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ts’
c
re
a
ti
v
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m
a
th
e
m
a
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l
th
in
k
in
g
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K
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w
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s
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Gam
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Dan
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tan
d
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[
1
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–
[
3
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.
T
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an
d
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lectiv
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in
k
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[
4
]
–
[
6
]
.
Evaluation Warning : The document was created with Spire.PDF for Python.
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crea
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1699
I
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m
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[
7
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.
W
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[
8
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r
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7
2
[
9
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Similar
ly
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T
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s
in
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MSS)
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f
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4
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o
u
t
o
f
5
8
p
ar
ticip
a
tin
g
co
u
n
tr
ies
[
1
0
]
.
T
h
ese
f
in
d
in
g
s
h
ig
h
lig
h
t
th
e
u
r
g
en
t
n
ee
d
f
o
r
in
s
tr
u
ctio
n
al
i
n
n
o
v
atio
n
th
at
ca
n
p
r
o
m
o
te
d
ee
p
er
r
ea
s
o
n
in
g
an
d
s
tr
en
g
th
e
n
s
tu
d
en
ts
’
cr
ea
tiv
e
m
ath
em
atica
l
th
in
k
i
n
g
.
I
n
th
i
s
co
n
tex
t,
g
am
e
–
p
r
o
ject
-
b
ased
lear
n
i
n
g
(
GPB
L
)
is
p
ar
ticu
lar
ly
r
elev
a
n
t,
as
it
in
teg
r
ates
g
am
e
-
b
ased
ch
alle
n
g
es
with
p
r
o
ject
-
o
r
ien
ted
ac
tiv
ities
th
at
s
u
p
p
o
r
t
ex
p
lo
r
ati
o
n
an
d
m
ea
n
in
g
f
u
l
m
ath
em
atica
l
u
n
d
er
s
tan
d
i
n
g
.
T
h
er
ef
o
r
e,
m
ath
em
atics
in
s
tr
u
ctio
n
s
h
o
u
ld
s
h
if
t
f
r
o
m
p
r
e
d
o
m
in
an
tly
teac
h
er
-
ce
n
ter
ed
ap
p
r
o
ac
h
es
to
war
d
le
ar
n
in
g
en
v
ir
o
n
m
en
ts
th
at
p
r
o
v
id
e
g
r
ea
ter
o
p
p
o
r
tu
n
ities
f
o
r
s
tu
d
en
ts
to
ex
p
lo
r
e,
ex
p
er
im
en
t,
a
n
d
d
ev
elo
p
m
ath
em
atica
l th
in
k
in
g
.
C
r
ea
tiv
e
m
ath
em
atica
l
th
in
k
in
g
p
lay
s
a
s
tr
ateg
ic
r
o
le
in
eq
u
ip
p
in
g
s
tu
d
en
ts
to
ad
d
r
ess
co
m
p
lex
r
ea
l
-
wo
r
ld
p
r
o
b
lem
s
.
Ma
th
e
m
atica
l
cr
ea
tiv
ity
ex
ten
d
s
b
ey
o
n
d
th
e
a
b
ilit
y
to
g
en
er
ate
m
u
ltip
le
an
s
wer
s
;
it
en
co
m
p
ass
es
f
lex
ib
ilit
y
in
s
elec
tin
g
s
tr
ateg
ies,
o
r
ig
in
ality
in
co
n
s
tr
u
ctin
g
s
o
lu
tio
n
s
,
a
n
d
th
e
ca
p
ac
ity
t
o
m
ea
n
in
g
f
u
lly
d
e
v
elo
p
n
ew
id
e
as
[
1
1
]
–
[
1
3
]
.
E
m
p
ir
ical
s
tu
d
ie
s
h
av
e
d
e
m
o
n
s
tr
ated
t
h
at
m
ath
em
atica
l
cr
ea
tiv
ity
d
ev
elo
p
s
o
p
tim
ally
th
r
o
u
g
h
en
g
ag
em
en
t
with
n
o
n
-
r
o
u
tin
e
p
r
o
b
lem
s
,
o
p
en
-
e
n
d
ed
task
s
,
an
d
lear
n
i
n
g
ac
tiv
ities
th
at
allo
w
f
o
r
m
u
ltip
le
s
o
lu
ti
o
n
s
tr
ateg
ies
[
1
4
]
–
[
1
6
]
.
Nev
e
r
th
eless
,
r
esear
ch
co
n
s
is
ten
tly
s
h
o
ws
th
at
m
an
y
s
tu
d
en
ts
r
em
ain
at
an
im
itativ
e
lev
el
o
f
th
in
k
in
g
,
r
ely
in
g
h
ea
v
ily
o
n
teac
h
er
-
m
o
d
ele
d
p
r
o
ce
d
u
r
es
an
d
s
tr
u
g
g
lin
g
to
p
r
o
d
u
ce
o
r
ig
in
al
s
o
lu
tio
n
s
in
d
ep
en
d
en
tly
[
1
7
]
–
[
1
9
]
.
T
o
ad
d
r
ess
th
ese
ch
allen
g
es,
v
ar
io
u
s
in
s
tr
u
ctio
n
al
ap
p
r
o
ac
h
es
—
s
u
ch
as
in
q
u
ir
y
lear
n
i
n
g
,
d
is
co
v
er
y
lear
n
in
g
,
a
n
d
o
p
en
-
e
n
d
ed
p
r
o
b
lem
s
o
lv
in
g
—
h
av
e
b
ee
n
im
p
lem
en
ted
.
T
h
ese
ap
p
r
o
ac
h
es
h
av
e
b
ee
n
s
h
o
w
n
to
en
h
an
ce
ce
r
tain
d
im
en
s
io
n
s
o
f
cr
ea
tiv
e
m
ath
em
atica
l
th
in
k
i
n
g
,
p
ar
ticu
la
r
ly
f
lu
e
n
cy
an
d
f
l
ex
ib
ilit
y
[
2
0
]
,
[
2
1
]
.
Ho
wev
er
,
th
eir
ef
f
ec
tiv
en
ess
is
h
ig
h
ly
d
ep
en
d
e
n
t
o
n
task
d
esig
n
,
in
s
tr
u
ctio
n
al
co
n
tex
t,
an
d
th
e
co
n
tin
u
ity
o
f
s
tu
d
en
ts
’
l
e
a
r
n
i
n
g
e
x
p
e
r
i
e
n
c
e
s
[
2
2
]
,
[
2
3
]
.
I
n
th
is
ca
s
e
,
Su
b
an
j
i
et
a
l
.
[
1
9
]
c
o
n
c
e
p
t
u
a
l
i
z
e
d
c
r
e
a
t
i
v
e
m
ath
em
atica
l
th
in
k
in
g
as
a
p
r
o
g
r
ess
iv
e
co
g
n
itiv
e
d
ev
elo
p
m
en
tal
p
r
o
ce
s
s
co
n
s
is
tin
g
o
f
f
i
v
e
s
tag
es:
i
)
p
r
e
-
im
itatio
n
,
in
wh
ich
s
tu
d
en
ts
r
ec
o
g
n
ize
a
p
r
o
b
lem
b
u
t
a
r
e
u
n
ab
le
to
f
o
r
m
u
late
a
s
o
lu
tio
n
;
ii
)
im
itatio
n
,
w
h
er
e
s
tu
d
en
ts
r
ep
licate
s
o
lu
tio
n
s
tr
ateg
ies
d
em
o
n
s
tr
at
ed
b
y
t
h
e
teac
h
er
;
iii
)
m
o
d
if
i
ca
tio
n
,
wh
er
e
s
tu
d
e
n
ts
ad
ap
t
o
r
r
ef
in
e
ex
is
tin
g
s
tr
ateg
ies;
iv
)
co
m
b
in
atio
n
,
wh
er
e
s
tu
d
en
ts
in
teg
r
ate
two
o
r
m
o
r
e
s
tr
ateg
ies
to
g
en
er
ate
a
s
o
lu
tio
n
;
an
d
v
)
co
n
s
tr
u
ctio
n
,
wh
er
e
s
tu
d
e
n
ts
in
d
ep
en
d
en
tly
d
ev
elo
p
n
o
v
el
s
o
lu
tio
n
s
tr
ateg
ies.
Desp
ite
its
p
o
ten
tial,
em
p
ir
ical
im
p
lem
e
n
tatio
n
o
f
th
is
m
o
d
el
r
em
ain
s
lim
ited
,
e
s
p
ec
ially
at
th
e
elem
en
tar
y
e
d
u
ca
tio
n
lev
el
a
n
d
with
in
g
am
if
ied
lear
n
in
g
c
o
n
t
ex
ts
.
Alo
n
g
s
id
e
th
e
d
ig
it
al
tr
an
s
f
o
r
m
at
io
n
o
f
m
a
th
em
at
ic
s
ed
u
c
at
io
n
,
GP
B
L
h
a
s
em
er
g
ed
as
a
p
r
o
m
i
s
in
g
ap
p
r
o
a
ch
to
en
h
a
n
ce
s
tu
d
en
t
s
’
m
o
t
iv
a
tio
n
an
d
c
o
g
n
it
iv
e
en
g
ag
em
en
t.
A
s
y
s
t
e
m
at
ic
r
ev
i
ew
b
y
Ma
y
r
h
o
f
er
et
a
l
.
[
2
4
]
r
ep
o
r
ted
th
a
t
g
am
i
f
i
ca
t
io
n
in
m
a
th
em
a
ti
cs
l
ea
r
n
in
g
g
en
er
al
ly
p
r
o
d
u
c
es
p
o
s
i
tiv
e
ef
f
ec
t
s
o
n
s
tu
d
en
t
s
’
m
o
t
iv
a
tio
n
,
en
g
ag
em
en
t,
an
d
le
ar
n
in
g
o
u
tc
o
m
es
,
al
th
o
u
g
h
it
s
im
p
ac
t
d
e
p
en
d
s
s
tr
o
n
g
ly
o
n
in
s
tr
u
ct
io
n
a
l
d
e
s
ig
n
q
u
a
li
ty
.
Su
p
p
o
r
ti
n
g
ev
id
e
n
ce
a
ls
o
i
n
d
ica
te
s
th
at
g
am
if
ica
t
io
n
c
an
im
p
r
o
v
e
s
tu
d
en
t
s
’
m
ath
em
a
t
ica
l
ac
h
i
ev
em
en
t
a
n
d
l
ea
r
n
in
g
en
g
ag
e
m
en
t
co
m
p
a
r
ed
to
t
r
ad
it
io
n
a
l
in
s
tr
u
c
t
io
n
[
2
5
]
–
[
2
7
]
,
an
d
m
ay
co
n
tr
ib
u
te
to
th
e
d
ev
el
o
p
m
en
t
o
f
s
t
u
d
en
t
s
’
th
in
k
in
g
s
k
il
l
s
a
n
d
lea
r
n
in
g
m
o
tiv
at
io
n
[
2
8
]
.
I
n
b
r
o
ad
e
r
ed
u
ca
t
io
n
a
l
co
n
tex
t
s
,
t
ec
h
n
o
l
o
g
y
-
s
u
p
p
o
r
t
ed
g
a
m
if
ied
en
v
i
r
o
n
m
en
t
s
,
p
r
o
jec
t
-
b
as
ed
g
a
m
es
,
an
d
s
i
m
u
l
at
io
n
s
h
av
e
b
ee
n
f
o
u
n
d
to
f
o
s
ter
h
i
g
h
er
-
o
r
d
er
th
in
k
in
g
an
d
co
n
c
ep
tu
al
u
n
d
er
s
tan
d
in
g
th
r
o
u
g
h
au
th
en
t
ic
l
ea
r
n
in
g
ex
p
er
ien
ce
s
[
2
9
]
–
[
3
1
]
.
Mo
r
e
o
v
er
,
i
n
t
eg
r
a
tin
g
v
i
s
u
al
a
n
d
ex
p
lo
r
a
to
r
y
te
ch
n
o
l
o
g
i
es
s
u
ch
a
s
au
g
m
en
ted
r
ea
li
ty
ca
n
s
tr
en
g
th
en
co
n
ce
p
tu
a
l
u
n
d
er
s
t
an
d
in
g
th
r
o
u
g
h
c
o
n
t
ex
tu
a
l
v
i
s
u
a
li
za
ti
o
n
an
d
ex
p
lo
r
at
io
n
[
3
2
]
,
[
3
3
]
.
Ho
w
ev
er
,
g
am
if
ica
t
io
n
th
at
r
el
ie
s
m
a
in
ly
o
n
p
o
in
t
s
,
s
c
o
r
e
s
,
a
n
d
co
m
p
et
i
tio
n
m
ay
f
a
i
l
t
o
p
r
o
m
o
t
e
h
ig
h
er
-
o
r
d
er
th
in
k
in
g
u
n
l
e
s
s
i
t
is
m
e
a
n
in
g
f
u
lly
a
lig
n
ed
wi
th
p
ed
ag
o
g
ica
l g
o
al
s
[
3
4
]
,
[
3
5
]
.
Pro
ject
-
b
ased
lear
n
i
n
g
(
PjB
L
)
h
as
b
ee
n
wid
el
y
r
ec
o
g
n
ize
d
as
an
ef
f
ec
tiv
e
in
s
tr
u
cti
o
n
al
a
p
p
r
o
ac
h
to
s
u
p
p
o
r
t
cr
ea
tiv
ity
,
p
r
o
b
lem
-
s
o
lv
in
g
s
k
ills
,
an
d
co
llab
o
r
atio
n
.
Pre
v
io
u
s
s
tu
d
ies
h
av
e
r
ep
o
r
ted
th
at
PjB
L
ca
n
im
p
r
o
v
e
s
tu
d
e
n
ts
’
cr
ea
tiv
e
th
in
k
in
g
an
d
m
ath
em
atica
l
p
r
o
b
l
em
-
s
o
lv
in
g
co
m
p
ar
ed
to
c
o
n
v
en
tio
n
al
in
s
tr
u
ctio
n
[
1
2
]
,
[
3
6
]
,
[
3
7
]
.
W
h
en
s
u
p
p
o
r
ted
b
y
tech
n
o
l
o
g
y
,
PjB
L
also
p
r
o
v
id
es
m
o
r
e
m
ea
n
in
g
f
u
l
lear
n
in
g
ex
p
er
ien
ce
s
th
r
o
u
g
h
c
o
n
tex
tu
al
ac
tiv
ities
an
d
p
r
o
d
u
ct
-
o
r
ien
ted
task
s
.
Ho
wev
er
,
m
o
s
t
s
tu
d
ies
s
till
d
is
cu
s
s
PjB
L
an
d
g
am
if
icatio
n
s
ep
a
r
ately
,
a
n
d
o
n
ly
lim
ited
r
esear
ch
h
as
e
x
am
in
ed
h
o
w
b
o
th
ap
p
r
o
ac
h
es
ca
n
b
e
in
teg
r
ated
in
to
o
n
e
lear
n
in
g
d
esig
n
to
s
p
ec
if
ic
ally
d
ev
elo
p
s
tu
d
e
n
ts
’
cr
ea
tiv
e
m
ath
em
atica
l
th
in
k
in
g
,
esp
ec
i
ally
in
elem
e
n
tar
y
g
eo
m
etr
y
lea
r
n
in
g
.
T
h
e
r
e
f
o
r
e
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atica
l
th
in
k
i
n
g
s
k
ills
p
r
io
r
to
th
e
in
ter
v
en
ti
o
n
.
T
h
e
p
r
etest
was
ad
m
in
is
ter
ed
to
all
s
tu
d
en
ts
in
b
o
th
g
r
o
u
p
s
u
n
d
er
th
e
s
am
e
tim
e
allo
ca
tio
n
.
T
h
e
p
r
etest
in
s
tr
u
m
en
t
co
n
s
is
ted
o
f
f
i
v
e
o
p
en
-
en
d
e
d
q
u
esti
o
n
s
d
esig
n
ed
t
o
ass
ess
in
d
icato
r
s
o
f
cr
ea
ti
v
e
m
at
h
em
atica
l
th
in
k
in
g
.
T
h
e
n
ex
t
s
tag
e
was
th
e
in
s
tr
u
ctio
n
al
in
ter
v
en
t
io
n
.
I
n
th
is
p
h
ase,
th
e
ex
p
er
i
m
en
tal
g
r
o
u
p
r
ec
eiv
e
d
g
eo
m
e
tr
y
in
s
tr
u
ctio
n
u
s
in
g
a
Py
th
o
n
-
b
ased
e
d
u
ca
tio
n
al
g
am
e
in
teg
r
ated
in
to
t
h
e
lear
n
i
n
g
m
o
d
u
le.
Alth
o
u
g
h
th
e
g
am
e
was
d
esig
n
ed
f
o
r
s
o
lid
g
eo
m
etr
y
to
p
ics,
th
is
s
tu
d
y
s
p
ec
if
ically
f
o
c
u
s
ed
o
n
t
h
e
s
u
r
f
ac
e
ar
ea
o
f
a
cy
lin
d
er
.
At
th
e
b
eg
in
n
i
n
g
o
f
th
e
less
o
n
,
s
tu
d
en
ts
wer
e
f
ir
s
t in
tr
o
d
u
ce
d
to
th
e
lear
n
in
g
m
o
d
u
le
u
s
ed
in
th
e
s
tu
d
y
.
I
n
ad
d
i
tio
n
,
s
tu
d
en
ts
in
th
e
ex
p
er
im
en
tal
g
r
o
u
p
r
ec
eiv
e
d
t
ec
h
n
ical
g
u
id
a
n
ce
o
n
h
o
w
to
o
p
er
ate
th
e
g
am
e,
i
n
clu
d
in
g
h
o
w
to
s
elec
t
an
d
r
u
n
ea
ch
lev
el
an
d
h
o
w
to
in
p
u
t
v
a
lu
es (
th
e
r
ad
iu
s
an
d
h
eig
h
t o
f
t
h
e
cy
lin
d
er
)
.
T
o
illu
s
tr
ate
th
e
s
tep
-
by
-
s
tep
t
h
e
Py
th
o
n
-
b
ased
g
am
e,
th
e
s
eq
u
en
ce
o
f
lear
n
in
g
lev
els
is
s
u
m
m
ar
ized
in
Fig
u
r
e
1
.
Stu
d
en
ts
ex
p
lo
r
ed
th
e
lea
r
n
in
g
co
n
ten
t
th
r
o
u
g
h
f
iv
e
s
eq
u
en
tial
lev
els:
i
)
o
b
s
er
v
in
g
a
th
r
ee
-
d
im
en
s
io
n
al
cy
lin
d
er
m
o
d
el
d
is
p
lay
ed
in
th
e
g
am
e;
at
th
is
lev
el,
s
tu
d
en
ts
g
e
n
er
ate
d
in
itial
id
ea
s
an
d
s
tim
u
lated
f
lu
en
cy
b
y
d
escr
ib
in
g
th
e
ch
ar
ac
ter
is
tics
o
f
th
e
cy
lin
d
er
an
d
id
en
tify
i
n
g
its
k
ey
elem
en
ts
(
r
ad
iu
s
,
d
iam
eter
,
an
d
h
eig
h
t)
;
ii
)
ex
p
lo
r
in
g
th
e
cy
lin
d
er
th
r
o
u
g
h
3
D
v
is
u
aliza
tio
n
b
y
ad
ju
s
tin
g
in
p
u
t
v
alu
es
(
r
ad
iu
s
an
d
h
eig
h
t)
an
d
o
b
s
er
v
in
g
c
h
an
g
es
in
th
e
m
o
d
el,
th
er
e
b
y
s
u
p
p
o
r
tin
g
f
lu
e
n
cy
th
r
o
u
g
h
r
ep
ea
ted
tr
ials
an
d
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t
J
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&
R
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I
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N:
2252
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8
8
2
2
Ho
w
g
a
me
–
p
r
o
ject
lea
r
n
in
g
e
n
h
a
n
ce
s
crea
tive
th
in
kin
g
in
el
eme
n
ta
r
y
g
eo
metry?
(
A
n
to
n
P
r
a
yitn
o
)
1701
ex
p
er
im
en
tatio
n
;
iii)
id
en
tify
i
n
g
g
eo
m
etr
ic
f
o
r
m
s
th
r
o
u
g
h
n
et
r
ep
r
esen
tatio
n
s
b
y
u
n
f
o
ld
i
n
g
th
e
cy
lin
d
er
an
d
ex
am
in
in
g
th
e
r
elatio
n
s
h
ip
b
e
twee
n
th
e
cu
r
v
ed
s
u
r
f
ac
e
an
d
a
r
ec
tan
g
u
lar
s
h
ap
e
,
wh
ich
s
u
p
p
o
r
ted
f
lex
ib
ilit
y
b
y
s
h
if
tin
g
s
tu
d
en
ts
’
r
ea
s
o
n
i
n
g
f
r
o
m
th
r
ee
-
d
im
e
n
s
io
n
al
v
is
u
aliza
tio
n
to
two
-
d
im
en
s
io
n
al
r
ep
r
esen
tatio
n
;
iv
)
an
aly
zin
g
r
elatio
n
s
h
ip
s
a
m
o
n
g
g
eo
m
etr
ic
elem
e
n
ts
,
p
a
r
ticu
lar
ly
th
e
r
elatio
n
s
h
ip
b
et
wee
n
th
e
r
ec
tan
g
le’
s
len
g
th
an
d
t
h
e
cir
cle’
s
cir
cu
m
f
er
en
ce
,
as
well
as
th
e
r
elatio
n
s
h
ip
b
etwe
en
th
e
r
ec
ta
n
g
le’
s
wid
th
an
d
th
e
cy
lin
d
er
’
s
h
eig
h
t,
th
er
e
b
y
s
tr
e
n
g
th
en
in
g
f
lex
ib
ilit
y
as st
u
d
en
ts
wer
e
r
eq
u
ir
ed
to
c
o
n
n
ec
t
m
u
ltip
le
co
n
ce
p
ts
an
d
r
ep
r
esen
tatio
n
s
;
an
d
v
)
co
n
s
tr
u
ctin
g
t
h
e
f
o
r
m
u
la
an
d
s
o
lv
in
g
p
r
o
b
lem
s
.
At
L
ev
el
5
,
s
tu
d
en
ts
co
m
p
leted
a
g
r
o
u
p
-
b
ased
p
r
o
ject
task
i
n
wh
ich
th
ey
d
e
r
iv
ed
th
e
s
u
r
f
ac
e
ar
ea
f
o
r
m
u
la
o
f
a
cy
li
n
d
er
b
ased
o
n
th
eir
ex
p
lo
r
atio
n
o
f
th
e
cy
lin
d
e
r
n
et
in
th
e
g
am
e
an
d
r
ec
o
r
d
ed
th
ei
r
f
in
d
i
n
g
s
o
n
a
wo
r
k
s
h
ee
t.
T
h
ey
th
en
ap
p
lied
th
e
d
er
iv
ed
f
o
r
m
u
la
to
s
o
lv
e
th
e
g
iv
en
p
r
o
b
lem
s
.
Fig
u
r
e
1
.
Gam
e
le
v
els f
o
r
lear
n
in
g
cy
lin
d
er
s
u
r
f
ac
e
ar
ea
Me
an
wh
ile,
th
e
co
n
tr
o
l
g
r
o
u
p
r
ec
eiv
ed
d
ir
ec
t
in
s
tr
u
ctio
n
u
s
in
g
th
e
s
am
e
lear
n
in
g
m
o
d
u
l
e
an
d
th
e
s
am
e
to
p
ic
(
s
u
r
f
ac
e
ar
ea
o
f
c
y
lin
d
er
s
)
,
b
u
t
with
o
u
t
th
e
Py
th
o
n
-
b
ased
g
am
e
m
ed
ia.
Dir
e
ct
in
s
tr
u
ctio
n
was
im
p
lem
en
ted
th
r
o
u
g
h
a
teac
h
er
-
ce
n
ter
ed
lear
n
i
n
g
s
eq
u
en
ce
co
n
s
is
tin
g
o
f
ex
p
lan
atio
n
o
f
cy
lin
d
e
r
elem
e
n
ts
an
d
n
et
r
ep
r
esen
tatio
n
,
teac
h
er
d
em
o
n
s
tr
atio
n
o
f
th
e
s
u
r
f
a
ce
ar
ea
f
o
r
m
u
la
d
e
r
iv
atio
n
o
n
th
e
b
o
ar
d
,
g
u
id
e
d
p
r
ac
tice
th
r
o
u
g
h
s
am
p
le
p
r
o
b
l
em
s
,
an
d
in
d
iv
id
u
al
e
x
er
cises
u
s
in
g
q
u
esti
o
n
s
.
Stu
d
e
n
ts
p
r
i
m
ar
ily
f
o
llo
wed
th
e
teac
h
er
’
s
p
r
o
ce
d
u
r
al
ex
p
la
n
atio
n
an
d
s
o
lv
ed
task
s
u
s
in
g
th
e
f
o
r
m
u
la
p
r
o
v
id
e
d
,
with
lim
ited
o
p
p
o
r
tu
n
ities
f
o
r
ex
p
lo
r
atio
n
an
d
i
n
d
ep
e
n
d
en
t
s
tr
ateg
y
co
n
s
tr
u
ctio
n
.
T
h
e
i
n
ter
v
en
tio
n
was
co
n
d
u
cte
d
o
v
er
th
r
ee
m
ee
tin
g
s
with
in
a
two
-
wee
k
p
er
i
o
d
,
wi
th
ea
ch
m
ee
tin
g
last
in
g
2
×3
5
m
in
u
tes.
B
o
th
g
r
o
u
p
s
r
ec
eiv
e
d
th
e
s
am
e
n
u
m
b
er
an
d
f
r
eq
u
e
n
cy
o
f
in
s
tr
u
ctio
n
al
s
ess
io
n
s
.
Af
ter
th
e
c
o
m
p
letio
n
o
f
th
e
in
s
tr
u
ctio
n
al
i
n
ter
v
en
t
io
n
,
a
p
o
s
t
-
test
was
ad
m
in
is
ter
ed
to
b
o
th
g
r
o
u
p
s
to
m
ea
s
u
r
e
s
tu
d
en
ts
’
im
p
r
o
v
em
e
n
t in
cr
ea
tiv
e
m
at
h
em
atica
l th
in
k
in
g
s
k
ills
.
Data
wer
e
co
llected
u
s
in
g
a
c
r
ea
tiv
e
m
ath
em
atica
l
th
in
k
in
g
test
ad
m
in
is
ter
ed
as
b
o
th
a
p
r
etest
an
d
a
p
o
s
t
-
test
to
s
tu
d
en
ts
in
th
e
e
x
p
er
im
en
tal
a
n
d
co
n
tr
o
l
g
r
o
u
p
s
.
T
o
en
s
u
r
e
c
o
n
ten
t
v
alid
ity
,
th
e
in
s
tr
u
m
en
t
was
r
ev
iewe
d
b
y
e
x
p
er
ts
in
m
ath
e
m
atics
ed
u
ca
tio
n
an
d
ed
u
ca
ti
o
n
al
tech
n
o
lo
g
y
.
E
x
p
er
t
ju
d
g
m
en
t
was
co
n
d
u
cted
u
s
in
g
a
v
alid
atio
n
s
h
ee
t
ev
a
lu
atin
g
th
e
r
elev
an
ce
o
f
ite
m
s
to
lear
n
in
g
o
b
jectiv
es,
c
lar
ity
o
f
wo
r
d
i
n
g
,
ap
p
r
o
p
r
iaten
ess
f
o
r
lear
n
er
s
,
a
n
d
alig
n
m
e
n
t
with
cr
ea
tiv
e
t
h
in
k
in
g
i
n
d
icato
r
s
.
T
h
e
r
eliab
ilit
y
o
f
t
h
e
in
s
tr
u
m
en
t
was
ex
am
in
ed
th
r
o
u
g
h
in
ter
n
a
l
co
n
s
is
ten
cy
an
aly
s
is
,
an
d
t
h
e
co
ef
f
icien
t
in
d
icate
d
th
at
t
h
e
test
h
ad
ac
ce
p
tab
le
r
eliab
ilit
y
f
o
r
r
esear
ch
p
u
r
p
o
s
es.
T
h
e
s
tu
d
e
n
ts
’
s
co
r
es
wer
e
th
en
p
r
o
ce
s
s
ed
an
d
q
u
an
titativ
ely
an
aly
ze
d
u
s
in
g
J
ASP
s
o
f
twar
e.
T
h
e
in
itial
s
tag
e
o
f
an
aly
s
is
in
v
o
lv
ed
d
e
s
cr
ip
tiv
e
s
tatis
tics
to
ex
am
in
e
th
e
m
ea
n
s
co
r
es,
s
tan
d
ar
d
d
ev
iatio
n
s
,
an
d
ch
a
n
g
es
in
p
r
etest
an
d
p
o
s
t
-
test
r
esu
lts
with
in
ea
ch
g
r
o
u
p
.
Nex
t,
t
h
e
ass
u
m
p
tio
n
s
f
o
r
p
ar
am
etr
ic
an
al
y
s
is
wer
e
test
e
d
,
in
clu
d
i
n
g
a
n
o
r
m
ality
test
u
s
in
g
th
e
Sh
ap
ir
o
–
W
ilk
test
an
d
a
h
o
m
o
g
e
n
eity
o
f
v
ar
ian
ce
test
u
s
in
g
L
ev
e
n
e’
s
t
est.
Af
ter
th
e
ass
u
m
p
tio
n
s
wer
e
m
et,
th
e
d
ata
wer
e
an
aly
ze
d
u
s
in
g
a
two
-
f
ac
to
r
r
ep
ea
te
d
m
ea
s
u
r
es
an
aly
s
is
o
f
v
ar
ian
ce
(
ANOV
A)
,
co
n
s
is
tin
g
o
f
a
with
in
-
s
u
b
ject
f
ac
to
r
(
tim
e:
p
r
etest
an
d
p
o
s
t
-
test
)
an
d
a
b
etwe
en
-
s
u
b
ject
f
ac
to
r
(
g
r
o
u
p
:
e
x
p
er
im
e
n
tal
an
d
c
o
n
tr
o
l)
.
T
h
is
an
aly
s
is
was
co
n
d
u
cted
to
ex
am
i
n
e:
i
)
d
if
f
e
r
en
ce
s
in
s
tu
d
en
ts
’
cr
e
ativ
e
m
ath
em
atica
l
th
i
n
k
in
g
b
ef
o
r
e
an
d
af
ter
th
e
i
n
ter
v
en
t
io
n
(
m
ai
n
ef
f
ec
t
o
f
tim
e)
;
ii
)
d
if
f
er
en
ce
s
b
etwe
en
th
e
ex
p
er
im
en
tal
an
d
co
n
tr
o
l g
r
o
u
p
s
(
m
ain
ef
f
ec
t
o
f
g
r
o
u
p
)
;
an
d
iii
)
d
if
f
er
en
ce
s
in
th
e
p
atter
n
o
f
im
p
r
o
v
em
e
n
t
b
etwe
en
th
e
two
g
r
o
u
p
s
(
in
te
r
ac
tio
n
ef
f
ec
t
o
f
tim
e×
g
r
o
u
p
)
.
T
o
s
tr
en
g
th
en
th
e
in
ter
p
r
etatio
n
o
f
th
e
r
esu
lts
,
ef
f
ec
t
s
ize
was
r
ep
o
r
ted
u
s
in
g
eta
s
q
u
ar
ed
(
η
²)
.
I
f
a
s
ig
n
if
ican
t
in
ter
ac
tio
n
ef
f
ec
t
was
f
o
u
n
d
,
f
u
r
th
e
r
a
n
aly
s
es
wer
e
c
o
n
d
u
cted
u
s
in
g
p
o
s
t
h
o
c
p
air
wis
e
c
o
m
p
ar
is
o
n
s
to
id
en
tif
y
s
p
ec
if
ic
d
if
f
er
en
ce
s
b
etwe
en
co
n
d
itio
n
s
.
All stati
s
tical
te
s
ts
wer
e
p
er
f
o
r
m
ed
at
a
s
ig
n
if
ican
ce
lev
el
o
f
0
.
0
5
.
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
.
15
,
No
.
2
,
Ap
r
il
20
2
6
:
1
6
9
8
-
1
7
1
0
1702
3.
RE
SU
L
T
S AN
D
D
I
SCU
SS
I
O
N
3
.
1
.
Dev
el
o
pm
ent
o
f
g
a
m
e
–
p
ro
j
ec
t
-
ba
s
ed
lea
rning
T
h
is
s
tu
d
y
co
m
m
en
ce
d
with
th
e
d
ev
elo
p
m
en
t
an
d
p
r
e
p
ar
ati
o
n
o
f
a
GPB
L
in
s
tr
u
ctio
n
al
in
ter
v
en
tio
n
d
esig
n
ed
to
s
tim
u
late
s
tu
d
en
ts
’
cr
ea
tiv
e
m
ath
em
atica
l
th
in
k
i
n
g
in
s
o
lv
in
g
n
o
n
-
r
o
u
tin
e
p
r
o
b
lem
s
in
g
e
o
m
etr
y
,
p
ar
ticu
lar
ly
th
e
c
o
n
ce
p
t
o
f
th
e
s
u
r
f
ac
e
ar
ea
o
f
a
c
y
lin
d
er
.
T
h
e
lear
n
in
g
m
ed
ia
wer
e
d
ev
elo
p
ed
u
s
in
g
th
e
Py
th
o
n
p
r
o
g
r
am
m
in
g
la
n
g
u
a
g
e
th
r
o
u
g
h
T
h
o
n
n
y
,
en
a
b
lin
g
s
tu
d
en
ts
t
o
d
ir
ec
tly
in
te
r
a
ct
with
g
e
o
m
etr
ic
v
is
u
aliza
tio
n
s
p
r
esen
ted
in
a
l
ev
el
-
b
ased
ed
u
ca
tio
n
al
g
am
e
f
o
r
m
at.
T
h
e
d
ash
b
o
ar
d
o
f
t
h
e
Py
th
o
n
-
b
ased
GPB
L
m
ed
ia
u
s
ed
in
th
is
s
tu
d
y
is
s
h
o
wn
in
Fig
u
r
e
2
.
As
illu
s
tr
ated
in
Fig
u
r
e
2
,
t
h
e
in
ter
f
ac
e
p
r
esen
ts
s
ev
e
r
al
s
t
ag
es
o
f
ex
p
l
o
r
atio
n
th
at
g
u
id
e
s
tu
d
en
ts
g
r
ad
u
ally
to
war
d
co
n
ce
p
tu
al
u
n
d
er
s
tan
d
in
g
.
Fig
u
r
e
2
(
a
)
d
is
p
lay
s
th
e
in
itial
v
is
u
ali
za
tio
n
o
f
a
th
r
ee
-
d
im
en
s
io
n
al
cy
lin
d
er
th
at
allo
ws
s
tu
d
en
ts
to
o
b
s
er
v
e
th
e
o
b
ject
as
a
co
m
p
lete
g
eo
m
etr
ic
s
o
lid
.
Fig
u
r
e
2
(
b
)
s
h
o
ws
an
in
ter
ac
tiv
e
cy
lin
d
er
m
o
d
el
in
wh
ich
s
tu
d
e
n
ts
ca
n
m
an
ip
u
late
th
e
r
ad
i
u
s
an
d
h
ei
g
h
t
v
alu
es
to
ex
p
l
o
r
e
h
o
w
ch
an
g
es
in
th
ese
p
ar
am
e
ter
s
af
f
ec
t
th
e
v
is
u
aliza
tio
n
.
F
ig
u
r
e
2
(
c)
p
r
esen
ts
th
e
n
et
r
ep
r
esen
tatio
n
o
f
th
e
cy
lin
d
er
af
ter
u
n
f
o
ld
in
g
th
e
th
r
ee
-
d
im
en
s
io
n
al
m
o
d
el.
T
h
r
o
u
g
h
th
is
v
is
u
aliza
tio
n
,
s
tu
d
en
ts
b
eg
in
to
r
ec
o
g
n
ize
th
e
g
eo
m
etr
ic
co
m
p
o
n
en
ts
f
o
r
m
in
g
th
e
c
y
lin
d
er
.
Fig
u
r
e
2
(
d
)
h
ig
h
lig
h
ts
th
e
id
en
tific
atio
n
o
f
th
e
later
al
s
u
r
f
a
ce
as
a
r
ec
tan
g
u
lar
s
h
ap
e
d
er
i
v
e
d
f
r
o
m
th
e
cu
r
v
ed
s
u
r
f
ac
e
o
f
th
e
cy
lin
d
e
r
.
Fig
u
r
e
2
(
e)
f
u
r
th
er
d
em
o
n
s
tr
ates
th
e
r
elatio
n
s
h
ip
b
etwe
en
th
e
r
ec
t
an
g
u
lar
later
al
s
u
r
f
ac
e
an
d
th
e
cir
cu
lar
b
ase
an
d
to
p
o
f
t
h
e
cy
lin
d
er
.
Fin
ally
,
Fig
u
r
e
2
(
f
)
illu
s
tr
ates
th
e
s
tag
e
in
wh
ich
s
tu
d
en
ts
co
n
s
tr
u
ct
th
e
s
u
r
f
ac
e
ar
ea
f
o
r
m
u
la
o
f
th
e
cy
lin
d
er
b
ased
o
n
th
e
r
elatio
n
s
h
ip
s
th
ey
h
av
e
o
b
s
er
v
ed
d
u
r
in
g
th
e
ex
p
lo
r
atio
n
p
r
o
ce
s
s
.
(
a)
(
b
)
(
c)
(
d
)
(
e)
(f)
Fig
u
r
e
2
.
Dash
b
o
ar
d
o
f
th
e
Py
th
o
n
-
b
ased
GPB
L
m
ed
ia
d
ev
e
lo
p
ed
u
s
in
g
T
h
o
n
n
y
:
(
a)
o
b
s
er
v
in
g
th
e
c
y
lin
d
er
s
h
ap
e,
(
b
)
u
n
f
o
ld
in
g
th
e
cy
lin
d
er
in
to
its
n
et,
(
c
)
id
en
tif
y
in
g
th
e
later
al
s
u
r
f
ac
e
f
o
r
m
,
(
d
)
th
e
later
al
s
u
r
f
ac
e
is
a
r
ec
tan
g
le;
its
len
g
th
is
th
e
b
a
s
e
cir
cu
m
f
er
en
ce
a
n
d
its
wid
th
is
th
e
cy
lin
d
er
h
eig
h
t,
(
e
)
an
al
y
zin
g
th
e
b
ase
an
d
to
p
s
u
r
f
ac
es,
an
d
(
f
)
co
n
s
t
r
u
ctin
g
th
e
s
u
r
f
ac
e
ar
ea
f
o
r
m
u
la
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
Ho
w
g
a
me
–
p
r
o
ject
lea
r
n
in
g
e
n
h
a
n
ce
s
crea
tive
th
in
kin
g
in
el
eme
n
ta
r
y
g
eo
metry?
(
A
n
to
n
P
r
a
yitn
o
)
1703
T
h
e
lear
n
in
g
m
e
d
ia
wer
e
s
tr
u
ctu
r
ed
in
t
o
s
ev
er
al
p
r
o
g
r
ess
iv
e
an
d
e
x
p
lo
r
ato
r
y
lev
els.
L
ev
el
1
g
u
id
e
d
s
tu
d
en
ts
to
v
is
u
ally
o
b
s
er
v
e
th
e
cy
lin
d
er
as
a
co
m
p
lete
th
r
ee
-
d
im
en
s
io
n
al
s
o
lid
.
L
ev
el
2
f
ac
ilit
ated
ex
p
lo
r
atio
n
o
f
t
h
e
cy
lin
d
er
th
r
o
u
g
h
in
ter
ac
tiv
e
th
r
ee
-
d
im
en
s
io
n
al
v
is
u
aliza
tio
n
,
wh
ich
co
u
ld
b
e
u
n
f
o
ld
e
d
in
t
o
a
n
et
r
ep
r
esen
tatio
n
.
L
e
v
el
3
d
ir
ec
ted
s
tu
d
en
ts
to
id
en
tify
th
e
later
al
s
u
r
f
ac
e
o
f
th
e
c
y
lin
d
er
th
r
o
u
g
h
t
h
e
n
et
v
is
u
aliza
tio
n
,
lead
in
g
t
h
em
to
d
is
co
v
er
th
at
th
e
later
al
s
u
r
f
ac
e
f
o
r
m
s
a
r
ec
ta
n
g
le.
At
L
ev
el
4
,
s
tu
d
en
ts
wer
e
e
n
c
o
u
r
ag
e
d
to
an
al
y
ze
th
e
r
elatio
n
s
h
ip
b
etwe
en
th
e
later
al
s
u
r
f
ac
e
an
d
th
e
b
ase
an
d
to
p
o
f
th
e
c
y
lin
d
er
,
r
ep
r
esen
ted
as
two
co
n
g
r
u
en
t
cir
cles.
T
h
is
ex
p
lo
r
atio
n
was
r
ein
f
o
r
ce
d
th
r
o
u
g
h
ex
p
lo
r
atio
n
p
r
o
m
p
ts
,
wh
ich
g
u
id
ed
s
tu
d
en
ts
to
r
elate
th
e
r
a
d
iu
s
o
f
th
e
cir
cu
la
r
b
ase
to
its
cir
cu
m
f
er
e
n
ce
an
d
th
e
h
eig
h
t
o
f
th
e
cy
lin
d
e
r
to
th
e
wid
th
o
f
th
e
r
ec
tan
g
u
lar
late
r
al
s
u
r
f
ac
e.
At
L
ev
el
5
,
s
tu
d
en
ts
wer
e
r
eq
u
ir
ed
to
co
n
s
tr
u
ct
th
e
f
o
r
m
u
la
f
o
r
th
e
s
u
r
f
ac
e
ar
ea
o
f
a
cy
lin
d
e
r
b
y
in
teg
r
atin
g
th
e
ar
ea
s
o
f
th
e
later
al
s
u
r
f
ac
e,
th
e
b
ase,
an
d
th
e
to
p
.
T
h
r
o
u
g
h
th
is
p
r
o
ce
s
s
,
th
e
f
o
r
m
u
la
was
d
er
iv
ed
th
r
o
u
g
h
co
n
ce
p
tu
al
r
ea
s
o
n
in
g
r
ath
er
th
an
m
em
o
r
izatio
n
.
T
o
s
u
p
p
o
r
t
th
e
in
s
tr
u
ctio
n
al
im
p
lem
en
tat
io
n
,
th
e
r
esear
ch
er
s
p
r
e
p
ar
e
d
co
m
p
lem
en
ta
r
y
in
s
tr
u
m
en
ts
,
in
clu
d
in
g
a
c
r
ea
tiv
e
m
ath
e
m
atica
l
th
in
k
i
n
g
test
an
d
v
alid
atio
n
s
h
ee
ts
f
o
r
b
o
th
th
e
lear
n
in
g
m
ed
ia
an
d
th
e
ass
ess
m
en
t in
s
tr
u
m
en
t
s
.
3
.
2
.
E
x
pert
v
a
lid
a
t
io
n o
f
t
he
m
edia
T
h
e
in
s
tr
u
ctio
n
al
p
r
o
to
ty
p
e
was
s
u
b
s
eq
u
en
tly
v
alid
ated
b
y
ex
p
er
ts
in
m
ath
em
atics
ed
u
ca
tio
n
an
d
ed
u
ca
tio
n
al
tec
h
n
o
lo
g
y
t
o
en
s
u
r
e
co
n
ten
t
a
p
p
r
o
p
r
iaten
ess
,
c
lar
ity
o
f
in
s
tr
u
ctio
n
s
,
v
is
u
al
d
esig
n
q
u
ality
,
an
d
m
ater
ial
ac
cu
r
ac
y
.
T
h
e
r
esu
lts
o
f
th
e
ex
p
er
t
v
alid
atio
n
o
f
th
e
p
r
o
ject
-
b
ased
g
am
e
p
r
o
to
t
y
p
e
ar
e
p
r
esen
ted
in
T
ab
le
2
.
T
h
e
v
alid
atio
n
r
esu
lts
in
d
icate
th
at
th
e
av
er
ag
e
f
ea
s
ib
ilit
y
p
er
ce
n
tag
e
ex
ce
e
d
ed
9
0
%,
s
u
g
g
esti
n
g
th
at
th
e
p
r
o
t
o
ty
p
e
was
h
ig
h
ly
f
ea
s
ib
le
f
o
r
im
p
lem
en
tatio
n
with
f
if
th
-
g
r
a
d
e
s
tu
d
en
ts
at
I
s
lam
ic
E
lem
en
tar
y
Sch
o
o
l
o
f
Hasy
im
Ash
ar
i,
Ma
lan
g
.
T
h
e
lear
n
i
n
g
i
n
ter
v
en
tio
n
was
c
o
n
d
u
cte
d
ac
r
o
s
s
s
ev
er
al
s
ess
io
n
s
,
b
eg
in
n
in
g
with
an
in
tr
o
d
u
ctio
n
to
th
e
g
am
e
m
ec
h
an
ics,
f
o
llo
wed
b
y
ac
tiv
e
en
g
ag
em
en
t
in
co
m
p
letin
g
m
ath
em
atica
l
p
r
o
ject
m
is
s
io
n
s
,
an
d
co
n
clu
d
in
g
with
r
ef
lectio
n
an
d
ev
alu
atio
n
o
f
s
tu
d
en
ts
’
wo
r
k
.
3
.
3
.
Descript
iv
e
a
na
ly
s
is
o
f
c
re
a
t
iv
e
m
a
t
hema
t
ica
l t
hin
k
i
ng
T
o
q
u
an
titativ
ely
d
escr
ib
e
s
tu
d
en
ts
’
c
r
ea
tiv
e
m
ath
e
m
atica
l
th
in
k
in
g
a
b
ilit
ies
in
b
o
th
t
h
e
ex
p
er
im
en
tal
a
n
d
co
n
tr
o
l
g
r
o
u
p
s
b
ased
o
n
p
r
etest
an
d
p
o
s
ttes
t
r
esu
lts
,
d
escr
ip
tiv
e
s
tatis
ti
cs
wer
e
co
m
p
u
ted
.
T
h
ese
s
tati
s
tics
in
clu
d
ed
th
e
n
u
m
b
er
o
f
p
ar
ticip
an
ts
,
m
ea
n
s
co
r
es,
s
tan
d
ar
d
d
ev
iatio
n
s
,
s
tan
d
ar
d
er
r
o
r
s
,
an
d
co
ef
f
icien
ts
o
f
v
ar
iatio
n
.
T
h
e
d
escr
ip
tiv
e
an
al
y
s
is
was
u
s
ed
to
e
x
am
in
e
th
e
eq
u
i
v
alen
ce
o
f
s
tu
d
e
n
ts
’
in
itial
ab
ilit
ies
b
etwe
en
th
e
two
g
r
o
u
p
s
an
d
t
o
id
en
tif
y
ch
an
g
es
in
cr
ea
tiv
e
m
ath
e
m
atica
l
th
in
k
in
g
f
o
llo
win
g
th
e
im
p
lem
en
tatio
n
o
f
d
if
f
er
e
n
t
in
s
tr
u
ctio
n
al
ap
p
r
o
ac
h
es.
T
h
e
r
e
s
u
lts
o
f
th
e
d
escr
ip
tiv
e
s
tatis
ti
cs
ar
e
p
r
esen
ted
in
T
ab
le
3
.
Descr
ip
tiv
ely
,
at
th
e
p
r
etest
s
tag
e,
th
e
m
ea
n
cr
ea
tiv
e
m
ath
em
atica
l
th
in
k
in
g
s
co
r
e
o
f
th
e
ex
p
er
im
en
tal
g
r
o
u
p
(
M=
7
2
.
6
4
,
SD=5
.
2
4
9
)
was
co
m
p
ar
ab
le
to
th
at
o
f
th
e
co
n
tr
o
l
g
r
o
u
p
(
M=
7
1
.
9
5
,
SD=3
.
6
2
2
)
,
in
d
icatin
g
th
at
b
o
th
g
r
o
u
p
s
h
ad
e
q
u
iv
alen
t
in
itial
ab
ilit
ies.
Fo
llo
win
g
th
e
i
n
s
tr
u
ctio
n
al
in
ter
v
en
tio
n
,
b
o
t
h
g
r
o
u
p
s
d
em
o
n
s
tr
ated
s
co
r
e
im
p
r
o
v
em
en
ts
;
h
o
wev
er
,
th
e
in
c
r
ea
s
e
in
th
e
ex
p
er
im
en
tal
g
r
o
u
p
was
s
u
b
s
tan
tially
g
r
ea
ter
.
T
h
e
p
o
s
t
-
test
m
ea
n
s
co
r
e
o
f
th
e
ex
p
er
im
en
tal
g
r
o
u
p
in
cr
ea
s
ed
to
8
1
.
9
3
(
SD=4
.
1
9
3
)
,
wh
er
ea
s
th
e
c
o
n
tr
o
l
g
r
o
u
p
r
ea
ch
ed
a
m
ea
n
s
co
r
e
o
f
7
4
.
3
3
(
SD=3
.
2
1
3
)
.
No
r
m
ality
test
in
g
u
s
in
g
t
h
e
Sh
a
p
ir
o
–
W
ilk
test
in
d
icate
d
th
at
all
d
at
asets
wer
e
n
o
r
m
ally
d
is
tr
ib
u
ted
(
p
>
0
.
0
5
)
.
I
n
ad
d
itio
n
,
L
ev
en
e’
s
test
co
n
f
ir
m
ed
th
at
th
e
v
ar
ian
ce
s
b
etwe
en
g
r
o
u
p
s
wer
e
h
o
m
o
g
en
e
o
u
s
f
o
r
b
o
th
p
r
etest
an
d
p
o
s
t
-
tes
t
s
co
r
es
(
p
>0
.
0
5
)
,
in
d
icatin
g
th
at
t
h
e
h
o
m
o
g
e
n
ei
ty
ass
u
m
p
tio
n
was
s
atis
f
ied
.
T
h
is
tr
en
d
is
also
clea
r
ly
illu
s
tr
ated
in
Fig
u
r
e
3
.
As
s
h
o
wn
in
Fig
u
r
e
3
,
th
e
ex
p
er
im
en
tal
g
r
o
u
p
d
e
m
o
n
s
tr
a
tes
a
s
teep
er
in
cr
ea
s
e
f
r
o
m
p
r
etest
to
p
o
s
t
-
test
co
m
p
ar
ed
to
th
e
c
o
n
tr
o
l
g
r
o
u
p
.
T
h
is
s
u
g
g
ests
th
at
s
tu
d
en
ts
wh
o
lear
n
e
d
th
r
o
u
g
h
th
e
GPB
L
m
o
d
el
ex
p
er
ien
ce
d
a
s
tr
o
n
g
er
im
p
r
o
v
em
en
t in
c
r
e
ativ
e
m
ath
em
atica
l th
in
k
in
g
th
an
th
o
s
e
wh
o
r
ec
eiv
ed
d
ir
ec
t i
n
s
tr
u
ctio
n
.
T
ab
le
2
.
E
x
p
er
t
v
alid
atio
n
r
es
u
lts
o
f
th
e
p
r
o
ject
-
b
ased
g
a
m
e
p
r
o
to
ty
p
e
Ev
a
l
u
a
t
i
o
n
a
s
p
e
c
t
O
b
t
a
i
n
e
d
sc
o
r
e
M
a
x
i
m
u
m
sc
o
r
e
P
e
r
c
e
n
t
a
g
e
(
%)
C
a
t
e
g
o
r
y
C
o
n
t
e
n
t
a
p
p
r
o
p
r
i
a
t
e
n
e
ss
56
60
9
3
.
3
V
a
l
i
d
V
i
su
a
l
d
e
s
i
g
n
45
50
9
0
.
0
V
a
l
i
d
M
a
t
e
r
i
a
l
q
u
a
l
i
t
y
47
50
9
4
.
0
V
a
l
i
d
T
ab
le
3
.
Descr
ip
tiv
e
s
tatis
tics
o
f
cr
ea
tiv
e
m
at
h
em
atica
l th
in
k
in
g
C
r
e
a
t
i
v
e
t
h
i
n
k
i
n
g
a
b
i
l
i
t
y
G
r
o
u
p
N
M
e
a
n
SD
SE
C
o
e
f
f
i
c
i
e
n
t
o
f
v
a
r
i
a
t
i
o
n
S
h
a
p
i
r
o
-
W
i
l
k
P
-
v
a
l
u
e
o
f
S
h
a
p
i
r
o
-
W
i
l
k
Le
v
e
n
e
’
s
F
p
P
r
e
t
e
s
t
Ex
p
e
r
i
m
e
n
t
a
l
42
7
2
.
6
4
5
.
2
4
9
0
.
8
1
0
0
.
0
7
2
0
.
9
2
9
0
.
0
6
2
6
.
4
1
0
0
.
0
5
3
C
o
n
t
r
o
l
42
7
1
.
9
5
3
.
6
2
2
0
.
5
5
9
0
.
0
5
0
0
.
9
6
6
0
.
2
3
5
6
.
4
1
0
0
.
0
5
3
P
o
st
-
t
e
st
Ex
p
e
r
i
m
e
n
t
a
l
42
8
1
.
9
3
4
.
1
9
3
0
.
6
4
7
0
.
0
5
1
0
.
9
5
9
0
.
1
3
8
4
.
7
3
8
0
.
0
9
2
C
o
n
t
r
o
l
42
7
4
.
3
3
3
.
2
1
3
0
.
4
9
6
0
.
0
4
3
0
.
9
5
1
0
.
0
7
0
4
.
7
3
8
0
.
0
9
2
N
o
t
e
:
N
=
n
u
m
b
e
r
o
f
p
a
r
t
i
c
i
p
a
n
t
s;
S
D
=
st
a
n
d
a
r
d
d
e
v
i
a
t
i
o
n
;
S
E
=
st
a
n
d
a
r
d
e
r
r
o
r
;
F
=
F
s
t
a
t
i
s
t
i
c
f
r
o
m
Le
v
e
n
e
’
s t
e
st
;
p
=
s
i
g
n
i
f
i
c
a
n
c
e
v
a
l
u
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
.
15
,
No
.
2
,
Ap
r
il
20
2
6
:
1
6
9
8
-
1
7
1
0
1704
F
i
g
u
r
e
3
.
P
r
e
t
es
t
a
n
d
p
o
s
t
-
t
es
t
m
e
a
n
s
c
o
r
e
s
o
f
s
t
u
d
e
n
ts
’
c
r
e
a
tiv
e
t
h
i
n
k
i
n
g
i
n
t
h
e
e
x
p
e
r
i
m
e
n
t
al
a
n
d
c
o
n
t
r
o
l
g
r
o
u
p
s
3
.
4
.
I
nfe
re
ntia
l
a
na
ly
s
is
us
in
g
re
pea
t
ed
m
ea
s
u
re
s
ANO
VA
T
o
ex
am
in
e
th
e
s
tatis
tical
s
i
g
n
if
ican
ce
o
f
c
h
an
g
es
in
s
tu
d
en
ts
’
cr
ea
tiv
e
m
ath
em
atica
l
th
in
k
in
g
ab
ilit
ies
o
v
er
tim
e
an
d
to
co
m
p
ar
e
im
p
r
o
v
em
en
t
p
atter
n
s
b
e
twee
n
g
r
o
u
p
s
,
a
r
ep
ea
ted
m
ea
s
u
r
es
ANOV
A
wa
s
co
n
d
u
cte
d
.
T
h
e
with
in
-
s
u
b
jec
ts
’
ef
f
ec
ts
ar
e
p
r
esen
ted
in
T
ab
le
3
,
wh
ile
th
e
b
etwe
en
-
s
u
b
jects
ef
f
ec
ts
ar
e
p
r
esen
ted
in
T
a
b
le
4
.
As
s
h
o
wn
in
T
a
b
le
4
,
t
h
e
an
aly
s
is
r
ev
ea
led
a
s
ig
n
if
ican
t
m
ain
ef
f
ec
t
o
f
tim
e,
F(1
,
8
2
)
=1
7
6
.
9
6
,
p
<
0
.
0
0
1
,
η
²=
0
.
2
6
2
,
in
d
icatin
g
a
s
u
b
s
tan
ti
al
im
p
r
o
v
em
en
t
in
s
tu
d
e
n
ts
’
c
r
ea
tiv
e
m
ath
em
atica
l
th
in
k
in
g
f
r
o
m
p
r
etest
to
p
o
s
t
-
t
est
ac
r
o
s
s
b
o
th
g
r
o
u
p
s
.
Mo
r
eo
v
er
,
a
s
ig
n
if
ican
t
in
ter
ac
tio
n
e
f
f
ec
t
b
etwe
en
tim
e
an
d
g
r
o
u
p
was
f
o
u
n
d
,
F(1
,
8
2
)
=6
1
.
9
8
,
p
<
0
.
0
0
1
,
η
²=
0
.
0
9
2
,
s
u
g
g
esti
n
g
th
at
th
e
m
ag
n
itu
d
e
o
f
im
p
r
o
v
em
e
n
t
d
if
f
er
ed
s
ig
n
if
ican
tly
b
etwe
en
th
e
ex
p
er
im
en
tal
a
n
d
c
o
n
tr
o
l
g
r
o
u
p
s
.
T
h
is
in
ter
ac
tio
n
e
f
f
e
ct
in
d
icate
s
th
at
th
e
in
s
tr
u
ctio
n
al
m
o
d
el
p
lay
ed
a
n
im
p
o
r
tan
t r
o
le
in
s
h
ap
in
g
s
tu
d
en
ts
’
lear
n
in
g
g
ain
s
.
T
ab
le
4
.
W
ith
in
-
s
u
b
jects e
f
f
ec
ts
o
f
r
ep
ea
ted
m
ea
s
u
r
es
ANOVA
C
a
ses
S
u
m
o
f
s
q
u
a
r
e
s
Df
M
e
a
n
sq
u
a
r
e
F
P
Η²
C
r
e
a
t
i
v
e
t
h
i
n
k
i
n
g
a
b
i
l
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3
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am
in
e
th
e
p
atter
n
o
f
ch
a
n
g
e
in
s
tu
d
en
ts
’
cr
ea
tiv
e
m
ath
em
atica
l
th
in
k
in
g
ab
ilit
ies
m
o
r
e
d
ee
p
ly
,
a
p
o
ly
n
o
m
ial
co
n
tr
ast
an
aly
s
i
s
was
co
n
d
u
cted
.
T
h
e
r
esu
lts
ar
e
p
r
esen
ted
in
T
a
b
le
7
.
T
h
e
r
esu
lts
in
d
icate
th
at
all
test
ed
co
n
tr
asts
wer
e
s
ta
tis
t
ically
s
ig
n
if
ican
t
(
p
<
0
.
0
0
1
)
,
d
em
o
n
s
tr
atin
g
a
co
n
s
is
ten
t
an
d
m
ea
n
in
g
f
u
l
p
atter
n
o
f
ch
an
g
e
in
cr
ea
tiv
e
m
at
h
em
atica
l
th
in
k
in
g
ab
ilit
ies.
I
n
C
o
n
tr
ast
1
,
th
e
p
o
s
itiv
e
esti
m
at
e
(
3
.
3
6
5
)
in
d
icate
s
th
at
th
e
im
p
r
o
v
em
e
n
t
f
r
o
m
p
r
etest
to
p
o
s
t
-
test
was
s
u
b
s
tan
t
ially
s
tr
o
n
g
er
in
th
e
e
x
p
er
im
e
n
tal
g
r
o
u
p
th
a
n
i
n
th
e
co
n
tr
o
l
g
r
o
u
p
.
T
h
is
f
in
d
i
n
g
alig
n
s
with
th
e
co
n
tr
ast
co
ef
f
icien
ts
,
wh
ich
r
ev
ea
l
d
iv
e
r
g
en
t
tr
ajec
to
r
ies
o
f
ch
an
g
e
b
etwe
en
th
e
two
g
r
o
u
p
s
,
with
th
e
ex
p
er
im
e
n
tal
g
r
o
u
p
e
x
h
ib
itin
g
a
m
o
r
e
p
r
o
n
o
u
n
ce
d
in
c
r
ea
s
e.
Fu
r
th
er
m
o
r
e
,
C
o
n
tr
ast
s
2
a
n
d
3
y
ield
e
d
s
ig
n
if
ica
n
t
n
eg
ati
v
e
esti
m
ates
(
−3
.
4
5
2
an
d
−6
.
3
1
4
,
r
esp
ec
tiv
ely
)
,
in
d
icatin
g
d
if
f
er
en
ce
s
in
b
o
t
h
th
e
d
ir
ec
tio
n
an
d
m
ag
n
itu
d
e
o
f
c
h
an
g
e
ac
r
o
s
s
g
r
o
u
p
s
an
d
m
ea
s
u
r
em
e
n
t
o
cc
asio
n
s
.
T
h
ese
r
esu
lts
f
u
r
th
er
co
n
f
ir
m
th
at
GPB
L
p
r
o
d
u
ce
d
a
d
is
tin
ct
an
d
m
o
r
e
r
o
b
u
s
t
p
atter
n
o
f
im
p
r
o
v
em
e
n
t in
cr
ea
tiv
e
m
ath
em
atica
l th
in
k
in
g
c
o
m
p
ar
e
d
to
d
ir
ec
t in
s
tr
u
ctio
n
.
T
ab
le
7
.
Po
ly
n
o
m
ial
co
n
tr
ast an
aly
s
is
o
f
cr
ea
tiv
e
m
ath
em
atic
al
th
in
k
in
g
C
o
m
p
a
r
i
so
n
Est
i
m
a
t
e
SE
df
t
p
1
3
.
3
6
5
0
.
5
2
8
8
2
.
0
0
6
.
3
7
4
<
0
.
0
0
1
2
-
3
.
4
5
2
0
.
4
3
9
8
2
.
0
0
-
7
.
8
7
3
<
0
.
0
0
1
3
-
6
.
3
1
4
0
.
7
3
3
8
2
.
0
0
-
8
.
6
1
1
<
0
.
0
0
1
3
.
8
.
Dis
cus
s
io
n
T
h
e
f
in
d
in
g
s
s
u
g
g
est
th
at
GPB
L
p
r
o
v
id
es
a
m
ea
n
in
g
f
u
l
lear
n
in
g
e
n
v
ir
o
n
m
en
t
t
h
at
s
u
p
p
o
r
ts
th
e
d
ev
elo
p
m
e
n
t
o
f
s
tu
d
e
n
ts
’
cr
ea
tiv
e
m
ath
em
atica
l
th
i
n
k
in
g
in
g
eo
m
etr
y
p
r
o
b
lem
s
o
lv
i
n
g
.
B
y
in
teg
r
atin
g
g
am
e
-
b
ased
ex
p
lo
r
atio
n
with
p
r
o
ject
-
b
ased
task
s
,
GPB
L
en
co
u
r
a
g
es
s
tu
d
en
ts
to
a
ctiv
ely
in
v
esti
g
ate
g
eo
m
etr
ic
co
n
ce
p
ts
th
r
o
u
g
h
v
is
u
aliza
tio
n
,
o
b
ject
m
an
ip
u
latio
n
,
an
d
r
ef
lectiv
e
r
ea
s
o
n
in
g
r
a
th
er
th
an
r
el
y
in
g
o
n
m
em
o
r
ized
f
o
r
m
u
las.
T
h
is
lear
n
in
g
m
ec
h
a
n
is
m
en
ab
les
s
tu
d
en
ts
to
co
n
s
tr
u
ct
co
n
ce
p
tu
al
u
n
d
er
s
tan
d
in
g
p
r
o
g
r
ess
iv
ely
an
d
to
g
e
n
er
at
e
m
o
r
e
f
lex
i
b
le
an
d
o
r
ig
in
al
s
o
lu
tio
n
s
tr
ateg
ies.
T
h
ese
r
esu
lts
alig
n
with
th
e
s
y
s
tem
atic
liter
atu
r
e
r
ev
iew
b
y
Ma
y
r
h
o
f
e
r
et
a
l
.
[
2
4
]
,
wh
ich
em
p
h
asized
th
at
g
a
m
if
icatio
n
ca
n
p
o
s
itiv
ely
in
f
lu
en
ce
m
o
tiv
atio
n
an
d
c
o
g
n
itiv
e
en
g
ag
e
m
en
t
w
h
en
i
n
s
tr
u
ctio
n
al
d
esig
n
s
g
o
b
ey
o
n
d
p
o
i
n
t
-
b
ased
a
n
d
s
co
r
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
.
15
,
No
.
2
,
Ap
r
il
20
2
6
:
1
6
9
8
-
1
7
1
0
1706
d
r
iv
en
s
y
s
tem
s
.
C
o
n
s
is
ten
t
with
s
ev
er
al
s
tu
d
ies
[
2
5
]
–
[
2
7
]
,
th
e
p
r
esen
t
s
tu
d
y
f
u
r
t
h
er
in
d
icate
s
th
at
g
am
if
icatio
n
b
ec
o
m
es
m
o
r
e
ef
f
ec
tiv
e
wh
en
it
is
em
b
ed
d
e
d
in
ch
allen
g
i
n
g
p
r
o
b
lem
-
s
o
l
v
in
g
ac
tiv
ities
th
at
p
r
o
m
o
te
ex
p
l
o
r
atio
n
an
d
m
eta
co
g
n
itiv
e
awa
r
e
n
ess
.
T
h
er
ef
o
r
e,
th
e
g
am
e
c
o
m
p
o
n
en
t
i
n
GP
B
L
f
u
n
ctio
n
e
d
n
o
t
m
er
ely
as
an
en
g
ag
em
e
n
t
en
h
an
ce
r
b
u
t
as
a
s
ca
f
f
o
ld
in
g
to
o
l
th
at
f
ac
ilit
ated
d
ee
p
er
co
n
ce
p
tu
al
co
n
s
tr
u
ctio
n
an
d
cr
ea
tiv
e
r
ea
s
o
n
in
g
.
Mo
r
e
s
p
ec
if
ically
,
th
e
lev
el
-
b
ased
g
am
e
s
tr
u
ctu
r
e
with
in
GPB
L
p
lay
ed
a
cr
u
cial
r
o
le
in
s
h
ap
in
g
s
tu
d
en
ts
’
cr
ea
tiv
e
th
i
n
k
in
g
p
r
o
ce
s
s
es.
L
ev
els
1
an
d
2
s
u
p
p
o
r
ted
s
tu
d
e
n
ts
in
u
n
d
er
s
tan
d
i
n
g
th
e
cy
lin
d
e
r
as
a
th
r
ee
-
d
i
m
en
s
io
n
al
o
b
ject
t
h
r
o
u
g
h
v
is
u
al
ex
p
l
o
r
atio
n
,
w
h
er
ea
s
L
ev
els
3
an
d
4
en
co
u
r
ag
ed
s
tu
d
en
ts
to
co
n
n
ec
t
th
e
n
et
r
ep
r
esen
tatio
n
with
k
ey
g
eo
m
etr
ic
elem
e
n
ts
o
f
th
e
cy
lin
d
er
,
s
u
ch
as
r
ad
iu
s
,
h
eig
h
t,
an
d
cir
cu
m
f
er
en
ce
.
T
h
ese
s
tag
es
en
ab
led
s
tu
d
e
n
ts
to
o
b
s
er
v
e
th
e
r
elatio
n
s
h
ip
s
am
o
n
g
g
eo
m
etr
ic
co
m
p
o
n
en
ts
m
o
r
e
c
o
n
cr
etely
th
r
o
u
g
h
d
y
n
am
ic
v
is
u
aliza
tio
n
an
d
im
m
e
d
iate
f
ee
d
b
ac
k
f
r
o
m
in
p
u
t
ch
an
g
es.
T
h
is
lear
n
in
g
m
ec
h
an
is
m
alig
n
s
with
t
h
e
s
ev
er
al
f
in
d
in
g
s
[
2
9
]
,
[
3
0
]
,
[
3
3
]
,
wh
o
em
p
h
asized
th
e
im
p
o
r
tan
ce
o
f
au
t
h
en
tic
co
n
tex
ts
,
s
im
u
latio
n
,
a
n
d
i
n
ter
ac
tiv
e
v
is
u
aliza
tio
n
in
s
tr
en
g
th
en
in
g
co
g
n
itiv
e
p
r
o
ce
s
s
in
g
an
d
co
n
ce
p
tu
a
l
u
n
d
er
s
tan
d
i
n
g
.
T
h
e
v
is
u
aliza
tio
n
o
f
th
e
cy
lin
d
e
r
an
d
its
n
e
t
th
r
o
u
g
h
th
e
g
am
e
p
r
o
v
i
d
ed
a
m
o
r
e
m
ea
n
in
g
f
u
l
lear
n
in
g
ex
p
er
ien
ce
co
m
p
ar
e
d
to
co
n
v
e
n
tio
n
al
in
s
tr
u
ctio
n
,
wh
ich
o
f
ten
ten
d
s
to
b
e
p
r
o
c
ed
u
r
al.
At
L
ev
el
5
,
s
tu
d
en
ts
wer
e
r
eq
u
i
r
ed
to
co
n
s
tr
u
ct
th
e
cy
lin
d
er
s
u
r
f
ac
e
ar
ea
f
o
r
m
u
la
b
ased
o
n
th
e
n
et
th
e
y
ex
p
lo
r
ed
an
d
th
en
ap
p
ly
it
to
p
r
o
b
lem
s
o
lv
in
g
t
h
r
o
u
g
h
co
llab
o
r
ativ
e
p
r
o
ject
task
s
.
T
h
is
p
r
o
ject
s
tag
e
r
ep
r
esen
ts
a
k
ey
f
ea
tu
r
e
d
is
tin
g
u
is
h
in
g
th
e
p
r
esen
t
s
tu
d
y
f
r
o
m
m
o
r
e
g
en
e
r
al
g
am
if
i
ca
tio
n
im
p
lem
en
tatio
n
s
.
Stu
d
en
ts
wer
e
n
o
t
o
n
ly
co
m
p
letin
g
g
a
m
e
m
is
s
io
n
s
b
u
t
also
p
r
o
d
u
cin
g
a
m
ea
n
in
g
f
u
l
co
g
n
itiv
e
p
r
o
d
u
ct
in
th
e
f
o
r
m
o
f
a
d
er
iv
e
d
f
o
r
m
u
la
a
n
d
p
r
o
b
lem
-
s
o
lv
in
g
s
tr
ateg
ies.
T
h
is
f
in
d
in
g
s
tr
en
g
t
h
en
s
th
e
ar
g
u
m
en
ts
[
2
7
]
,
[
2
8
]
,
wh
o
r
e
p
o
r
ted
th
at
g
am
if
icatio
n
p
o
s
itiv
ely
af
f
ec
ts
th
e
co
g
n
itiv
e
d
o
m
ain
wh
en
d
esig
n
ed
to
p
r
o
m
o
te
d
ec
is
io
n
m
ak
in
g
a
n
d
p
r
o
b
lem
s
o
lv
in
g
.
Ho
wev
er
,
th
is
s
tu
d
y
ex
ten
d
s
th
o
s
e
f
in
d
i
n
g
s
b
y
d
e
m
o
n
s
tr
atin
g
th
at
wh
e
n
g
am
if
i
ca
tio
n
is
in
teg
r
ated
with
PjB
L
,
it
s
im
p
ac
t
b
ec
o
m
es
s
tr
o
n
g
er
b
ec
au
s
e
s
tu
d
en
ts
ar
e
en
co
u
r
a
g
ed
to
o
r
g
an
ize
id
ea
s
,
ju
s
tify
r
ea
s
o
n
in
g
,
an
d
co
n
s
tr
u
ct
m
ath
em
atica
l stra
teg
ies m
o
r
e
in
d
e
p
en
d
e
n
tly
.
Usi
n
g
th
e
in
d
icato
r
s
o
f
cr
ea
tiv
e
m
ath
em
atica
l
th
in
k
in
g
,
t
h
e
GPB
L
m
ec
h
an
is
m
p
r
o
v
id
es
a
clea
r
ex
p
lan
atio
n
o
f
h
o
w
f
lu
e
n
cy
,
f
lex
ib
ilit
y
,
an
d
o
r
ig
in
ality
d
ev
elo
p
ed
th
r
o
u
g
h
o
u
t
th
e
lear
n
in
g
p
r
o
ce
s
s
.
Flu
en
cy
was
p
r
im
ar
ily
s
tim
u
lated
in
L
ev
els
1
an
d
2
,
wh
e
r
e
s
tu
d
en
ts
o
b
s
er
v
ed
a
th
r
ee
-
d
im
en
s
io
n
al
cy
lin
d
er
m
o
d
el
an
d
r
ep
ea
ted
ly
ex
p
e
r
im
en
ted
with
d
if
f
er
en
t
in
p
u
t
v
alu
es
(
r
ad
i
u
s
an
d
h
eig
h
t)
,
en
a
b
lin
g
th
em
to
g
en
er
ate
m
u
ltip
le
p
o
s
s
ib
le
s
o
lu
tio
n
s
tep
s
.
Flex
ib
ilit
y
was
d
ev
elo
p
e
d
in
L
ev
els
3
an
d
4
,
as
s
tu
d
e
n
ts
wer
e
r
eq
u
ir
ed
to
s
h
if
t
ac
r
o
s
s
v
ar
io
u
s
r
e
p
r
esen
tatio
n
s
,
in
cl
u
d
in
g
th
r
ee
-
d
im
e
n
s
io
n
al
v
is
u
ali
za
tio
n
,
n
et
r
ep
r
esen
tatio
n
,
a
n
d
r
elatio
n
al
a
n
aly
s
is
am
o
n
g
g
e
o
m
etr
ic
elem
en
ts
b
ef
o
r
e
co
n
s
tr
u
ctin
g
th
e
f
o
r
m
u
la.
T
h
ese
ac
tiv
ities
en
co
u
r
ag
ed
s
tu
d
en
ts
to
ap
p
r
o
ac
h
th
e
co
n
ce
p
t
f
r
o
m
d
if
f
er
e
n
t
p
e
r
s
p
ec
tiv
es
an
d
a
p
p
ly
d
iv
er
s
e
s
tr
ateg
ies.
Me
an
wh
ile,
o
r
ig
i
n
ality
em
er
g
ed
m
o
s
t
s
tr
o
n
g
ly
in
L
ev
el
5
,
wh
en
s
t
u
d
en
ts
wer
e
r
e
q
u
ir
ed
to
d
e
r
iv
e
th
e
s
u
r
f
ac
e
ar
ea
f
o
r
m
u
la
b
ased
o
n
t
h
eir
o
wn
ex
p
lo
r
atio
n
d
u
r
in
g
th
e
p
r
o
jec
t
s
tag
e
r
ath
er
t
h
an
m
e
m
o
r
izin
g
o
r
co
p
y
in
g
f
o
r
m
u
las
p
r
o
v
id
ed
b
y
th
e
teac
h
e
r
.
T
h
is
m
ec
h
an
is
m
is
co
n
s
is
ten
t
with
B
ar
an
et
a
l
.
[
3
8
]
an
d
C
o
x
b
ill
et
a
l.
[
1
1
]
,
wh
o
a
r
g
u
e
d
th
at
m
ath
em
atica
l
cr
ea
tiv
ity
d
ev
elo
p
s
th
r
o
u
g
h
p
r
o
b
lem
-
s
o
lv
in
g
s
itu
atio
n
s
th
at
allo
w
m
u
ltip
le
s
tr
ateg
ies.
Activ
ities
s
u
ch
as
u
n
f
o
ld
i
n
g
cy
lin
d
er
n
ets,
lin
k
in
g
cir
cle
cir
cu
m
f
er
en
ce
to
th
e
len
g
th
o
f
th
e
later
al
s
u
r
f
ac
e
,
an
d
in
d
ep
en
d
en
tl
y
f
o
r
m
u
latin
g
th
e
s
u
r
f
ac
e
ar
ea
eq
u
atio
n
r
ep
r
esen
t
n
o
n
-
r
o
u
tin
e
task
s
alig
n
ed
with
p
r
ev
io
u
s
s
tu
d
ies
[
1
3
]
,
[
1
4
]
,
wh
o
em
p
h
asized
t
h
e
ef
f
ec
tiv
e
n
ess
o
f
o
p
en
-
en
d
e
d
p
r
o
b
lem
s
i
n
f
o
s
ter
in
g
f
lex
ib
ilit
y
an
d
o
r
ig
in
ality
.
Fu
r
th
er
m
o
r
e
,
th
e
f
in
d
in
g
s
clar
if
y
th
at
g
am
if
icatio
n
co
m
b
i
n
e
d
with
PjB
L
p
r
o
d
u
ce
s
a
m
o
r
e
co
n
s
is
ten
t
im
p
ac
t
th
an
s
u
p
er
f
icial
g
am
i
f
icatio
n
d
esig
n
s
.
Sev
er
al
p
r
ev
i
o
u
s
s
tu
d
ies
h
av
e
r
ep
o
r
ted
th
at
g
am
if
icatio
n
ef
f
ec
ts
o
n
h
ig
h
er
-
o
r
d
e
r
th
i
n
k
in
g
ten
d
to
b
e
m
o
d
er
ate
o
r
u
n
s
tab
le
wh
en
g
a
m
if
icatio
n
f
o
cu
s
es
p
r
im
ar
ily
o
n
ex
ter
n
al
elem
en
ts
s
u
ch
as
p
o
in
ts
,
r
ewa
r
d
s
,
o
r
c
o
m
p
etitio
n
.
I
n
th
e
p
r
e
s
en
t
s
tu
d
y
,
th
e
g
am
e
was
n
o
t
im
p
lem
en
ted
as
an
is
o
lated
m
o
tiv
atio
n
al
to
o
l
b
u
t
as
p
ar
t
o
f
a
PjB
L
s
eq
u
en
ce
th
at
r
eq
u
ir
ed
ex
p
lo
r
atio
n
,
c
o
n
ce
p
tu
al
co
n
s
tr
u
ctio
n
,
an
d
ap
p
licatio
n
in
p
r
o
b
lem
s
o
lv
in
g
.
As
a
r
esu
lt,
s
tu
d
en
ts
wer
e
en
co
u
r
a
g
ed
to
en
g
a
g
e
in
d
ee
p
er
th
in
k
in
g
p
r
o
ce
s
s
es,
wh
ich
is
co
n
s
is
ten
t
with
p
r
e
v
io
u
s
s
tu
d
ies
[
2
9
]
,
[
3
0
]
,
[
3
3
]
,
r
eg
ar
d
in
g
th
e
im
p
o
r
tan
ce
o
f
au
t
h
en
tic
an
d
s
im
u
latio
n
-
b
ased
lear
n
in
g
d
esig
n
s
to
s
tr
en
g
th
en
co
n
ce
p
t
u
al
u
n
d
e
r
s
tan
d
in
g
.
I
n
ad
d
itio
n
,
Su
b
an
ji
et
a
l
.
[
1
9
]
f
i
v
e
-
s
tag
e
m
o
d
el
o
f
cr
ea
tiv
e
m
ath
em
atica
l
th
in
k
in
g
p
r
o
v
id
e
s
a
d
ev
elo
p
m
en
tal
len
s
to
e
x
p
lain
h
o
w
GPB
L
s
u
p
p
o
r
ted
s
tu
d
en
ts
’
p
r
o
g
r
ess
io
n
f
r
o
m
im
itatio
n
to
war
d
co
n
s
tr
u
ctio
n
.
At
th
e
b
eg
in
n
i
n
g
,
s
tu
d
e
n
ts
ten
d
ed
to
o
p
e
r
ate
at
th
e
im
itatio
n
s
tag
e,
r
ep
licatin
g
s
o
lu
tio
n
s
tr
ateg
ies
d
em
o
n
s
tr
ated
b
y
th
e
teac
h
er
wh
e
n
d
escr
ib
i
n
g
c
y
lin
d
er
p
r
o
p
er
ties
an
d
d
eter
m
i
n
in
g
s
u
r
f
ac
e
ar
ea
.
T
h
is
co
n
d
itio
n
is
co
n
s
is
ten
t
with
th
e
f
in
d
i
n
g
s
o
f
Su
s
ilo
wati
et
a
l
.
[
1
8
]
,
wh
o
r
ep
o
r
ted
th
at
m
an
y
elem
e
n
tar
y
s
ch
o
o
l
s
tu
d
en
ts
r
em
ai
n
at
t
h
e
im
itatio
n
s
tag
e
in
cr
ea
ti
v
e
m
ath
em
atica
l
th
in
k
in
g
.
Ho
wev
er
,
th
r
o
u
g
h
g
am
e
-
b
ased
ac
tiv
ities
th
at
all
o
wed
s
tu
d
en
ts
t
o
m
a
n
ip
u
late
r
ad
iu
s
a
n
d
h
eig
h
t
v
alu
es
an
d
d
ir
ec
tly
o
b
s
er
v
e
ch
an
g
es
in
o
u
tc
o
m
es,
s
tu
d
en
t
s
b
eg
an
to
m
o
v
e
in
to
th
e
m
o
d
if
icatio
n
s
tag
e
b
y
a
d
ap
tin
g
a
n
d
r
ef
in
i
n
g
ex
is
tin
g
s
tr
ateg
ies
b
ased
o
n
th
eir
e
x
p
lo
r
atio
n
s
.
W
h
en
s
tu
d
e
n
ts
u
n
f
o
ld
ed
th
e
cy
lin
d
e
r
a
n
d
ex
am
in
e
d
its
n
et
r
ep
r
esen
tatio
n
,
th
e
m
o
d
if
icati
o
n
p
r
o
ce
s
s
b
ec
am
e
m
o
r
e
e
v
id
en
t
as
th
e
y
r
e
v
is
ed
th
eir
u
n
d
er
s
tan
d
in
g
b
y
s
h
if
tin
g
f
r
o
m
a
th
r
ee
-
d
im
e
n
s
io
n
al
p
e
r
s
p
ec
tiv
e
to
a
two
-
d
im
en
s
io
n
al
r
ep
r
esen
tatio
n
.
Fu
r
th
er
m
o
r
e,
d
u
r
in
g
th
e
an
aly
s
is
s
tag
e,
s
tu
d
en
ts
d
e
m
o
n
s
tr
ated
ch
ar
ac
ter
is
tics
o
f
th
e
co
m
b
in
a
tio
n
s
tag
e
b
y
in
teg
r
atin
g
two
o
r
m
o
r
e
co
n
ce
p
ts
o
r
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
Ho
w
g
a
me
–
p
r
o
ject
lea
r
n
in
g
e
n
h
a
n
ce
s
crea
tive
th
in
kin
g
in
el
eme
n
ta
r
y
g
eo
metry?
(
A
n
to
n
P
r
a
yitn
o
)
1707
s
tr
ateg
ies,
s
u
ch
as
lin
k
in
g
th
e
cir
cu
m
f
er
en
ce
o
f
a
cir
cle
with
th
e
ar
ea
o
f
a
r
ec
tan
g
le
t
o
ex
p
lain
th
e
later
al
s
u
r
f
ac
e
an
d
r
elatin
g
it
t
o
th
e
b
ase
ar
ea
.
Fin
ally
,
th
r
o
u
g
h
th
e
p
r
o
ject
task
,
s
tu
d
e
n
ts
r
ea
ch
ed
th
e
co
n
s
tr
u
ctio
n
s
tag
e,
in
d
ep
en
d
e
n
tly
d
ev
el
o
p
in
g
s
o
lu
tio
n
s
tr
ateg
ies
b
y
d
er
iv
in
g
th
e
s
u
r
f
ac
e
a
r
ea
f
o
r
m
u
la
f
r
o
m
th
e
n
et
s
tr
u
ctu
r
e
an
d
a
p
p
ly
in
g
it to
s
o
l
v
e
p
r
o
b
lem
s
.
C
o
m
p
ar
ed
to
in
q
u
ir
y
a
n
d
d
is
co
v
er
y
lear
n
in
g
a
p
p
r
o
ac
h
es
[
2
0
]
–
[
2
2
]
,
th
is
s
tu
d
y
s
u
g
g
ests
th
at
g
am
e
-
b
ased
tech
n
o
l
o
g
y
in
teg
r
atio
n
o
f
f
er
s
ad
d
ed
v
alu
e
th
r
o
u
g
h
d
y
n
am
ic
v
is
u
aliza
tio
n
an
d
im
m
ed
iate
f
ee
d
b
ac
k
.
I
n
th
is
s
tu
d
y
,
in
q
u
ir
y
a
n
d
d
is
co
v
er
y
p
r
o
ce
s
s
es
s
till
o
cc
u
r
r
ed
;
h
o
wev
er
,
th
e
y
wer
e
e
n
r
ich
ed
b
y
d
ig
ital
m
ed
ia
th
at
allo
wed
s
tu
d
en
ts
to
ex
p
lo
r
e
co
n
ce
p
ts
m
o
r
e
co
n
c
r
etely
an
d
i
n
ter
ac
tiv
ely
.
T
h
ese
f
in
d
in
g
s
s
u
p
p
o
r
t
th
e
r
esear
ch
by
Sh
aw
et
a
l
.
[
2
3
]
,
wh
o
em
p
h
asized
th
at
m
ath
em
atica
l
cr
ea
tiv
ity
ca
n
b
e
im
p
r
o
v
e
d
th
r
o
u
g
h
o
p
en
-
en
d
e
d
task
s
th
at
allo
w
r
ef
lectio
n
an
d
id
e
a
ex
p
lo
r
atio
n
,
p
ar
ticu
la
r
ly
w
h
en
s
tu
d
en
ts
ar
e
g
i
v
en
o
p
p
o
r
tu
n
ities
to
ac
tiv
ely
co
n
s
tr
u
ct
k
n
o
wled
g
e.
B
ased
o
n
th
e
o
v
er
all
r
esu
lts
,
th
e
m
ajo
r
co
n
tr
i
b
u
tio
n
o
f
th
is
s
tu
d
y
lies
in
its
ex
p
licit
in
te
g
r
atio
n
o
f
g
am
if
icatio
n
,
PjB
L
,
an
d
t
h
e
d
ev
elo
p
m
en
t
o
f
cr
ea
tiv
e
m
ath
e
m
atica
l
th
in
k
in
g
with
in
o
n
e
c
o
h
er
en
t
in
s
tr
u
ctio
n
al
d
esig
n
.
Mo
s
t
p
r
ev
io
u
s
s
tu
d
ies
h
av
e
ex
am
in
e
d
g
a
m
if
icatio
n
,
g
am
e
-
b
ased
lear
n
i
n
g
,
o
r
PjB
L
s
ep
ar
ately
.
T
h
is
s
tu
d
y
d
em
o
n
s
tr
ates
th
at
th
e
s
y
n
er
g
y
o
f
th
ese
th
r
ee
co
m
p
o
n
en
ts
p
r
o
d
u
ce
s
a
s
tr
o
n
g
er
ef
f
ec
t
o
n
s
tu
d
en
ts
’
cr
ea
tiv
e
m
ath
em
atica
l
th
in
k
in
g
,
as
r
ef
lecte
d
in
th
e
m
o
d
er
at
e
-
to
-
lar
g
e
ef
f
ec
t
s
ize
v
alu
es.
Desp
ite
th
e
p
o
s
itiv
e
r
esu
lts
,
th
is
s
tu
d
y
h
as
s
ev
er
al
lim
itatio
n
s
.
T
h
e
s
am
p
le
in
v
o
lv
ed
o
n
l
y
8
4
s
tu
d
en
ts
f
r
o
m
o
n
e
s
ch
o
o
l,
a
n
d
th
e
in
ter
v
en
tio
n
last
ed
o
n
ly
two
wee
k
s
,
wh
ich
m
ay
lim
it
g
en
er
aliza
b
ilit
y
an
d
lo
n
g
-
ter
m
c
o
n
clu
s
io
n
s
.
T
ea
ch
er
ef
f
ec
ts
m
ay
also
h
av
e
in
f
lu
en
ce
d
lear
n
in
g
o
u
tco
m
es.
I
n
ad
d
itio
n
,
th
is
s
tu
d
y
d
id
n
o
t
an
aly
ze
f
lu
en
cy
,
f
lex
ib
ilit
y
,
an
d
o
r
ig
in
ality
s
e
p
ar
ately
n
o
r
m
ap
s
tu
d
e
n
ts
’
p
r
o
ce
s
s
es
to
Su
b
an
ji
et
a
l
.
[
1
9
]
f
iv
e
-
s
tag
e
m
o
d
el.
Fu
tu
r
e
r
esear
ch
s
h
o
u
l
d
in
v
o
l
v
e
lar
g
e
r
m
u
lti
-
s
ch
o
o
l
s
am
p
l
es,
lo
n
g
er
in
ter
v
e
n
tio
n
s
,
d
ee
p
er
in
d
icato
r
-
b
ased
an
aly
s
is
,
an
d
m
ay
e
x
p
lo
r
e
em
er
g
in
g
tech
n
o
lo
g
ies s
u
ch
as a
u
g
m
en
ted
r
ea
lity
t
o
en
r
ic
h
GPB
L
im
p
lem
en
tatio
n
.
4.
CO
NCLU
SI
O
N
B
ased
o
n
th
e
an
aly
s
es,
th
e
i
m
p
lem
en
tatio
n
o
f
GPB
L
is
e
f
f
ec
tiv
e
in
en
h
an
ci
n
g
s
tu
d
en
t
s
’
cr
ea
tiv
e
m
ath
em
atica
l
th
in
k
in
g
in
g
eo
m
etr
ic
p
r
o
b
lem
s
o
lv
in
g
.
I
n
s
tr
u
ctio
n
th
at
in
teg
r
ates
g
am
if
icatio
n
with
p
r
o
ject
-
b
ased
ac
tiv
ities
en
co
u
r
ag
es
s
tu
d
en
ts
to
ac
tiv
ely
en
g
a
g
e
in
co
n
ce
p
tu
al
ex
p
lo
r
atio
n
,
en
ab
l
in
g
th
em
to
m
o
v
e
b
ey
o
n
d
p
r
o
ce
d
u
r
al
ex
ec
u
ti
o
n
to
war
d
th
e
c
o
n
s
tr
u
ctio
n
o
f
u
n
d
er
s
tan
d
in
g
th
r
o
u
g
h
f
lex
ib
le,
o
r
ig
in
al,
an
d
m
ea
n
in
g
f
u
l
r
ea
s
o
n
in
g
.
T
h
ese
f
in
d
in
g
s
d
ir
ec
tly
ad
d
r
ess
th
e
r
esear
ch
o
b
jectiv
e
b
y
d
em
o
n
s
tr
atin
g
th
at
im
p
r
o
v
em
e
n
ts
in
cr
ea
tiv
e
m
at
h
em
atica
l
th
in
k
in
g
wer
e
s
u
b
s
tan
tially
g
r
ea
ter
in
th
e
e
x
p
er
im
en
tal
g
r
o
u
p
th
an
i
n
th
e
g
r
o
u
p
r
ec
ei
v
in
g
d
ir
ec
t
in
s
tr
u
ctio
n
.
T
h
e
p
r
im
a
r
y
co
n
t
r
ib
u
tio
n
o
f
th
is
s
tu
d
y
lies
in
s
tr
en
g
th
en
in
g
em
p
ir
ical
ev
id
en
ce
t
h
at
th
e
s
y
n
er
g
y
b
etwe
en
g
am
if
icatio
n
an
d
PjB
L
r
ep
r
esen
ts
a
r
ele
v
an
t
a
n
d
e
f
f
e
ctiv
e
p
ed
a
g
o
g
ical
d
esig
n
in
m
at
h
em
atics
ed
u
c
atio
n
,
p
ar
ticu
la
r
ly
f
o
r
f
o
s
ter
in
g
s
tu
d
en
ts
’
h
ig
h
er
-
o
r
d
e
r
t
h
in
k
in
g
s
k
ills
.
B
y
p
o
s
itio
n
in
g
g
am
e
m
ec
h
an
ics
a
s
co
g
n
itiv
e
to
o
ls
r
ath
er
th
a
n
m
er
ely
m
o
tiv
atio
n
al
elem
e
n
ts
,
th
e
GPB
L
ap
p
r
o
ac
h
o
f
f
er
s
a
s
tr
u
ctu
r
e
d
p
ath
wa
y
f
o
r
s
u
p
p
o
r
tin
g
cr
ea
tiv
e
m
ath
em
a
tical
th
in
k
in
g
in
class
r
o
o
m
p
r
a
ctice
.
M
ath
em
atics
teac
h
er
s
ar
e
en
co
u
r
ag
e
d
to
im
p
lem
en
t
GPB
L
b
y
u
tili
zin
g
ac
ce
s
s
ib
le
an
d
lo
w
-
co
s
t
d
ig
ital
to
o
ls
,
s
u
ch
as
Py
th
o
n
-
b
ased
ap
p
licatio
n
s
(
T
h
o
n
n
y
)
,
G
e
o
G
e
b
r
a
,
o
r
o
t
h
e
r
i
n
t
e
r
a
c
t
i
v
e
v
i
s
u
a
li
z
a
ti
o
n
m
ed
ia.
I
n
s
tr
u
ctio
n
ca
n
b
e
d
esig
n
ed
with
s
tep
-
by
-
s
tep
s
ca
f
f
o
ld
in
g
,
s
tar
tin
g
f
r
o
m
o
b
s
er
v
in
g
3
D
s
h
a
p
es,
u
n
f
o
ld
in
g
,
an
d
ex
am
in
in
g
n
ets,
an
aly
zin
g
r
el
atio
n
s
h
ip
s
am
o
n
g
g
eo
m
etr
ic
elem
en
ts
,
an
d
f
in
ally
c
o
n
s
tr
u
ctin
g
f
o
r
m
u
las
an
d
ap
p
ly
in
g
th
em
in
p
r
o
ject
task
s
.
T
h
r
o
u
g
h
th
is
p
r
o
ce
s
s
,
th
e
g
am
e
f
u
n
ctio
n
s
as
a
lear
n
in
g
to
o
l
th
at
s
u
p
p
o
r
ts
co
n
ce
p
tu
al
u
n
d
er
s
tan
d
i
n
g
an
d
r
ea
s
o
n
in
g
,
r
ath
er
t
h
an
s
er
v
in
g
as
en
ter
tain
m
en
t
alo
n
e.
Desp
ite
th
e
p
o
s
itiv
e
o
u
tco
m
es,
th
is
s
tu
d
y
h
as
s
ev
er
al
lim
itatio
n
s
.
I
t
d
id
n
o
t
ex
am
in
e
th
e
d
ev
elo
p
m
en
t
o
f
ea
ch
cr
ea
tiv
e
m
ath
em
atica
l
th
in
k
in
g
in
d
icat
o
r
(
f
l
u
e
n
c
y
,
f
l
e
x
i
b
i
l
it
y
,
a
n
d
o
r
ig
i
n
a
l
i
t
y
)
s
e
p
a
r
a
te
l
y
.
I
n
ad
d
itio
n
,
th
e
co
n
ten
t
s
co
p
e
was
lim
ited
,
th
e
s
am
p
le
s
ize
was
r
elativ
ely
s
m
all,
an
d
th
e
i
n
ter
v
en
tio
n
p
er
io
d
was
r
elativ
ely
s
h
o
r
t.
T
h
e
r
ef
o
r
e
,
f
u
tu
r
e
r
esear
ch
is
r
ec
o
m
m
en
d
ed
to
im
p
lem
en
t
GPB
L
o
v
er
a
lo
n
g
er
d
u
r
atio
n
,
in
v
o
l
v
e
m
u
ltip
le
s
ch
o
o
ls
with
d
iv
er
s
e
s
tu
d
en
t
ch
ar
ac
te
r
is
tics
,
an
d
ex
ten
d
its
ap
p
licatio
n
to
o
th
er
m
ath
em
atics
to
p
ics
b
ey
o
n
d
g
e
o
m
etr
y
.
Mo
r
eo
v
er
,
th
ese
f
in
d
i
n
g
s
ar
e
co
n
s
is
ten
t
with
I
n
d
o
n
esia
’
s
cu
r
r
en
t
c
u
r
r
icu
lu
m
d
ir
ec
tio
n
,
wh
ich
em
p
h
asizes
co
m
p
eten
cy
-
b
ased
lear
n
i
n
g
an
d
th
e
d
ev
el
o
p
m
en
t
o
f
2
1
s
t
-
ce
n
tu
r
y
s
k
ills
.
T
h
u
s
,
GPB
L
m
ay
s
er
v
e
as
a
p
r
ac
tical
in
s
tr
u
ctio
n
al
m
o
d
el
t
o
s
u
p
p
o
r
t
s
tu
d
en
ts
’
cr
ea
tiv
ity
,
cr
itical
th
i
n
k
in
g
,
an
d
p
r
o
b
lem
-
s
o
lv
in
g
ab
ilit
ies.
ACK
NO
WL
E
DG
M
E
N
T
S
T
h
e
au
th
o
r
s
wo
u
ld
lik
e
to
ex
p
r
ess
th
eir
s
in
ce
r
e
g
r
atitu
d
e
to
all
s
u
r
v
ey
an
d
in
ter
v
iew
p
ar
ti
cip
an
ts
f
o
r
th
eir
v
alu
ab
le
tim
e
a
n
d
c
o
n
tr
ib
u
tio
n
s
.
T
h
eir
s
u
p
p
o
r
t is g
r
ea
tly
ap
p
r
ec
iated
.
F
UNDING
I
NF
O
R
M
A
T
I
O
N
T
h
e
au
th
o
r
s
d
ec
lar
e
th
at
th
is
r
esear
ch
r
ec
eiv
ed
n
o
ex
te
r
n
al
f
u
n
d
in
g
.
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