
ev
ices
ca
n
h
av
e
a
n
e
g
ativ
e
im
p
ac
t
o
n
r
ea
d
in
g
co
m
p
r
e
h
e
n
s
io
n
.
I
n
a
d
d
itio
n
,
to
o
m
u
c
h
tim
e
s
p
en
t
in
f
r
o
n
t
o
f
a
s
cr
e
en
ca
n
also
lead
to
f
atig
u
e
a
n
d
a
d
ec
lin
e
in
s
tu
d
en
ts
'
ey
e
h
ea
lth
.
Seco
n
d
ly
,
le
ar
n
in
g
to
d
r
aw
with
elec
tr
o
n
i
c
d
ev
ices
is
o
f
ten
u
n
ab
le
to
d
ev
elo
p
d
r
awin
g
s
k
i
lls
p
r
o
p
er
ly
.
T
h
e
u
s
e
o
f
tab
let
s
o
r
co
m
p
u
ter
s
f
o
r
d
r
awin
g
d
o
es
n
o
t
p
r
o
v
id
e
th
e
s
am
e
ex
p
er
ien
ce
as
u
s
in
g
p
a
p
er
an
d
p
e
n
cil,
wh
ich
ca
n
b
e
d
etr
im
en
tal
to
t
h
e
d
ev
elo
p
m
en
t
o
f
s
tu
d
en
ts
'
ar
tis
tic
s
k
ills
an
d
v
is
u
al
s
en
s
ib
ilit
ies
[
2
7
]
.
T
h
u
s
,
wh
ile
tech
n
o
lo
g
y
p
r
o
v
id
es
ac
ce
s
s
to
d
ig
ital
r
eso
u
r
ce
s
an
d
f
lex
ib
ilit
y
in
lear
n
in
g
,
th
e
o
p
p
o
s
itio
n
lie
s
in
its
p
o
ten
tial
n
e
g
ativ
e
im
p
ac
t
o
n
h
ea
lth
an
d
s
p
ec
if
ic
s
k
ills
,
s
u
ch
as
d
r
awin
g
ab
ilit
y
an
d
r
ea
d
in
g
s
k
ills
.
I
n
m
at
h
em
atics
lear
n
in
g
,
t
h
er
e
ar
e
two
d
if
f
er
en
t
v
iews
o
n
th
e
in
teg
r
atio
n
o
f
tech
n
o
lo
g
y
in
v
is
u
al
r
ep
r
esen
tatio
n
.
T
h
e
p
o
s
itiv
e
v
iew
em
p
h
asizes
th
at
th
e
u
s
e
o
f
tech
n
o
lo
g
y
s
u
ch
as
Geo
Ge
b
r
a
s
o
f
twar
e
o
r
3
D
m
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ca
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m
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m
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Ho
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Evaluation Warning : The document was created with Spire.PDF for Python.
I
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p
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I
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N:
2088
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8
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2
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3
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ased
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en
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ed
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8
7
0
8
I
n
t J E
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C
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,
Vo
l.
14
,
No
.
6
,
Dec
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b
e
r
20
24
:
6
4
1
1
-
6
4
2
2
6414
with
s
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d
2
.
T
h
e
th
r
ee
m
eth
o
d
s
ar
e
u
s
ed
in
th
e
h
o
p
e
th
at
th
ey
ca
n
co
r
r
ec
t
ea
ch
o
th
er
s
o
th
at
th
e
ass
u
m
p
tio
n
ca
n
b
e
p
r
o
v
e
n
s
tead
ily
.
T
h
e
d
ec
is
io
n
-
m
ak
i
n
g
cr
it
er
ia
f
o
r
p
r
o
v
in
g
th
e
ass
u
m
p
tio
n
is
th
at
th
e
d
ata
i
s
s
aid
to
b
e
s
u
b
s
tan
tially
h
ete
r
o
g
en
eo
u
s
if
th
e
v
al
u
e
o
f
2
is
b
etwe
en
7
5
%
an
d
1
0
0
%
[
3
8
]
,
p
-
v
alu
e
Q
<
0
.
0
5
,
an
d
Ʈ
2
is
m
o
r
e
th
an
ze
r
o
[
3
9
]
.
Fig
u
r
e
1
.
PR
I
SMA
d
ata
s
elec
tio
n
Fu
r
th
er
m
o
r
e
,
th
e
ass
u
m
p
tio
n
to
b
e
p
r
o
v
e
n
is
th
e
f
r
ee
d
o
m
o
f
d
ata
f
r
o
m
p
u
b
licatio
n
b
ias.
I
n
m
eta
-
an
aly
s
is
,
th
is
p
r
o
o
f
is
a
m
u
s
t
as
a
g
u
ar
a
n
tee
th
at
th
e
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ata
a
n
d
an
al
y
s
is
r
esu
lts
ca
n
b
e
tr
u
s
ted
.
T
h
e
p
r
o
o
f
o
f
f
r
ee
d
o
m
f
r
o
m
p
u
b
licatio
n
will
u
s
e
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m
eth
o
d
s
,
n
am
ely
id
e
n
tific
atio
n
o
f
f
u
n
n
el
p
l
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t
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ase
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in
t
d
is
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ib
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n
an
d
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er
en
ce
o
f
th
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th
e
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ile
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e
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lcu
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ased
o
n
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ass
in
g
g
r
ad
e.
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is
f
r
ee
f
r
o
m
p
u
b
licatio
n
b
ias
wh
en
th
e
d
is
tr
ib
u
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o
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ts
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h
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el
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y
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m
e
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ical
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d
t
h
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ile
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s
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alu
e
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g
r
ea
ter
th
a
n
th
e
cr
iter
io
n
o
f
5
+
10
wh
er
e
is
th
e
n
u
m
b
er
o
f
d
ata
[
4
0
]
.
2
.
4
.
Det
er
m
ini
ng
ef
f
ec
t
s
ize
Af
ter
th
e
p
r
er
eq
u
is
ite
ass
u
m
p
tio
n
s
h
av
e
b
ee
n
p
r
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v
en
,
th
e
a
n
aly
s
is
p
r
o
ce
s
s
ca
n
co
n
tin
u
e
to
f
in
d
th
e
to
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f
ec
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s
ize.
T
h
is
r
esear
ch
is
a
m
eta
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aly
s
is
o
f
s
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r
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p
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co
n
tr
o
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g
r
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p
co
m
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ar
is
o
n
m
o
d
els
[
4
1
]
.
T
h
e
s
tan
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ar
d
iza
tio
n
p
r
o
ce
s
s
aim
s
to
e
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u
aliz
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ata
t
h
at
h
as
h
ig
h
ly
v
a
r
iab
le
ag
g
r
e
g
ates.
T
h
e
r
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lts
o
f
s
tan
d
ar
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ized
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ata
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aly
s
is
will
ev
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ally
f
in
d
an
ef
f
ec
t
s
ize
(
)
.
T
o
m
in
im
ize
th
e
b
ias
o
f
th
e
e
f
f
ec
t
s
ize
ca
lcu
latio
n
r
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lts
,
t
h
e
d
a
ta
is
tr
an
s
f
o
r
m
ed
[
4
2
]
.
T
h
e
an
aly
s
is
an
d
ca
lcu
latio
n
p
r
o
ce
s
s
ar
e
ass
is
ted
b
y
th
e
m
eta
an
d
m
eta
p
h
o
r
p
ac
k
a
g
es
in
R
s
o
f
twar
e.
T
h
e
r
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lts
o
f
th
e
ef
f
ec
t
s
ize
f
in
d
in
g
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ca
n
b
e
class
if
ied
as
i
)
n
o
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f
ec
t
f
o
r
E
S
b
el
o
w
0
.
1
9
,
ii
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s
m
all
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ec
t
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o
r
E
S
b
etwe
en
0
.
2
0
to
0
.
4
9
,
iii
)
m
ed
i
u
m
ef
f
ec
t
f
o
r
E
S
b
etwe
en
0
.
5
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9
,
iv
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la
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g
e
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ec
t f
o
r
E
S b
etwe
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0
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8
0
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2
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,
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d
v
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er
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e
e
f
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ec
t f
o
r
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S a
b
o
v
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1
.
3
0
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2
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5
.
M
o
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na
ly
s
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Mo
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aly
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s
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u
p
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f
o
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m
atio
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ter
p
r
etin
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th
e
f
in
d
in
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s
o
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th
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tal
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ize
ef
f
ec
t
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elate
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to
t
h
e
in
f
lu
en
ce
o
f
v
ir
t
u
al
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ep
r
esen
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in
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ath
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o
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en
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m
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r
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les,
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ely
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tu
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m
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ev
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s
ed
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d
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a
o
f
v
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al
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im
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lem
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tatio
n
.
An
aly
s
is
o
f
m
o
d
e
r
ato
r
v
ar
i
ab
les
u
tili
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s
an
an
aly
s
is
o
f
v
ar
ian
ce
m
o
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el.
T
h
e
r
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lts
o
f
th
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is
o
f
v
ar
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ce
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n
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e
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o
r
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t th
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in
ca
teg
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o
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le.
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th
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ata
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d
ar
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,
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at
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ased
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r
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f
ile,
it
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m
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at
th
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g
en
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s
s
o
th
at
th
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s
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r
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m
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ased
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u
s
t b
e
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u
p
p
o
r
ted
b
y
s
tatis
tical
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aly
s
is
.
3
.
1
.
P
ro
v
ing
t
he
a
s
s
um
ptio
n o
f
h
et
er
o
g
eneit
y
T
h
e
ass
u
m
p
tio
n
o
f
s
tatis
tical
h
eter
o
g
en
eity
ca
n
b
e
p
r
o
v
e
n
b
y
th
e
p
ar
am
eter
s
,
Ʈ
2
,
an
d
2
.
T
h
e
an
aly
s
is
r
esu
lts
s
h
o
w
th
at
th
e
v
alu
e
o
f
Ʈ
2
with
an
er
r
o
r
to
ler
an
ce
o
f
5
%
is
1
.
1
4
6
6
[
0
.
8
3
9
2
;
2
.
0
3
5
9
]
.
B
ased
o
n
th
e
r
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lts
o
f
th
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aly
s
is
,
Ʈ
2
>
0
,
wh
ich
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ea
n
s
th
at
th
e
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ata
is
h
eter
o
g
en
eo
u
s
.
T
h
u
s
,
b
ased
o
n
th
e
an
aly
s
is
m
o
d
el
Ʈ
2
s
h
o
ws
th
at
t
h
e
ass
u
m
p
tio
n
o
f
h
eter
o
g
e
n
eo
u
s
d
ata
i
s
p
r
o
v
e
n
.
T
h
e
r
esu
lts
o
f
t
h
e
a
n
aly
s
is
with
th
e
s
tatis
t
ical
m
o
d
el
2
with
an
er
r
o
r
to
ler
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ce
o
f
5
%
s
h
o
w
a
v
al
u
e
o
f
9
0
.
1
%
[
8
7
.
9
%;
9
1
.
9
%].
W
h
en
co
m
p
ar
e
d
with
th
e
class
if
icatio
n
o
f
d
ata
h
eter
o
g
en
eity
,
th
e
d
ata
in
th
is
s
tu
d
y
ar
e
in
clu
d
e
d
in
th
e
s
u
b
s
tan
tial
h
eter
o
g
en
e
o
u
s
ca
teg
o
r
y
b
ec
a
u
s
e
th
ey
ar
e
i
n
th
e
r
a
n
g
e
o
f
7
5
%
to
1
0
0
%.
T
h
e
n
b
ased
o
n
th
e
Q
p
a
r
am
eter
an
aly
s
is
,
it
is
f
o
u
n
d
th
at
t
h
e
p
-
v
alu
e
is
v
e
r
y
s
m
all,
wh
ich
is
c
lo
s
e
to
ze
r
o
.
B
ased
o
n
th
e
r
esu
lts
o
f
th
is
a
n
aly
s
is
,
it
is
p
r
o
v
en
t
h
at
th
e
d
ata
is
h
eter
o
n
.
B
ased
o
n
th
e
t
h
r
ee
p
r
o
o
f
s
o
f
ass
u
m
p
tio
n
s
,
t
h
e
d
ata
is
p
r
o
v
en
to
b
e
s
tatis
t
ically
h
eter
o
g
en
eo
u
s
s
o
th
at
th
e
r
an
d
o
m
e
f
f
ec
t m
o
d
el
s
elec
tio
n
in
th
e
m
eta
-
a
n
aly
s
is
is
ap
p
r
o
p
r
iate.
3
.
2
.
P
ro
o
f
o
f
f
re
edo
m
f
r
o
m
bia
s
e
d pub
l
ica
t
io
n
Data
f
r
ee
d
o
m
f
r
o
m
b
ias
p
u
b
licatio
n
is
an
o
th
er
th
in
g
th
at
m
u
s
t
b
e
p
r
o
v
en
.
T
h
is
p
r
o
o
f
is
u
s
ed
to
g
u
ar
an
tee
th
at
th
e
d
ata
an
d
a
n
aly
s
is
r
esu
lts
ca
n
b
e
tr
u
s
ted
.
Pro
o
f
o
f
f
r
ee
d
o
m
f
r
o
m
p
u
b
li
ca
tio
n
b
ias
u
s
es
th
e
in
ter
p
r
etatio
n
o
f
th
e
f
u
n
n
el
p
l
o
t
an
d
co
m
p
ar
es
th
e
Fail
-
Saf
e
v
al
u
e
with
t
h
e
m
in
im
u
m
s
ta
n
d
ar
d
.
Fig
u
r
e
2
is
an
im
ag
e
o
f
th
e
f
u
n
n
el
p
lo
t.
B
ased
o
n
th
e
f
u
n
n
el
p
lo
t,
th
e
d
is
tr
ib
u
tio
n
o
f
p
o
in
ts
ca
n
n
o
t
s
h
o
w
p
er
f
ec
t
s
y
m
m
et
r
y
.
T
h
u
s
,
b
ased
o
n
th
e
f
ig
u
r
e,
t
h
e
in
ter
p
r
etatio
n
r
esu
lts
ca
n
b
e
d
is
p
u
te
d
.
B
ased
o
n
th
is
,
it
is
n
ec
ess
ar
y
to
co
n
d
u
ct
a
f
ail
-
s
af
e
-
b
ased
p
r
o
o
f
to
d
em
o
n
s
tr
ate
f
r
ee
d
o
m
f
r
o
m
p
u
b
licatio
n
b
ias.
T
h
e
f
ail
-
s
af
e
ca
lcu
latio
n
r
esu
lt
s
in
1
6
,
7
1
4
(
5
%
co
n
f
id
en
ce
in
ter
v
al)
.
T
h
e
v
al
u
e
lim
it
to
b
e
s
aid
t
o
b
e
f
r
ee
f
r
o
m
p
u
b
licatio
n
b
ias
is
N
>
5
+1
with
th
e
in
f
o
r
m
atio
n
N
is
th
e
f
ail
-
s
af
e
v
alu
e
a
n
d
is
th
e
am
o
u
n
t
o
f
d
ata.
T
h
is
r
esear
ch
in
v
o
lv
es
5
4
d
ata
s
o
th
a
t
5
+1
=2
7
1
s
o
t
h
at
th
e
v
alu
e
is
f
ar
b
elo
w
1
6
7
1
4
.
B
ased
o
n
th
e
f
ail
-
s
af
e
m
eth
o
d
,
it
ca
n
b
e
c
o
n
clu
d
ed
th
at
th
e
d
ata
is
f
r
ee
f
r
o
m
p
u
b
licatio
n
b
ias.
T
h
is
m
ea
n
s
th
at
th
e
d
ata
a
n
d
r
esu
lts
o
f
m
eta
-
an
aly
s
is
r
elate
d
to
th
e
im
p
ac
t
o
f
v
ir
tu
al
r
ep
r
esen
tatio
n
i
n
h
el
p
in
g
s
tu
d
en
ts
lear
n
m
ath
em
ati
cs in
th
is
s
tu
d
y
ca
n
b
e
t
r
u
s
ted
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8
7
0
8
I
n
t J E
lec
&
C
o
m
p
E
n
g
,
Vo
l.
14
,
No
.
6
,
Dec
em
b
e
r
20
24
:
6
4
1
1
-
6
4
2
2
6416
Fig
u
r
e
2
.
Fu
n
n
el
p
lo
t
3
.
3
.
T
o
t
a
l
ef
f
ec
t
s
ize
Af
ter
th
e
ass
u
m
p
tio
n
o
f
h
eter
o
g
en
eity
was
p
r
o
v
en
as
th
e
b
asis
f
o
r
th
e
co
r
r
ec
t
s
elec
tio
n
o
f
th
e
r
an
d
o
m
e
f
f
ec
t
m
o
d
el
an
d
f
r
ee
d
o
m
f
r
o
m
p
u
b
licatio
n
b
ias
was
also
p
r
o
v
en
,
th
e
an
aly
s
is
en
ter
ed
th
e
s
tag
e
o
f
ca
lcu
latin
g
th
e
ef
f
ec
t
s
ize
o
f
ea
ch
s
tu
d
y
an
d
th
e
t
o
tal
ef
f
e
ct
s
ize.
T
h
e
m
eta
-
an
aly
s
is
s
h
o
wed
th
at
th
e
to
tal
ef
f
ec
t size
with
th
e
r
an
d
o
m
ef
f
ec
t m
o
d
el
was 1
.
1
7
6
1
[
0
.
8
8
0
2
; 1
.
4
7
1
9
]
with
a
p
-
v
alu
e
<
0
.
0
0
1
(
9
5
% c
o
n
f
id
en
ce
in
ter
v
al)
.
T
h
ese
r
esu
lts
ca
n
b
e
tr
an
s
lated
th
at
t
h
er
e
is
a
s
ig
n
if
ican
t
ef
f
ec
t
o
f
v
ir
tu
al
r
ep
r
es
en
tatio
n
in
h
elp
in
g
s
tu
d
en
ts
lear
n
m
ath
em
atics.
T
h
e
ef
f
ec
t
s
ize
o
f
1
.
1
7
6
1
ca
n
b
e
ca
teg
o
r
ized
as
h
av
in
g
a
lar
g
e
ef
f
ec
t
[
3
9
]
.
T
h
e
r
esu
lts
o
f
th
e
an
aly
s
is
o
f
ea
ch
s
tu
d
y
an
d
t
h
e
to
tal
ca
n
b
e
s
ee
n
in
th
e
f
o
r
est p
lo
t sh
o
wn
in
Fig
u
r
e
3
.
Fig
u
r
e
3
.
Fo
r
est
p
l
o
t
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J E
lec
&
C
o
m
p
E
n
g
I
SS
N:
2088
-
8
7
0
8
Th
e
p
o
ten
tia
l o
f v
ir
tu
a
l rep
r
esen
ta
tio
n
s
to
h
elp
s
tu
d
e
n
ts
in
lea
r
n
in
g
ma
th
ema
tics
(
Yu
r
iz
ka
Melia
S
a
r
i
)
6417
Vir
tu
al
r
ep
r
esen
tatio
n
h
as
g
r
ea
t
p
o
ten
tial
to
f
ac
ilit
ate
th
e
lear
n
in
g
p
r
o
ce
s
s
o
f
m
ath
e
m
atics
f
o
r
s
tu
d
en
ts
.
Var
io
u
s
tec
h
n
o
lo
g
ie
s
s
u
ch
as
Ge
o
Geb
r
a
s
o
f
twar
e,
3
D
m
o
d
els,
an
d
Go
o
g
le
Sit
e
ap
p
licatio
n
s
ca
n
b
e
u
s
ed
to
ex
p
l
o
r
e
m
ath
e
m
atica
l
co
n
ce
p
ts
v
is
u
ally
an
d
in
ter
ac
ti
v
ely
.
Fo
r
e
x
am
p
le,
Ge
o
Geb
r
a
allo
ws
th
e
cr
ea
tio
n
o
f
im
ag
es
o
r
v
is
u
aliza
tio
n
s
th
at
ca
n
h
elp
s
tu
d
en
ts
u
n
d
er
s
tan
d
m
ath
em
atica
l
p
r
o
b
lem
s
m
o
r
e
clea
r
ly
,
f
ac
ilit
ate
p
r
o
b
lem
s
o
lv
in
g
,
an
d
m
a
k
e
m
ath
em
atica
l
co
n
ce
p
ts
m
o
r
e
ac
ce
s
s
ib
le
[
4
3
]
.
T
h
e
u
s
e
o
f
3
D
m
o
d
els
in
b
u
ild
in
g
s
p
ac
e
lear
n
in
g
also
allo
ws
s
t
u
d
en
ts
to
ex
p
lo
r
e
c
o
n
ce
p
ts
in
r
ea
l
tim
e,
wh
ile
th
e
Go
o
g
le
Sit
e
ap
p
licatio
n
ca
n
in
teg
r
ate
v
is
u
al
r
ep
r
esen
tatio
n
s
o
f
m
ath
em
atica
l
m
o
d
els
with
tech
n
o
lo
g
y
,
cr
ea
tin
g
a
m
o
r
e
in
ter
esti
n
g
an
d
en
jo
y
ab
le
lear
n
in
g
e
x
p
er
ie
n
c
e
[
2
0
]
.
W
ith
th
is
ap
p
r
o
ac
h
,
s
tu
d
en
ts
ca
n
d
ev
el
o
p
a
d
ee
p
er
u
n
d
er
s
tan
d
in
g
o
f
m
ath
em
atica
l
co
n
ce
p
ts
th
r
o
u
g
h
v
is
u
al
r
ep
r
esen
tatio
n
s
th
at
p
r
o
v
id
e
a
r
ea
l
an
d
co
n
cr
ete
p
ict
u
r
e
o
f
th
e
lear
n
in
g
m
ater
ial.
3
.
4
.
M
o
dera
t
o
r
v
a
ria
bles
a
na
ly
s
is
B
ased
o
n
th
e
to
tal
ef
f
ec
t
s
i
ze
an
aly
s
is
,
it
s
h
o
ws
th
at
th
er
e
is
a
p
o
s
itiv
e
in
f
l
u
en
ce
o
f
v
ir
t
u
al
r
ep
r
esen
tatio
n
in
m
ak
in
g
it
ea
s
ier
f
o
r
s
tu
d
en
ts
to
lear
n
m
ath
em
atics.
T
h
e
f
in
d
i
n
g
s
wer
e
d
ee
p
en
e
d
b
y
an
aly
zin
g
m
o
d
er
ato
r
v
ar
iab
les
to
g
et
m
o
r
e
d
etailed
f
in
d
in
g
s
.
T
h
e
v
ar
iab
les an
aly
ze
d
wer
e
t
h
e
co
n
tin
en
t w
h
er
e
th
e
r
esear
ch
was
co
n
d
u
cted
(
Af
r
ica,
Asi
a,
an
d
E
u
r
o
p
e)
,
th
e
n
u
m
b
er
o
f
s
tu
d
e
n
ts
in
o
n
e
class
r
ep
r
esen
ted
b
y
th
e
n
u
m
b
e
r
o
f
r
esear
c
h
s
am
p
l
es
(
s
m
all,
m
ed
iu
m
,
an
d
lar
g
e)
,
th
e
th
em
e
o
f
t
h
e
m
ath
em
ati
cs
m
ater
ial
s
tu
d
ied
(
g
eo
m
etr
y
,
ca
lcu
lu
s
,
tr
ig
o
n
o
m
etr
y
,
an
d
alg
eb
r
a)
,
th
e
lev
el
o
f
s
tu
d
en
ts
wh
o
r
ec
eiv
ed
th
e
tr
ea
tm
en
t
(
ju
n
io
r
h
ig
h
s
ch
o
o
l,
h
ig
h
s
ch
o
o
l,
an
d
u
n
i
v
er
s
ity
)
,
co
m
p
ete
n
cies
tar
g
eted
in
lear
n
in
g
(
attitu
d
e,
k
n
o
wled
g
e,
an
d
s
k
ills
)
,
ap
p
licatio
n
s
u
s
ed
to
d
is
p
lay
r
ep
r
esen
tatio
n
s
(
Geo
Geb
r
a
,
C
ad
r
i,
Sk
etch
p
ad
,
Geo
H
ep
ta,
an
d
Gr
ap
h
n
g
C
alcu
lato
r
)
,
an
d
d
ev
ices
u
s
ed
in
th
e
lear
n
in
g
p
r
o
ce
s
s
(
co
m
p
u
ter
s
,
s
m
ar
tp
h
o
n
es,
an
d
ca
lcu
l
ato
r
s
)
.
Fig
u
r
e
4
is
a
f
o
r
est p
lo
t o
f
th
e
m
o
d
er
ato
r
v
a
r
iab
le
an
aly
s
is
r
esu
lts
.
Fig
u
r
e
4
.
I
n
ter
ac
tio
n
a
n
d
f
o
r
es
t p
lo
t o
f
m
o
d
er
at
o
r
v
a
r
iab
les
B
ased
o
n
th
e
d
ata,
it
ap
p
ea
r
s
t
h
at
th
e
u
tili
za
tio
n
o
f
v
ir
tu
al
r
e
p
r
esen
tatio
n
in
m
ath
e
m
atics
lear
n
in
g
is
m
o
s
tly
d
o
n
e
in
Asi
a.
M
o
r
e
t
h
an
5
0
%
o
f
th
e
d
ata
co
m
es
f
r
o
m
Asi
a.
T
h
is
is
d
u
e
t
o
th
e
r
ap
id
g
r
o
wth
o
f
tech
n
o
lo
g
y
an
d
th
e
a
d
o
p
tio
n
o
f
in
n
o
v
atio
n
in
Asi
an
co
u
n
t
r
ies.
C
o
u
n
tr
ies
s
u
ch
as
J
ap
an
,
So
u
th
Ko
r
ea
an
d
C
h
in
a
h
av
e
b
ec
o
m
e
ce
n
ter
s
o
f
tech
n
o
lo
g
ical
in
n
o
v
atio
n
,
in
clu
d
in
g
in
th
e
d
ev
el
o
p
m
e
n
t
o
f
m
ath
em
atics
ap
p
licatio
n
s
th
at
u
tili
ze
v
i
r
tu
a
l
r
ep
r
esen
tatio
n
s
[
4
4
]
.
T
h
is
s
u
g
g
ests
th
at
tech
n
o
lo
g
ical
ad
v
a
n
ce
s
in
Asi
a
h
av
e
en
co
u
r
a
g
ed
t
h
e
u
s
e
o
f
v
ir
tu
al
r
ep
r
esen
tatio
n
s
in
m
at
h
em
ati
cs
lear
n
in
g
.
I
n
ad
d
itio
n
,
e
d
u
c
atio
n
al
p
o
licies
in
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8
7
0
8
I
n
t J E
lec
&
C
o
m
p
E
n
g
,
Vo
l.
14
,
No
.
6
,
Dec
em
b
e
r
20
24
:
6
4
1
1
-
6
4
2
2
6418
s
o
m
e
Asi
an
co
u
n
tr
ies
also
s
u
p
p
o
r
t
th
e
in
te
g
r
atio
n
o
f
tech
n
o
l
o
g
y
in
t
h
e
m
ath
em
atics
cu
r
r
icu
lu
m
.
Fo
r
e
x
am
p
le,
th
e
u
s
e
o
f
v
ir
tu
al
r
ea
lity
an
d
d
ig
ital m
ath
to
o
ls
ca
n
b
e
in
teg
r
ated
in
an
ef
f
o
r
t to
im
p
r
o
v
e
m
ath
em
atica
l c
o
n
ce
p
t
u
n
d
er
s
tan
d
i
n
g
s
k
ills
[
4
4
]
–
[
4
6
]
.
Ho
wev
er
,
wh
en
v
iewe
d
f
r
o
m
th
e
ef
f
ec
t,
th
e
p
o
s
itiv
e
ef
f
ec
t
o
f
v
ir
tu
al
r
ep
r
esen
tatio
n
o
n
v
ar
io
u
s
co
n
tin
en
ts
d
o
es
n
o
t
s
h
o
w
an
y
s
ig
n
if
ican
t
d
if
f
er
en
ce
.
T
h
e
co
n
cl
u
s
io
n
is
b
ased
o
n
th
e
p
-
v
alu
e
=
0
.
2
5
>
0
.
0
5
(
9
5
%
co
n
f
id
en
ce
in
ter
v
al)
.
T
h
u
s
,
v
ir
tu
al
r
ep
r
esen
tatio
n
ca
n
b
e
r
ec
o
m
m
en
d
ed
to
b
e
im
p
lem
en
ted
in
all
lo
ca
tio
n
s
.
T
h
is
is
b
ased
o
n
th
e
ch
ar
ac
ter
is
tics
o
f
m
ath
em
atics lea
r
n
in
g
wh
ich
h
as u
n
iv
er
s
al
p
r
in
cip
les
an
d
ca
n
b
e
a
p
p
lied
eq
u
ally
in
v
ar
io
u
s
g
lo
b
al
co
n
t
ex
ts
.
I
n
a
d
d
itio
n
,
th
e
c
h
ar
ac
ter
is
tics
o
f
s
tu
d
en
ts
in
d
if
f
er
en
t
co
n
tin
en
ts
ca
n
also
b
e
s
im
ilar
in
r
esp
o
n
s
e
to
ce
r
t
ain
lear
n
in
g
m
et
h
o
d
s
,
esp
ec
ia
lly
with
ad
ap
tatio
n
an
d
p
er
s
o
n
aliza
tio
n
in
th
e
u
s
e
o
f
v
ir
tu
al
r
ep
r
esen
tatio
n
s
.
W
h
ile
cu
ltu
r
al
an
d
co
n
tex
t
u
al
asp
ec
ts
r
em
ain
im
p
o
r
tan
t,
th
ese
r
esu
lts
p
r
o
v
i
d
e
a
r
atio
n
al
b
asis
f
o
r
r
ec
o
m
m
en
d
in
g
th
e
u
s
e
o
f
v
ir
tu
al
r
e
p
r
esen
tatio
n
s
in
all
lo
ca
tio
n
s
in
an
ef
f
o
r
t to
im
p
r
o
v
e
m
ath
em
atics lea
r
n
in
g
g
lo
b
a
lly
.
T
h
e
s
ec
o
n
d
m
o
d
e
r
ato
r
v
ar
ia
b
le
is
s
am
p
le
s
ize.
T
h
is
v
ar
iab
le
s
h
o
ws
th
e
n
u
m
b
er
o
f
s
tu
d
e
n
ts
in
o
n
e
tr
ea
tm
en
t.
T
h
e
r
esu
lt
o
f
th
e
m
o
d
er
ato
r
v
ar
iab
le
an
aly
s
is
s
h
o
ws
th
at
th
e
p
-
v
alu
e
=
0
.
0
6
>
0
.
0
5
(
9
5
%
co
n
f
id
e
n
ce
in
ter
v
al)
s
o
th
er
e
is
n
o
s
ig
n
if
i
ca
n
t
d
if
f
e
r
en
ce
in
t
h
e
ef
f
ec
t
o
f
u
s
in
g
v
ir
tu
al
r
ep
r
esen
tatio
n
in
s
m
all,
m
ed
i
u
m
,
an
d
lar
g
e
s
am
p
le
s
izes.
T
h
is
r
atio
ca
n
b
e
attr
ib
u
ted
to
th
e
f
ac
t
th
at
th
e
u
s
e
o
f
v
ir
tu
al
r
ep
r
esen
tatio
n
in
m
ath
em
atics
lear
n
in
g
h
as
a
ten
d
en
cy
to
in
cr
ea
s
e
lea
r
n
in
g
in
d
e
p
en
d
e
n
ce
[
4
7
]
,
[
4
8
]
.
W
ith
v
ir
tu
al
r
ep
r
esen
tatio
n
s
,
s
tu
d
en
ts
ca
n
en
g
ag
e
in
lea
r
n
in
g
ac
tiv
ities
u
s
in
g
th
eir
o
wn
d
ev
ices,
r
eg
a
r
d
less
o
f
th
e
n
u
m
b
e
r
o
f
s
tu
d
en
ts
in
a
lear
n
in
g
s
ess
io
n
[
4
9
]
.
I
n
th
is
co
n
tex
t,
th
e
in
s
ig
n
if
ican
t
d
if
f
er
e
n
ce
in
ef
f
ec
t
ac
r
o
s
s
d
if
f
er
en
t
s
am
p
le
s
izes
m
ay
b
e
d
u
e
t
o
t
h
e
ad
ap
tiv
e
n
atu
r
e
o
f
u
s
in
g
v
i
r
tu
al
r
ep
r
esen
tatio
n
s
.
Stu
d
en
ts
,
wh
eth
er
in
s
m
all,
m
ed
iu
m
o
r
lar
g
e
g
r
o
u
p
s
,
ca
n
ef
f
ec
tiv
ely
en
g
ag
e
in
m
ath
e
m
atics
lear
n
in
g
ac
tiv
ities
wit
h
th
e
s
am
e
lev
el
o
f
in
d
ep
en
d
en
ce
,
th
a
n
k
s
to
t
h
e
ad
v
an
tag
es
o
f
v
ir
tu
al
r
ep
r
ese
n
tatio
n
tech
n
o
lo
g
y
th
at
s
u
p
p
o
r
ts
in
d
iv
id
u
alize
d
lear
n
in
g
.
Ma
th
em
atics
h
as
m
an
y
lear
n
in
g
th
em
es.
B
ased
o
n
th
e
d
ata
co
llected
,
th
er
e
ar
e
4
th
em
es
th
at
ar
e
s
tu
d
ied
b
y
u
tili
zin
g
v
ir
tu
al
r
e
p
r
esen
tatio
n
,
n
am
ely
g
eo
m
etr
y
,
ca
lcu
lu
s
,
tr
ig
o
n
o
m
etr
y
,
an
d
alg
eb
r
a.
Alth
o
u
g
h
th
ey
h
av
e
d
if
f
er
e
n
t
ch
ar
ac
ter
s
b
etwe
en
th
em
es
in
m
ath
em
at
ics,
th
ey
ar
e
in
ter
r
elate
d
with
ea
ch
o
th
er
.
Am
o
n
g
th
e
f
o
u
r
th
em
es,
th
e
g
eo
m
etr
y
th
em
e
is
th
e
m
o
s
t
co
m
m
o
n
th
em
e
in
m
ath
em
atics
[
5
0
]
.
T
h
e
u
s
e
o
f
v
ir
tu
al
r
ep
r
esen
tatio
n
in
lear
n
in
g
m
ath
em
atics
is
d
o
m
in
ated
b
y
g
eo
m
etr
y
b
ec
au
s
e
it
is
in
ac
co
r
d
an
ce
with
its
n
atu
r
e.
I
n
th
is
co
n
tex
t,
v
ir
tu
al
r
e
p
r
es
en
tatio
n
is
m
ain
ly
f
o
cu
s
ed
o
n
th
e
u
s
e
o
f
s
im
u
latio
n
s
to
v
is
u
alize
g
eo
m
etr
ic
o
b
jects
an
d
co
n
ce
p
ts
in
2
D
an
d
3
D
[
5
1
]
.
Geo
m
etr
y
p
r
o
v
id
es
ad
v
an
tag
es
in
th
e
f
o
r
m
o
f
c
o
n
cr
ete
v
is
u
aliza
tio
n
o
f
s
p
ac
e,
s
o
g
eo
m
etr
ic
o
b
ject
s
s
u
ch
as
tr
ian
g
les
an
d
cir
cle
s
ca
n
b
e
clea
r
ly
s
h
o
wn
in
a
v
ir
tu
al
en
v
ir
o
n
m
en
t
[
1
2
]
.
T
h
e
co
n
ce
p
t
o
f
g
eo
m
et
r
y
also
o
p
e
n
s
u
p
o
p
p
o
r
tu
n
iti
es
f
o
r
in
ten
s
e
in
ter
ac
tio
n
,
allo
win
g
s
tu
d
e
n
ts
to
d
ir
ec
tly
ex
p
lo
r
e
g
e
o
m
etr
ic
o
b
jects
an
d
o
b
s
er
v
e
ch
a
n
g
es
in
p
ar
am
eter
s
.
I
n
ad
d
itio
n
,
th
e
s
im
u
latio
n
o
f
g
eo
m
etr
ic
ev
en
ts
s
u
ch
as
s
h
if
ts
,
r
o
tatio
n
s
,
an
d
s
ca
le
ch
an
g
es
b
ec
o
m
e
r
elev
a
n
t
in
v
ir
t
u
al
r
ep
r
esen
tatio
n
,
m
ak
in
g
it
an
ef
f
ec
tiv
e
to
o
l
f
o
r
u
n
d
er
s
tan
d
in
g
th
ese
co
n
c
ep
ts
.
I
n
th
e
s
im
u
latio
n
o
f
2
D
an
d
3
D
o
b
jects,
g
eo
m
etr
y
p
r
o
v
id
es
th
e
b
asis
f
o
r
a
r
ich
an
d
im
m
er
s
iv
e
lear
n
i
n
g
ex
p
er
ien
ce
.
T
h
u
s
,
th
e
u
s
e
o
f
g
eo
m
etr
y
th
e
m
es
in
v
ir
tu
al
r
ep
r
esen
tatio
n
all
o
ws
f
o
r
m
o
r
e
r
ea
l,
i
n
ter
ac
t
iv
e
lear
n
in
g
o
f
m
ath
e
m
atics
an
d
d
ee
p
e
n
s
th
e
u
n
d
er
s
tan
d
i
n
g
o
f
g
eo
m
etr
ic
co
n
ce
p
ts
.
T
h
e
r
esu
lts
o
f
th
e
an
aly
s
is
s
h
o
w
th
at
th
e
p
-
v
alu
e=
0
.
0
6
>
0
.
0
5
(
9
5
%
co
n
f
id
en
ce
in
ter
v
al
)
s
o
th
at
t
h
er
e
is
n
o
s
ig
n
if
ican
t
d
if
f
er
en
ce
in
th
e
ef
f
ec
t
o
f
u
s
in
g
v
ir
t
u
al
r
ep
r
esen
tatio
n
o
n
th
e
th
em
es
o
f
g
eo
m
etr
y
,
ca
lc
u
lu
s
,
tr
ig
o
n
o
m
et
r
y
,
an
d
alg
eb
r
a.
T
h
is
is
b
ec
au
s
e
m
ath
em
atics
h
as
a
n
at
u
r
e
as
an
ab
s
tr
ac
t
s
cien
c
e,
wh
er
e
th
e
u
s
e
o
f
v
ir
tu
al
r
ep
r
esen
tatio
n
aim
s
to
f
ac
ilit
ate
th
e
u
n
d
er
s
tan
d
in
g
o
f
th
ese
ab
s
tr
ac
t
co
n
ce
p
ts
b
y
s
tu
d
en
ts
wh
o
ar
e
lear
n
in
g
m
at
h
em
atics
[
5
2
]
,
[
5
3
]
.
I
n
th
is
co
n
tex
t
,
th
e
in
s
ig
n
i
f
ican
t
d
if
f
er
e
n
ce
b
etwe
en
m
ath
em
atica
l
th
em
es
in
th
e
in
f
lu
en
ce
o
f
v
ir
tu
al
r
ep
r
es
en
tatio
n
s
ca
n
b
e
ex
p
lain
ed
b
y
th
e
f
ac
t
th
at
th
e
ab
s
tr
ac
t
n
atu
r
e
o
f
m
ath
em
atics
is
u
n
if
o
r
m
ly
r
ef
lecte
d
th
r
o
u
g
h
v
i
s
u
al
r
ep
r
esen
tatio
n
s
,
r
esu
ltin
g
in
u
n
if
o
r
m
an
al
y
s
is
r
esu
lts
.
I
n
its
im
p
lem
e
n
tatio
n
,
v
ir
tu
al
r
ep
r
esen
tatio
n
h
elp
s
f
ac
ilit
ate
th
e
lear
n
in
g
o
f
m
ath
em
atics
at
v
ar
i
o
u
s
lev
els.
T
h
e
an
aly
s
is
s
h
o
wed
p
-
v
alu
e=
0
.
1
1
>
0
.
0
5
(
9
5
%
co
n
f
id
en
ce
in
ter
v
al)
.
T
h
e
m
ea
n
i
n
g
o
f
th
e
a
n
aly
s
is
r
esu
lts
is
th
at
v
ir
tu
al
r
ep
r
esen
tatio
n
p
r
o
v
id
es
e
q
u
ally
g
o
o
d
ass
is
tan
ce
to
s
tu
d
en
ts
in
lear
n
in
g
m
ath
e
m
atics
at
ju
n
io
r
h
ig
h
s
ch
o
o
l,
h
ig
h
s
ch
o
o
l
an
d
u
n
iv
er
s
ity
lev
els.
At
t
h
e
ju
n
io
r
h
ig
h
s
ch
o
o
l
lev
el,
m
ath
em
atica
l
m
ater
ials
ar
e
g
en
e
r
ally
m
o
r
e
c
o
n
cr
ete,
v
is
u
al
r
ep
r
esen
tatio
n
s
ca
n
h
elp
s
tu
d
en
ts
u
n
d
e
r
s
tan
d
co
n
ce
p
ts
b
etter
[
5
4
]
.
I
n
h
ig
h
s
ch
o
o
l,
th
e
lev
el
o
f
ab
s
tr
ac
tio
n
o
f
th
e
m
ater
ial
in
cr
ea
s
es,
b
u
t
v
ir
tu
al
r
ep
r
esen
tatio
n
r
em
ain
s
ef
f
ec
tiv
e
b
ec
au
s
e
it
is
ab
le
to
v
is
u
ally
r
ed
u
ce
th
e
co
m
p
le
x
ity
o
f
th
e
co
n
ce
p
t
[
5
5
]
.
At
th
e
u
n
iv
e
r
s
ity
lev
el,
wh
er
e
m
ath
em
atica
l
m
ater
ial
r
ea
ch
es
th
e
h
ig
h
est
l
ev
el
o
f
ab
s
tr
ac
tio
n
,
v
ir
tu
al
r
e
p
r
esen
tatio
n
s
r
em
ain
r
elev
a
n
t
b
y
p
r
o
v
id
in
g
v
is
u
al
im
ag
es
th
at
ass
i
s
t
s
tu
d
en
ts
in
u
n
d
er
s
tan
d
i
n
g
co
m
p
lex
co
n
ce
p
ts
.
T
h
u
s
,
th
e
u
n
if
o
r
m
ity
o
f
a
s
s
is
tan
ce
p
r
o
v
id
ed
b
y
v
ir
t
u
al
r
ep
r
esen
tatio
n
s
at
d
if
f
er
e
n
t
ed
u
ca
tio
n
al
lev
els
ca
n
b
e
ex
p
lain
e
d
th
r
o
u
g
h
th
eir
f
lex
ib
ilit
y
in
p
r
esen
tin
g
m
ath
em
atica
l
co
n
c
ep
ts
,
cr
ea
tin
g
a
lear
n
in
g
en
v
i
r
o
n
m
en
t
th
at
s
u
p
p
o
r
ts
s
tu
d
en
t
u
n
d
er
s
tan
d
in
g
at
ea
ch
lev
el
o
f
a
b
s
tr
ac
tio
n
.
I
n
a
lear
n
in
g
p
r
o
ce
s
s
,
th
e
co
m
p
eten
cy
d
o
m
ain
s
(
attitu
d
e,
k
n
o
wled
g
e
an
d
s
k
ills
)
s
h
o
u
ld
d
e
v
elo
p
in
a
b
alan
ce
d
m
an
n
er
.
T
h
u
s
,
teac
h
er
s
ar
e
co
n
s
id
er
ed
s
u
cc
ess
f
u
l
if
th
ey
ch
o
o
s
e
lear
n
in
g
m
eth
o
d
s
o
r
s
tr
ateg
ies
th
at
ca
n
d
ev
elo
p
all
th
r
ee
.
T
h
is
in
clu
d
es
th
e
im
p
lem
en
tatio
n
o
f
v
ir
tu
al
r
ep
r
esen
tatio
n
wh
ich
s
h
o
u
ld
b
e
a
b
le
to
Evaluation Warning : The document was created with Spire.PDF for Python.
I
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2088
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8
7
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Th
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p
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tia
l o
f v
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ta
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ma
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Yu
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ev
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lts
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s
is
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at
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p
-
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,
m
ea
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th
at
th
er
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n
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s
ig
n
if
ican
t
d
if
f
er
e
n
ce
in
th
e
ef
f
ec
t
o
f
v
ir
tu
al
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ep
r
esen
tatio
n
im
p
lem
e
n
tatio
n
o
n
th
e
d
e
v
elo
p
m
e
n
t
o
f
attitu
d
es,
k
n
o
wled
g
e,
a
n
d
s
k
ills
.
T
h
i
s
also
s
u
p
p
o
r
ts
th
at
lear
n
in
g
m
ath
e
m
atics u
s
in
g
v
i
r
tu
al
r
ep
r
esen
tatio
n
s
h
as si
g
n
if
ican
t b
en
ef
its
f
o
r
s
tu
d
e
n
t
m
o
ti
v
atio
n
[
5
6
]
.
On
e
o
f
th
e
b
en
ef
its
is
th
r
o
u
g
h
co
n
ce
p
t
v
is
u
aliza
tio
n
,
wh
ich
ca
n
m
a
k
e
lear
n
in
g
m
o
r
e
in
ter
esti
n
g
a
n
d
in
cr
ea
s
e
s
tu
d
e
n
t
en
g
ag
em
e
n
t.
I
n
ad
d
itio
n
,
v
ir
tu
al
r
ep
r
esen
tatio
n
s
ca
n
also
tr
an
s
f
o
r
m
ab
s
tr
ac
t
m
ath
em
atica
l
co
n
ce
p
ts
in
to
m
o
r
e
co
n
cr
ete
o
n
es,
th
u
s
h
el
p
in
g
s
t
u
d
en
ts
in
u
n
d
er
s
tan
d
i
n
g
c
o
m
p
lex
id
ea
s
[
1
3
]
.
Mo
r
eo
v
e
r
,
th
r
o
u
g
h
s
im
u
latio
n
an
d
in
ter
ac
tiv
ity
,
v
ir
tu
al
r
ep
r
esen
tatio
n
s
ca
n
also
d
ev
elo
p
s
tu
d
en
ts
'
p
r
ac
tical
s
k
il
ls
,
th
u
s
cr
ea
tin
g
a
h
o
lis
tic
lear
n
in
g
ex
p
er
ie
n
ce
[
5
3
]
.
Gi
v
en
th
ese
p
o
s
itiv
e
im
p
ac
ts
,
v
ir
tu
al
r
ep
r
esen
tatio
n
s
h
a
v
e
a
n
im
p
o
r
tan
t
r
o
le
in
im
p
r
o
v
in
g
s
tu
d
en
ts
'
m
o
tiv
atio
n
,
co
n
ce
p
t u
n
d
er
s
tan
d
in
g
,
an
d
s
k
ills
in
lear
n
in
g
m
ath
em
atics.
Am
o
n
g
th
e
s
o
f
twar
e
u
s
ed
i
n
th
e
im
p
lem
en
tatio
n
o
f
v
ir
tu
al
r
ep
r
esen
tatio
n
,
Ge
o
Geb
r
a
i
s
th
e
m
o
s
t
p
o
p
u
lar
s
o
f
twar
e.
6
8
.
5
%
o
f
th
e
d
ata
is
r
esear
ch
th
at
u
tili
ze
s
Geo
Geb
r
a
as
a
to
o
l
to
cr
ea
te
v
is
u
al
illu
s
tr
atio
n
s
.
Un
iq
u
ely
,
Geo
Ge
b
r
a
is
also
th
e
s
o
f
twar
e
th
at
h
as
th
e
b
est
in
f
lu
en
ce
co
m
p
ar
ed
to
o
th
er
s
o
f
twar
e.
Statis
tically
,
th
is
co
n
clu
s
io
n
is
b
ased
o
n
th
e
p
-
v
alu
e
<
0
.
0
0
1
<
0
.
0
5
(
9
5
%
co
n
f
id
en
ce
in
ter
v
al)
,
wh
ich
m
ea
n
s
th
at
th
er
e
is
a
s
ig
n
if
ican
t
d
if
f
er
e
n
ce
in
t
h
e
ef
f
ec
t
o
f
ap
p
licatio
n
s
o
n
th
e
ea
s
e
o
f
lear
n
i
n
g
m
at
h
em
atics.
T
h
is
co
n
clu
s
io
n
is
th
e
f
o
u
n
d
atio
n
f
o
r
id
e
n
tify
in
g
th
e
ef
f
ec
t
in
ea
ch
ca
teg
o
r
y
wh
ic
h
s
h
o
ws
th
at
Geo
Geb
r
a
is
th
e
s
o
f
twar
e
with
th
e
g
r
ea
test
ef
f
ec
t
co
m
p
ar
ed
to
o
th
er
s
o
f
twar
e.
Geo
Geb
r
a
h
as
ad
v
an
tag
es
in
s
ev
er
al
asp
ec
ts
th
at
d
is
tin
g
u
is
h
it
f
r
o
m
o
th
er
m
at
h
s
o
f
twar
e.
O
n
e
o
f
t
h
ese
ad
v
an
ta
g
es
is
th
e
ab
ilit
y
to
in
teg
r
ate
co
n
ce
p
ts
,
s
u
ch
as
g
eo
m
et
r
y
,
alg
eb
r
a,
a
n
d
ca
lcu
lu
s
,
wh
ic
h
g
iv
es
s
tu
d
en
ts
th
e
f
lex
ib
ilit
y
to
u
n
d
er
s
tan
d
an
d
ap
p
l
y
m
at
h
em
atica
l
co
n
ce
p
ts
h
o
lis
tically
[
5
7
]
.
W
ith
a
h
ig
h
lev
el
o
f
in
ter
ac
tiv
ity
a
n
d
d
y
n
am
ic
f
ea
tu
r
es,
Geo
Geb
r
a
allo
ws
s
tu
d
en
ts
to
in
ter
ac
t
d
ir
ec
tly
with
g
r
a
p
h
ic
v
is
u
aliza
tio
n
s
an
d
m
ath
em
atic
al
o
b
jects,
th
u
s
d
ee
p
en
i
n
g
co
n
ce
p
t
u
n
d
e
r
s
tan
d
in
g
[
5
8
]
.
T
h
e
d
is
co
v
er
y
lear
n
in
g
ap
p
r
o
ac
h
s
u
p
p
o
r
ted
b
y
Ge
o
Geb
r
a
also
p
r
o
v
id
es e
x
p
lo
r
atio
n
s
p
ac
e
f
o
r
s
tu
d
en
ts
in
ex
p
lo
r
in
g
m
ath
em
atica
l
co
n
c
ep
ts
in
d
ep
en
d
e
n
tly
,
wh
ich
in
tu
r
n
s
tim
u
lates
cr
ea
tiv
e
th
in
k
in
g
an
d
in
tu
itiv
e
u
n
d
er
s
tan
d
i
n
g
.
T
h
e
f
lex
i
b
ilit
y
o
f
u
s
in
g
th
is
s
o
f
twar
e
f
r
o
m
elem
en
tar
y
s
ch
o
o
l
to
co
ll
eg
e
lev
el
s
h
o
ws
it
s
ad
ap
tab
ilit
y
in
v
ar
io
u
s
lear
n
i
n
g
co
n
tex
ts
[
4
3
]
,
[
5
8
]
.
Mo
r
e
o
v
er
,
t
h
e
p
r
esen
ce
o
f
an
ac
tiv
e
co
m
m
u
n
ity
an
d
ab
u
n
d
a
n
t
lear
n
i
n
g
r
eso
u
r
ce
s
p
r
o
v
id
e
ad
d
itio
n
al
s
u
p
p
o
r
t,
m
ak
in
g
Geo
Ge
b
r
a
a
s
u
p
er
i
o
r
c
h
o
ice
in
im
p
r
o
v
in
g
s
tu
d
en
ts
'
co
n
ce
p
t u
n
d
er
s
tan
d
i
n
g
an
d
v
is
u
aliza
tio
n
s
k
ills
in
m
ath
em
atics lea
r
n
in
g
.
T
h
e
n
ex
t
v
ar
iab
le
an
aly
ze
d
is
th
e
d
ev
ice
u
s
ed
in
o
p
er
atin
g
t
h
e
s
o
f
twar
e.
T
h
is
v
ar
iab
le
is
i
m
p
o
r
tan
t
t
o
an
aly
ze
b
ec
au
s
e
it
ca
n
b
e
r
elat
ed
to
th
e
r
elev
a
n
ce
o
f
u
s
in
g
v
i
r
tu
al
r
ep
r
esen
tatio
n
t
o
m
ak
e
it
ea
s
ier
f
o
r
s
tu
d
en
ts
to
lear
n
m
ath
e
m
atics.
T
h
er
e
ar
e
th
r
ee
d
e
v
ices
id
en
tifie
d
a
s
b
ein
g
u
s
ed
,
n
am
ely
lap
t
o
p
s
,
s
m
ar
tp
h
o
n
es,
an
d
ca
lcu
lato
r
s
.
Am
o
n
g
th
e
th
r
e
e,
lap
to
p
/co
m
p
u
ter
is
th
e
d
ev
ic
e
th
at
h
as
th
e
g
r
ea
test
im
p
ac
t
am
o
n
g
o
th
er
s
.
T
h
is
co
n
clu
s
io
n
is
b
ased
o
n
th
e
p
-
v
alu
e
=
0
.
0
0
6
>
0
.
0
0
5
(
9
5
%
co
n
f
i
d
en
ce
in
ter
v
al)
.
T
h
is
m
ea
n
s
th
at
th
er
e
is
a
s
ig
n
if
ican
t
d
if
f
er
en
ce
in
th
e
im
p
ac
t
o
f
d
ev
ices
o
n
s
tu
d
en
ts
'
ea
s
e
o
f
lear
n
in
g
m
ath
em
atics.
T
h
e
an
aly
s
is
co
n
tin
u
ed
b
y
lo
o
k
in
g
at
th
e
ef
f
ec
t
v
alu
e.
B
ased
o
n
t
h
e
e
f
f
ec
t
v
alu
e
an
d
er
r
o
r
r
a
n
g
e,
th
e
m
o
s
t
ef
f
ec
tiv
e
lear
n
in
g
with
v
ir
tu
al
r
ep
r
esen
tatio
n
is
u
s
in
g
lap
to
p
.
T
h
e
u
s
e
o
f
v
ir
tu
al
r
ep
r
esen
tatio
n
o
n
lap
to
p
s
in
lear
n
in
g
h
as
s
ig
n
if
ican
t
ad
v
a
n
tag
es
co
m
p
ar
ed
t
o
th
e
u
s
e
o
f
o
th
er
d
e
v
ices
s
u
ch
as
tab
lets
o
r
s
m
ar
t
p
h
o
n
es.
O
n
e
o
f
th
e
m
ain
ad
v
an
tag
es
o
f
lap
to
p
s
is
th
eir
lar
g
e
r
s
cr
ee
n
,
wh
ic
h
allo
ws
f
o
r
m
o
r
e
d
etailed
an
d
clea
r
v
is
u
aliza
tio
n
[
5
9
]
.
T
h
is
ca
n
en
h
a
n
ce
s
tu
d
en
ts
'
v
is
u
al
ex
p
er
ien
ce
i
n
u
n
d
er
s
t
an
d
in
g
m
ath
em
atica
l
co
n
ce
p
t
s
.
I
n
ad
d
itio
n
,
th
e
k
ey
b
o
a
r
d
a
n
d
t
o
u
ch
p
ad
o
n
la
p
to
p
s
also
p
r
o
v
id
e
m
o
r
e
co
m
p
lete
an
d
p
r
ec
is
e
in
p
u
t
co
n
tr
o
l,
wh
ich
m
ak
es
it
ea
s
ier
to
in
ter
ac
t w
ith
m
ath
s
o
f
twar
e
[
5
9
]
.
4.
CO
NCLU
SI
O
N
T
h
e
r
esu
lts
o
f
th
is
s
tu
d
y
s
h
o
w
th
at
th
e
d
ata
co
llected
a
r
e
h
et
er
o
g
en
e
o
u
s
(
in
d
icatin
g
th
at
th
e
s
elec
tio
n
o
f
th
e
r
an
d
o
m
ef
f
ec
t
m
o
d
el
is
ap
p
r
o
p
r
iate)
an
d
f
r
ee
f
r
o
m
p
u
b
licatio
n
b
ias
(
in
d
icatin
g
th
a
t
th
e
r
esu
lts
o
f
th
e
an
aly
s
is
ca
n
b
e
tr
u
s
ted
)
.
T
h
e
r
esu
lts
o
f
th
e
an
aly
s
is
s
h
o
w
t
h
at
v
ir
tu
al
r
ep
r
esen
tatio
n
h
as
a
p
o
s
itiv
e
in
f
lu
en
ce
o
n
s
tu
d
e
n
ts
'
ea
s
e
o
f
lear
n
in
g
m
ath
em
atics.
T
h
e
ea
s
e
o
f
l
ea
r
n
in
g
is
m
an
if
ested
f
r
o
m
t
h
e
s
tu
d
en
t
lea
r
n
in
g
ac
h
iev
em
en
t
in
d
e
x
at
th
e
en
d
o
f
lear
n
i
n
g
.
Ov
e
r
all,
it
was
f
o
u
n
d
th
at
th
e
p
-
v
al
u
e
<
5
%
(
9
5
%
co
n
f
i
d
en
ce
in
ter
v
al)
an
d
th
e
to
tal
ef
f
ec
t
o
f
1
.
1
7
6
1
s
o
th
at
v
ir
tu
al
r
ep
r
es
en
tatio
n
h
as
a
s
ig
n
if
ican
t
in
f
lu
en
ce
an
d
f
alls
in
to
th
e
ca
teg
o
r
y
o
f
h
a
v
in
g
a
lar
g
e
ef
f
ec
t
in
in
f
lu
en
ci
n
g
m
ath
e
m
atics
lear
n
in
g
o
u
tco
m
es.
T
h
e
g
lo
b
al
co
n
cl
u
s
io
n
was
tr
an
s
lated
m
o
r
e
s
p
ec
if
ically
th
r
o
u
g
h
m
o
d
er
ato
r
v
ar
iab
le
s
.
T
h
e
an
aly
s
is
s
h
o
wed
th
at
i
)
im
p
lem
en
tatio
n
in
ea
ch
co
n
tin
en
t
s
h
o
wed
id
en
tic
al
p
o
s
itiv
e
ef
f
ec
ts
;
ii
)
th
e
n
u
m
b
er
o
f
s
tu
d
e
n
ts
s
u
b
jecte
d
to
th
e
tr
ea
tm
en
t
d
id
n
o
t
m
ak
e
a
d
if
f
er
e
n
ce
;
iii
)
th
e
m
es
in
m
ath
em
atics
wer
e
eq
u
ally
well
af
f
ec
ted
with
t
h
e
h
elp
o
f
v
ir
tu
al
r
ep
r
esen
tatio
n
;
iv
)
t
h
e
ef
f
ec
t
o
f
v
ir
tu
al
r
e
p
r
esen
tatio
n
in
ju
n
io
r
h
ig
h
s
ch
o
o
l,
h
ig
h
s
ch
o
o
l,
an
d
u
n
iv
er
s
ity
was
id
en
tical
;
v
)
th
e
d
ev
el
o
p
m
en
t
o
f
co
m
p
eten
cies
in
attitu
d
e
,
k
n
o
wled
g
e,
an
d
s
k
ills
was
eq
u
ally
g
o
o
d
;
vi
)
am
o
n
g
th
e
m
an
y
a
p
p
licatio
n
s
,
Geo
Geb
r
a
was
th
e
ap
p
licatio
n
t
h
at
h
ad
th
e
g
r
ea
test
im
p
ac
t
in
h
elp
in
g
s
tu
d
en
ts
u
n
d
er
s
tan
d
m
at
h
em
atics
s
u
b
ject
m
atter
;
an
d
v
ii
)
th
e
u
s
e
o
f
s
m
ar
tp
h
o
n
es
h
ad
a
g
r
ea
te
r
ef
f
ec
t
th
an
o
t
h
er
d
ev
ices
s
u
ch
as
co
m
p
u
ter
s
a
n
d
ca
lcu
lato
r
s
.
T
h
u
s
,
v
ir
tu
al
r
e
p
r
esen
tatio
n
lear
n
in
g
is
h
ig
h
l
y
r
ec
o
m
m
en
d
e
d
in
m
ath
em
atics
lear
n
in
g
.
T
o
p
r
o
d
u
ce
th
e
m
a
x
im
u
m
e
f
f
ec
t
in
u
n
d
er
s
tan
d
i
n
g
s
tu
d
en
ts
,
it
is
r
ec
o
m
m
en
d
e
d
to
u
s
e
m
o
b
ile
d
ev
ices a
n
d
Geo
Geb
r
a
s
o
f
twar
e.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
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N
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2
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8
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I
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&
C
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p
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g
,
Vo
l.
14
,
No
.
6
,
Dec
em
b
e
r
20
24
:
6
4
1
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6
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Un
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Su
r
ab
ay
a
an
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Natio
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R
esear
ch
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o
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Ag
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(
B
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o
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R
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d
o
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esia
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at
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m
o
s
t
o
f
th
e
au
th
o
r
s
.
RE
F
E
R
E
NC
E
S
[
1
]
M
.
M
a
r
si
g
i
t
e
t
a
l
.
,
“
C
o
n
st
r
u
c
t
i
n
g
ma
t
h
e
m
a
t
i
c
a
l
c
o
n
c
e
p
t
s
t
h
r
o
u
g
h
e
x
t
e
r
n
a
l
r
e
p
r
e
s
e
n
t
a
t
i
o
n
s
u
t
i
l
i
z
i
n
g
t
e
c
h
n
o
l
o
g
y
:
a
n
i
m
p
l
e
me
n
t
a
t
i
o
n
i
n
I
R
T
c
o
u
r
se
,
”
T
EM
J
o
u
r
n
a
l
,
v
o
l
.
9
,
n
o
.
1
,
p
p
.
3
1
7
–
3
2
6
,
2
0
2
0
,
d
o
i
:
1
0
.
1
8
4
2
1
/
TE
M
9
1
-
4
4
.
[
2
]
H
.
R
e
t
n
a
w
a
t
i
,
B
.
K
a
r
t
o
w
a
g
i
r
a
n
,
J.
A
r
l
i
n
w
i
b
o
w
o
,
a
n
d
E.
S
u
l
i
st
y
a
n
i
n
g
s
i
h
,
“
W
h
y
a
r
e
t
h
e
m
a
t
h
e
m
a
t
i
c
s
n
a
t
i
o
n
a
l
e
x
a
m
i
n
a
t
i
o
n
i
t
e
ms
d
i
f
f
i
c
u
l
t
a
n
d
w
h
a
t
i
s
t
e
a
c
h
e
r
s’
st
r
a
t
e
g
y
t
o
o
v
e
r
c
o
me
i
t
?
,
”
I
n
t
e
rn
a
t
i
o
n
a
l
J
o
u
rn
a
l
o
f
I
n
st
r
u
c
t
i
o
n
,
v
o
l
.
1
0
,
n
o
.
3
,
p
p
.
2
5
7
–
2
7
6
,
2
0
1
7
,
d
o
i
:
1
0
.
1
2
9
7
3
/
i
j
i
.
2
0
1
7
.
1
0
3
1
7
a
.
[
3
]
B
.
M
a
i
n
a
l
i
,
“
R
e
p
r
e
s
e
n
t
a
t
i
o
n
i
n
t
e
a
c
h
i
n
g
a
n
d
l
e
a
r
n
i
n
g
ma
t
h
e
m
a
t
i
c
s,
”
I
n
t
e
r
n
a
t
i
o
n
a
l
J
o
u
r
n
a
l
o
f
E
d
u
c
a
t
i
o
n
i
n
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
,
v
o
l
.
9
,
n
o
.
1
,
p
p
.
1
–
2
1
,
2
0
2
1
,
d
o
i
:
1
0
.
4
6
3
2
8
/
i
j
e
ms
t
.
1
1
1
1
.
[
4
]
J.
S
o
t
o
-
A
n
d
r
a
d
e
,
“
M
e
t
a
p
h
o
r
s
i
n
ma
t
h
e
ma
t
i
c
s
e
d
u
c
a
t
i
o
n
,
”
E
n
c
y
c
l
o
p
e
d
i
a
o
f
M
a
t
h
e
m
a
t
i
c
s
Ed
u
c
a
t
i
o
n
,
p
p
.
6
1
9
–
6
2
5
,
2
0
2
0
,
d
o
i
:
1
0
.
1
0
0
7
/
9
7
8
-
3
-
0
3
0
-
1
5
7
8
9
-
0
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1
1
3
.
[
5
]
A
.
A
l
a
m,
“
C
h
a
l
l
e
n
g
e
s
a
n
d
p
o
ssi
b
i
l
i
t
i
e
s
i
n
t
e
a
c
h
i
n
g
a
n
d
l
e
a
r
n
i
n
g
o
f
c
a
l
c
u
l
u
s:
a
c
a
se
st
u
d
y
o
f
I
n
d
i
a
,
”
J
o
u
r
n
a
l
f
o
r
t
h
e
E
d
u
c
a
t
i
o
n
o
f
G
i
f
t
e
d
Y
o
u
n
g
S
c
i
e
n
t
i
s
t
s
,
v
o
l
.
8
,
n
o
.
1
,
p
p
.
4
0
7
–
4
3
3
,
2
0
2
0
,
d
o
i
:
1
0
.
1
7
4
7
8
/
j
e
g
y
s.
6
6
0
2
0
1
.
[
6
]
U
.
T.
J
a
n
k
v
i
s
t
a
n
d
M
.
N
i
ss
,
“
U
p
p
e
r
s
e
c
o
n
d
a
r
y
sc
h
o
o
l
st
u
d
e
n
t
s’
d
i
f
f
i
c
u
l
t
i
e
s
w
i
t
h
ma
t
h
e
ma
t
i
c
a
l
m
o
d
e
l
l
i
n
g
,
”
I
n
t
e
r
n
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
Ed
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
1
,
n
o
.
4
,
p
p
.
4
6
7
–
4
9
6
,
2
0
2
0
,
d
o
i
:
1
0
.
1
0
8
0
/
0
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2
0
7
3
9
X
.
2
0
1
9
.
1
5
8
7
5
3
0
.
[
7
]
A
.
B
.
El
l
i
s,
E
.
Lo
c
k
w
o
o
d
,
E.
Ti
l
l
e
ma
,
a
n
d
K
.
M
o
o
r
e
,
“
G
e
n
e
r
a
l
i
z
a
t
i
o
n
a
c
r
o
ss
m
u
l
t
i
p
l
e
m
a
t
h
e
m
a
t
i
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