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A
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Sc
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an
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Mo
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I
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8822
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39
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1
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I
n
cl
u
d
in
g
w
r
iti
n
g
in
m
at
h
e
m
atic
s
c
u
r
r
icu
l
u
m
d
ev
iates
t
h
e
s
t
u
d
en
ts
’
f
o
cu
s
to
s
o
le
m
at
h
e
m
a
tical
co
m
p
u
tatio
n
an
d
g
iv
e
s
th
e
m
co
n
f
id
en
ce
to
e
x
p
r
ess
t
h
eir
id
ea
s
an
d
h
a
v
e
a
d
ee
p
er
u
n
d
er
s
tan
d
i
n
g
o
f
th
e
co
n
ten
t
[
1
3
]
.
L
iter
at
u
r
e
o
f
f
er
s
ev
id
en
ce
t
h
at
teac
h
er
s
,
asid
e
f
r
o
m
t
h
e
s
tu
d
e
n
ts
,
b
en
e
f
it
f
r
o
m
g
i
v
i
n
g
w
r
i
tin
g
ac
tiv
itie
s
li
k
e
j
o
u
r
n
al
in
g
to
t
h
eir
s
t
u
d
en
ts
.
P
o
w
e
ll
[
1
4
]
p
o
in
ted
o
u
t
t
h
at
“w
r
iti
n
g
to
lea
r
n
m
ath
e
m
atic
s
i
s
tr
an
s
f
o
r
m
ati
v
e
n
o
t
o
n
l
y
f
o
r
lear
n
er
s
b
u
t
f
o
r
in
s
tr
u
cto
r
s
a
s
w
ell”
(
p
.
1
7
4
)
.
T
h
is
is
s
u
p
p
o
r
ted
b
y
t
h
e
s
tu
d
y
o
f
A
d
d
is
o
n
[
15
]
w
h
ich
r
ev
ea
le
d
th
at
s
tu
d
e
n
ts
’
j
o
u
r
n
al
s
i
n
c
r
ea
s
e
th
e
teac
h
er
’
s
a
w
ar
en
es
s
o
f
h
i
s
s
t
u
d
en
ts
’
co
m
p
r
e
h
en
s
io
n
o
f
co
n
ce
p
ts
,
f
ee
lin
g
s
,
a
n
d
attitu
d
e
to
w
ar
d
s
th
e
s
u
b
j
ec
t.
I
t
s
ets
an
a
v
e
n
u
e
f
o
r
h
i
m
to
p
o
n
d
er
u
p
o
n
t
h
e
s
u
cc
e
s
s
es
an
d
c
h
alle
n
g
e
s
o
f
teac
h
in
g
m
at
h
e
m
ati
c
s
.
J
o
u
r
n
al
w
r
iti
n
g
o
f
th
e
s
t
u
d
e
n
ts
s
tr
e
n
g
t
h
e
n
s
th
e
lin
e
o
f
co
m
m
u
n
ica
tio
n
b
et
w
e
en
t
h
e
m
an
d
t
h
e
teac
h
er
[
1
5
-
16]
.
I
t
ass
is
ts
th
e
teac
h
er
i
n
p
la
n
n
i
n
g
i
n
s
tr
u
ctio
n
a
l
m
et
h
o
d
o
lo
g
ies
an
d
ass
e
s
s
m
e
n
t
o
f
h
is
s
t
u
d
en
ts
’
u
n
d
er
s
tan
d
i
n
g
[
7
]
an
d
ev
en
d
if
f
ic
u
ltie
s
[
3
]
.
No
t
o
n
l
y
d
o
es
it
p
r
o
v
id
e
th
e
teac
h
er
w
i
th
a
n
o
t
h
er
s
tr
ate
g
y
f
o
r
tar
g
et
s
t
u
d
en
t
s
to
co
m
m
u
n
icate
th
eir
t
h
o
u
g
h
ts
b
u
t
it
al
s
o
allo
w
s
th
e
teac
h
er
“
to
s
ta
y
i
n
to
u
ch
ev
e
n
w
it
h
t
h
e
m
o
s
t
q
u
iet
a
n
d
r
eser
v
ed
s
tu
d
e
n
t”
[
1
5
]
.
Ad
d
itio
n
all
y
,
j
o
u
r
n
a
l
w
r
iti
n
g
a
f
f
o
r
d
s
th
e
o
p
p
o
r
tu
n
it
y
f
o
r
th
e
s
t
u
d
en
t
s
to
co
m
m
u
n
i
ca
te
to
th
e
ir
teac
h
er
m
a
th
e
m
at
ical
id
ea
s
,
w
h
ic
h
i
s
o
n
e
o
f
th
e
g
o
als s
tr
e
s
s
ed
in
N
C
T
M’
s
s
ta
n
d
ar
d
s
[
1
7
]
.
Ho
w
e
v
er
,
m
o
s
t
o
f
t
h
e
s
e
s
tu
d
i
es
h
av
e
f
a
iled
to
f
o
cu
s
o
n
e
x
p
lo
r
in
g
t
h
e
f
ee
l
in
g
s
o
f
co
lleg
e
s
t
u
d
en
t
s
to
w
ar
d
s
r
ef
lec
tiv
e
j
o
u
r
n
al
w
r
i
t
in
g
(
R
J
W
)
an
d
h
o
w
s
u
c
h
h
as
an
ef
f
ec
t
o
n
th
e
m
esp
ec
iall
y
i
n
th
e
ir
m
at
h
e
m
atics
lear
n
in
g
.
L
i
ter
atu
r
e
alr
ea
d
y
p
r
o
v
id
es
e
n
o
u
g
h
d
ata
to
s
h
o
w
t
h
at
b
en
e
f
its
o
f
j
o
u
r
n
al
w
r
it
in
g
a
r
e
d
escr
ib
ed
b
ased
o
n
th
e
r
atin
g
s
o
f
th
e
teac
h
er
s
an
d
not
o
f
th
e
s
tu
d
e
n
ts
w
h
o
a
r
e
th
e
f
o
ca
l
p
ar
ticip
an
ts
i
n
th
e
ac
tiv
it
y
.
R
esear
c
h
h
as te
n
d
ed
to
f
o
c
u
s
o
n
h
o
w
t
h
e
ac
tiv
it
y
i
s
i
n
s
tr
u
m
en
ta
l in
th
e
s
u
cc
e
s
s
o
f
teac
h
in
g
a
n
d
lear
n
in
g
m
at
h
e
m
atics
a
s
ex
p
r
ess
ed
b
y
t
h
e
teac
h
er
s
.
Ho
w
t
h
e
s
tu
d
e
n
ts
ex
ac
t
l
y
f
ee
l
ab
o
u
t
R
J
W
,
its
s
tr
en
g
t
h
s
an
d
w
e
ak
n
e
s
s
e
s
,
i
s
s
till
n
o
t
co
m
p
lete
l
y
u
n
co
v
er
ed
.
A
d
d
iti
o
n
all
y
,
a
co
n
s
id
er
ab
le
am
o
u
n
t
o
f
r
esear
ch
h
a
s
b
ee
n
d
o
n
e
o
n
s
tu
d
e
n
ts
i
n
t
h
e
p
r
im
ar
y
s
c
h
o
o
l
to
in
v
es
tig
a
te
th
eir
le
v
els
o
f
ac
h
iev
e
m
e
n
t
an
d
m
etac
o
g
n
itio
n
i
n
e
x
p
r
ess
in
g
th
e
ir
n
ee
d
s
i
n
ter
m
s
o
f
m
at
h
e
m
a
tics
teac
h
i
n
g
a
n
d
lear
n
i
n
g
t
h
r
o
u
g
h
R
J
W
[
11
-
18
-
20]
.
Yet,
less
atte
n
tio
n
h
as
b
ee
n
p
aid
to
s
tu
d
y
in
g
s
tu
d
e
n
ts
i
n
th
e
co
lle
g
e
lev
el.
I
n
De
L
a
Salle
L
ip
a,
s
t
u
d
en
t
s
ar
e
h
o
n
ed
to
b
e
cr
itica
l
t
h
i
n
k
er
s
,
b
ein
g
o
n
e
o
f
E
x
p
ec
ted
L
a
s
allia
n
Gr
ad
u
ate
A
ttrib
u
tes.
I
n
t
h
e
at
te
m
p
ts
o
f
th
e
S
c
h
o
o
l
to
s
u
p
p
o
r
t
en
d
ea
v
o
r
s
t
h
at
w
ill
en
h
a
n
ce
cr
itical
t
h
i
n
k
in
g
s
k
il
ls
o
f
t
h
e
s
t
u
d
en
ts
,
i.e
.
p
r
o
m
o
tin
g
co
n
s
tr
u
ct
iv
i
s
t
’
s
teac
h
i
n
g
,
tr
a
n
s
f
o
r
m
ati
v
e
lear
n
in
g
,
a
n
d
d
is
cip
lin
e
-
b
ased
teac
h
i
n
g
ap
p
r
o
ac
h
es
(
D
L
SL
P
ed
ag
o
g
ical
Fra
m
e
w
o
r
k
)
,
e
f
f
o
r
ts
e
m
p
lo
y
i
n
g
s
tr
ateg
ies
t
h
at
will
tar
g
et
i
m
p
r
o
v
ed
co
g
n
iti
v
e
s
k
ills
w
ill
s
u
r
el
y
b
e
f
a
v
o
r
ab
le.
Desp
ite
h
o
w
t
h
e
l
iter
atu
r
e
co
n
v
e
y
s
t
h
at
j
o
u
r
n
al
-
k
ee
p
i
n
g
i
m
p
r
o
v
e
s
cr
itical
th
i
n
k
in
g
s
k
ill
s
o
f
th
e
s
tu
d
en
t
s
[
4
,
2
4
]
,
n
o
r
esear
ch
u
ti
lizin
g
R
J
W
in
m
a
th
e
m
atic
s
cl
ass
ap
p
ea
r
s
to
h
av
e
b
ee
n
co
n
d
u
ct
ed
in
De
L
a
Sall
e
L
ip
a
to
d
ate.
I
m
p
le
m
en
ta
tio
n
o
f
th
e
s
aid
ac
tiv
i
t
y
a
n
d
ex
p
o
s
itio
n
o
f
t
h
e
co
lleg
e
s
tu
d
e
n
ts
’
v
ie
w
s
a
n
d
s
en
t
i
m
e
n
t
s
o
n
t
h
e
u
s
e
o
f
R
J
W
as
a
m
ed
iu
m
to
lear
n
m
at
h
e
m
atic
s
r
e
m
ain
a
p
r
ess
i
n
g
a
n
d
ti
m
el
y
is
s
u
e.
Hen
ce
,
it
is
o
f
i
n
ter
est
to
ca
r
r
y
o
u
t
a
r
esear
ch
th
at
w
ill
g
i
v
e
a
clea
r
p
ictu
r
e
o
f
h
o
w
th
e
ac
ti
v
it
y
in
a
m
ath
e
m
at
ics cla
s
s
is
p
er
ce
iv
ed
b
y
th
e
s
t
u
d
en
ts
.
T
h
e
p
ap
er
f
o
cu
s
es
o
n
j
o
u
r
n
al
w
r
iti
n
g
as
a
r
eg
u
lar
m
at
h
e
m
at
ics
clas
s
ac
ti
v
it
y
.
I
t
ai
m
s
to
d
e
s
cr
ib
e
th
e
f
ee
li
n
g
s
o
f
t
h
e
s
t
u
d
en
t
s
ab
o
u
t
R
J
W
.
Sp
ec
i
f
icall
y
,
it
s
ee
k
s
to
an
s
w
er
t
h
e
f
o
llo
w
i
n
g
q
u
esti
o
n
s
.
1.
W
h
at
ar
e
th
e
th
o
u
g
h
t
s
an
d
f
ee
l
in
g
s
o
f
t
h
e
s
t
u
d
en
t
s
ab
o
u
t RJ
W
as a
m
a
th
e
m
atic
s
clas
s
ac
ti
v
it
y
?
2.
Do
th
e
s
t
u
d
en
t
s
f
i
n
d
j
o
u
r
n
al
wr
itin
g
h
elp
f
u
l?
I
f
y
e
s
,
in
w
h
at
w
a
y
?
3.
Ho
w
d
o
es r
ef
lectio
n
p
la
y
a
r
o
le
in
o
n
e
’
s
m
ath
e
m
at
ics lea
r
n
in
g
?
I
t
w
as
m
ad
e
clea
r
to
th
e
s
tu
d
en
ts
t
h
at
t
h
eir
j
o
u
r
n
al
w
r
iti
n
g
s
w
ill
n
o
t
b
e
ev
al
u
ated
as
p
ar
t
o
f
th
eir
g
r
ad
e
co
m
p
u
tat
io
n
.
T
h
at
m
ea
n
s
th
e
y
ar
e
f
r
ee
to
w
r
ite
a
n
d
ex
p
r
ess
th
e
m
s
e
lv
e
s
o
p
en
l
y
i
n
t
h
e
ac
ti
v
it
y
.
T
h
e
r
esu
lt
s
o
f
t
h
e
s
tu
d
y
f
o
cu
s
s
o
le
l
y
o
n
th
e
s
t
u
d
en
t
s
’
th
o
u
g
h
t
s
o
n
an
d
p
er
ce
iv
ed
e
f
f
ec
ts
o
f
R
J
W
.
I
t
is
n
o
t
u
s
ed
to
ev
alu
a
te
th
e
co
n
ten
t o
r
th
e
m
a
n
n
er
t
h
e
s
u
b
j
ec
t
w
as ta
u
g
h
t.
1
.
1
.
T
heo
re
t
ica
l F
ra
m
ew
o
rk
T
h
e
s
tu
d
y
is
an
c
h
o
r
ed
o
n
Ko
lb
’
s
E
x
p
er
ien
t
ial
L
ea
r
n
i
n
g
T
h
eo
r
y
(
E
L
T
)
.
T
h
e
th
eo
r
y
d
ef
i
n
es
lear
n
in
g
as
“
t
h
e
p
r
o
ce
s
s
w
h
er
eb
y
k
n
o
w
led
g
e
is
cr
ea
ted
th
r
o
u
g
h
t
h
e
tr
an
s
f
o
r
m
atio
n
o
f
ex
p
er
ien
ce
ac
co
r
d
in
g
to
Ko
lb
,
as
cited
b
y
Ko
lb
[
2
5
]
.
Kn
o
w
le
d
g
e
r
esu
l
ts
f
r
o
m
t
h
e
co
m
b
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L
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m
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p
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tr
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co
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cr
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x
p
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,
r
ef
lecti
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e
o
b
s
er
v
atio
n
,
ab
s
tr
ac
t
co
n
ce
p
t
u
aliza
tio
n
,
an
d
ac
ti
v
e
e
x
p
er
i
m
e
n
tatio
n
.
I
n
s
i
m
p
le
w
o
r
d
s
,
th
e
c
y
cle
to
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c
h
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p
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n
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
2
5
2
-
8822
I
J
E
R
E
Vo
l.
6
,
No
.
1
,
Ma
r
ch
20
1
7
:
38
–
44
40
ex
p
er
ien
ci
n
g
,
r
ef
lec
tin
g
,
th
i
n
k
in
g
,
a
n
d
ac
tin
g
i
n
a
r
ep
etitiv
e
p
r
o
ce
s
s
.
T
h
e
cy
cle
b
eg
i
n
s
w
it
h
o
n
e’
s
i
m
m
ed
iate
o
r
co
n
cr
ete
ex
p
er
ien
ce
.
T
h
e
l
ea
r
n
er
r
ef
lects
o
n
t
h
is
ex
p
er
ie
n
ce
to
g
iv
e
m
ea
n
i
n
g
to
s
u
c
h
[
2
6
]
.
T
h
is
r
ef
lectio
n
(
o
r
m
ea
n
in
g
)
i
s
d
ig
es
ted
in
to
ab
s
tr
ac
t
co
n
ce
p
ts
w
h
ich
b
r
ee
d
n
e
w
i
m
p
licatio
n
s
f
o
r
ac
t
io
n
.
T
h
ese
i
m
p
licatio
n
s
s
er
v
e
as d
r
iv
er
s
i
n
b
u
ild
i
n
g
n
e
w
e
x
p
er
ien
ce
s
[
2
5
]
.
Sv
i
n
ic
k
i
an
d
Di
x
o
n
[
2
6
]
,
in
th
eir
s
t
u
d
y
T
h
e
Ko
lb
Mo
d
el
Mo
d
if
ied
f
o
r
C
las
s
r
o
o
m
A
cti
v
itie
s
,
ad
d
ed
to
Ko
lb
’
s
m
o
d
el
ce
r
tain
ac
tiv
i
ties
th
at
s
u
p
p
o
r
t
th
e
d
if
f
er
en
t
p
h
ases
o
f
lear
n
i
n
g
in
t
h
e
c
y
cle
.
T
ea
ch
er
s
h
av
e
th
e
p
r
im
ar
y
r
o
le
o
f
co
n
s
tr
u
cti
n
g
lear
n
i
n
g
o
r
d
er
th
at
lead
s
h
is
s
tu
d
e
n
ts
t
h
r
o
u
g
h
o
u
t
t
h
e
w
h
o
le
c
y
cle.
I
n
a
m
at
h
e
m
a
tics
clas
s
f
o
r
ex
a
m
p
l
e,
a
teac
h
er
m
a
y
u
s
e
s
i
m
u
latio
n
s
o
r
g
a
m
es,
p
r
o
b
lem
s
et
s
,
a
n
d
ex
a
m
p
le
s
i
n
o
r
d
er
f
o
r
h
i
m
to
g
i
v
e
h
is
s
t
u
d
en
t
s
a
co
n
cr
ete
ex
p
er
ien
ce
.
Un
d
er
r
ef
lecti
v
e
o
b
s
er
v
at
io
n
,
d
is
c
u
s
s
i
o
n
,
b
r
ain
s
to
r
m
in
g
,
an
d
j
o
u
r
n
al
k
ee
p
in
g
ar
e
b
eliev
ed
to
co
m
p
el
t
h
e
s
t
u
d
en
t
s
to
“
r
ef
lec
t
o
n
th
e
ir
ex
p
er
ien
ce
s
an
d
th
e
ex
p
er
ie
n
ce
s
o
f
o
th
er
s
”
(
p
.
1
4
2
)
.
W
h
ile,
le
ctu
r
e,
m
o
d
el
-
b
u
i
ld
in
g
e
x
er
ci
s
es,
p
r
o
j
ec
ts
,
an
d
an
alo
g
ies
co
u
ld
f
o
s
ter
ab
s
tr
ac
t
co
n
ce
p
tu
aliza
tio
n
.
An
d
la
s
tl
y
,
s
i
m
u
lat
io
n
s
,
p
r
o
j
ec
ts
,
an
d
h
o
m
e
w
o
r
k
s
g
iv
e
s
t
h
e
s
t
u
d
en
t
s
i
n
f
er
en
ce
s
w
h
ic
h
t
h
e
y
co
u
ld
ap
p
ly
i
n
b
u
ild
i
n
g
n
e
w
e
x
p
er
ien
ce
s
.
T
h
e
co
n
cr
ete
ex
p
er
ien
ce
,
cr
af
ted
b
y
th
e
teac
h
er
s
,
is
g
i
v
e
n
to
th
e
s
t
u
d
en
t
s
f
o
r
th
e
latter
’
s
g
at
h
er
in
g
o
f
f
ac
ts
.
T
h
is
is
t
h
e
s
ta
g
e
f
o
r
w
h
ich
th
e
y
d
escr
ib
e
th
e
ac
t
u
al
e
v
en
t.
Fro
m
th
i
s
,
j
o
u
r
n
al
w
r
i
tin
g
s
ets
t
h
e
m
o
o
d
f
o
r
th
e
m
to
b
e
m
o
r
e
r
ef
lec
tiv
e
o
f
w
h
a
t
th
e
y
h
a
v
e
e
x
p
er
ien
ce
d
an
d
h
o
w
t
h
e
y
f
ee
l
ab
o
u
t
it.
Qu
esti
o
n
s
s
u
c
h
as
“
W
h
at
d
id
I
lear
n
?
”
an
d
“
W
h
at
d
o
I
n
ee
d
to
i
m
p
r
o
v
e
o
n
?
”
ca
n
s
er
v
e
as
g
u
id
es
in
th
e
s
t
u
d
en
t
s
’
r
ef
lec
tio
n
.
T
h
r
o
u
g
h
t
h
i
s
,
th
e
y
w
i
ll
f
i
n
d
m
ea
n
in
g
f
u
l
co
n
n
ec
tio
n
o
f
t
h
e
k
n
o
w
led
g
e
t
h
at
t
h
e
y
h
a
v
e
g
ai
n
ed
w
it
h
t
h
e
m
s
el
v
es
as
lear
n
er
s
.
C
ar
e
f
u
l
t
h
o
u
g
h
t
w
il
l
m
a
k
e
t
h
e
m
u
n
d
er
s
ta
n
d
a
n
d
ex
p
lai
n
w
h
y
th
i
n
g
s
h
ap
p
e
n
ed
.
T
h
is
lead
s
t
h
e
s
tu
d
e
n
ts
to
b
e
m
o
r
e
s
tr
ateg
ic
i
n
p
lan
n
in
g
w
h
at
to
d
o
n
ex
t
w
h
ich
en
ab
le
s
th
e
m
to
b
u
ild
a
n
e
w
e
x
p
er
ien
ce
.
2.
RE
S
E
ARCH
M
E
T
H
O
D
T
h
is
is
a
q
u
alita
tiv
e
t
y
p
e
o
f
r
esear
ch
s
tu
d
y
;
i
n
p
ar
tic
u
lar
,
a
ca
s
e
s
tu
d
y
.
I
t
d
o
cu
m
en
ts
th
o
u
g
h
t
s
a
n
d
f
ee
li
n
g
s
o
f
th
e
r
esp
o
n
d
e
n
ts
.
P
atto
n
an
d
C
o
ch
r
a
n
[
2
7]
w
r
o
te
th
at
q
u
alitati
v
e
r
esear
ch
i
s
t
y
p
if
ied
b
y
it
s
ai
m
s
r
elatin
g
to
u
n
d
er
s
tan
d
i
n
g
a
ce
r
tain
m
at
ter
an
d
m
eth
o
d
s
w
h
ic
h
g
e
n
er
ate
w
o
r
d
s
r
ath
er
th
an
n
u
m
b
er
s
.
I
t
in
v
o
lv
e
s
n
atu
r
al
is
tic
a
n
d
in
ter
p
r
etativ
e
ap
p
r
o
ac
h
to
th
e
w
o
r
ld
letti
n
g
th
e
r
esear
c
h
er
s
“
s
t
u
d
y
t
h
i
n
g
s
i
n
th
eir
n
atu
r
al
s
etti
n
g
,
atte
m
p
tin
g
to
m
a
k
e
s
en
s
e
o
f
,
o
r
to
in
ter
p
r
et,
p
h
en
o
m
e
n
a
i
n
ter
m
s
o
f
th
e
m
ea
n
i
n
g
s
p
eo
p
le
b
r
in
g
to
th
e
m
”
[
2
8
]
.
I
n
f
ir
s
t
w
ee
k
o
f
t
h
e
cla
s
s
es
in
t
h
e
f
ir
s
t se
m
e
s
ter
o
f
Sc
h
o
o
l Y
e
ar
2
0
1
6
-
2017
,
th
e
n
atu
r
e
a
n
d
p
u
r
p
o
s
e
o
f
th
is
s
tu
d
y
w
er
e
d
is
c
u
s
s
ed
w
it
h
th
e
s
t
u
d
en
t
s
.
L
i
k
e
w
i
s
e,
th
e
teac
h
er
-
r
esear
c
h
er
ask
ed
f
o
r
t
h
eir
co
n
s
e
n
t
to
b
e
p
ar
t o
f
th
e
s
t
u
d
y
.
T
h
e
p
ar
ticip
an
ts
i
n
t
h
e
s
t
u
d
y
w
er
e
t
h
e
s
i
x
t
h
ir
d
y
ea
r
B
ac
h
elo
r
o
f
Scie
n
ce
in
E
d
u
ca
tio
n
Ma
j
o
r
in
Ma
t
h
e
m
a
tics
s
tu
d
e
n
ts
o
f
De
L
a
Salle
L
ip
a
en
r
o
lled
in
Di
f
f
er
en
tial Calc
u
l
u
s
cla
s
s
.
R
J
W
w
as
in
tr
o
d
u
ce
d
to
th
e
s
tu
d
en
t
s
as
a
r
e
g
u
lar
class
ac
t
i
v
it
y
.
I
n
t
h
e
s
t
u
d
y
m
ad
e
b
y
C
ar
ter
[
29
]
,
ad
d
in
g
w
r
iti
n
g
to
r
ef
lecti
v
e
j
o
u
r
n
al
s
to
d
ail
y
r
o
u
t
in
e
i
n
m
at
h
e
m
a
tics
cla
s
s
e
n
h
an
ce
s
s
t
u
d
e
n
ts
’
t
h
i
n
k
i
n
g
s
k
ill
s
.
Usu
al
l
y
at
f
i
f
tee
n
m
i
n
u
te
s
b
e
f
o
r
e
th
e
e
n
d
o
f
ea
c
h
m
ee
ti
n
g
,
a
j
o
u
r
n
al
q
u
e
s
tio
n
o
r
p
r
o
m
p
t
was
p
r
o
v
id
ed
b
y
t
h
e
teac
h
er
-
r
esear
c
h
er
f
o
r
th
e
s
tu
d
en
ts
to
r
ef
lect
o
n
.
T
h
e
q
u
e
s
tio
n
o
r
p
r
o
m
p
t d
id
n
o
t
n
ec
es
s
ar
il
y
h
av
e
to
b
e
r
elate
d
to
th
e
to
p
ic
d
is
c
u
s
s
ed
d
u
r
in
g
t
h
e
d
a
y
.
R
at
h
er
,
i
t
w
a
s
co
m
m
o
n
l
y
s
o
m
eth
in
g
t
h
at
t
h
e
s
t
u
d
en
t
s
ar
e
ab
le
to
r
elate
to
o
r
ex
p
r
ess
th
e
m
s
el
v
es
o
n
.
I
n
ad
d
itio
n
,
th
e
teac
h
er
m
ad
e
s
u
r
e
th
at
th
r
o
u
g
h
th
e
p
r
o
m
p
t,
s
h
e
ca
n
p
r
o
b
e
o
n
th
e
s
tu
d
e
n
ts
’
f
ee
l
in
g
s
s
o
as
to
elic
it
f
ee
d
b
ac
k
s
f
r
o
m
th
e
m
.
A
s
s
ta
ted
b
y
Am
ir
&
L
ia
n
g
h
u
o
[
20
]
,
“
th
e
j
o
u
r
n
al
allo
w
u
s
(
teac
h
er
s
)
to
g
at
h
er
in
f
o
r
m
atio
n
o
n
o
u
r
p
u
p
ils
,
m
o
n
it
o
r
th
eir
u
n
d
er
s
ta
n
d
in
g
o
f
th
e
to
p
ic,
an
d
m
o
s
t
i
m
p
o
r
tan
tl
y
,
it
b
r
in
g
s
u
s
clo
s
er
to
o
u
r
p
u
p
ils
”
(
p
.
1
0
)
.
Fu
r
th
er
,
it
is
a
g
o
o
d
av
en
u
e
f
o
r
s
tu
d
e
n
ts
to
p
r
o
v
id
e
f
ee
d
b
ac
k
an
d
ex
p
r
e
s
s
th
eir
n
ee
d
s
in
ter
m
s
o
f
m
ath
e
m
at
ics tea
ch
in
g
a
n
d
lear
n
i
n
g
.
T
h
er
e
w
er
e
d
a
y
s
w
h
e
n
th
e
s
t
u
d
en
t
s
ar
e
ask
ed
to
s
h
ar
e
to
th
e
clas
s
t
h
e
en
tr
ie
s
t
h
e
y
h
a
v
e
in
t
h
eir
j
o
u
r
n
al.
A
d
d
is
o
n
[
15
]
p
o
in
te
d
o
u
t
th
at
h
av
in
g
s
tu
d
e
n
ts
s
h
ar
e
an
d
ex
p
lain
th
eir
w
r
it
ten
w
o
r
k
to
ea
c
h
o
th
er
w
o
u
ld
ca
u
s
e
s
t
u
d
en
t
s
to
t
h
i
n
k
ab
o
u
t
t
h
eir
o
w
n
s
o
lu
t
io
n
s
an
d
co
n
s
id
er
t
h
e
s
o
lu
tio
n
s
o
f
o
th
er
s
,
n
o
t
to
m
e
n
tio
n
th
e
li
k
eli
h
o
o
d
o
f
p
r
o
m
p
ti
n
g
m
o
r
e
d
etailed
w
o
r
k
o
n
t
h
eir
p
ar
t.
Feed
b
ac
k
s
ar
e
o
f
ten
t
i
m
es
p
r
o
v
id
ed
b
y
t
h
e
teac
h
er
as a
m
ea
n
s
to
g
iv
e
ap
p
r
ec
ia
tio
n
f
o
r
th
eir
e
n
tr
ies a
n
d
also
to
g
iv
e
e
n
co
u
r
ag
e
m
e
n
t to
th
o
s
e
w
h
o
n
ee
d
it.
T
h
e
d
ata
w
er
e
g
at
h
er
ed
th
r
o
u
g
h
a
f
o
c
u
s
g
r
o
u
p
d
is
c
u
s
s
io
n
(
FGD)
a
m
o
n
g
t
h
e
s
tu
d
e
n
t
p
ar
ticip
an
ts
s
ch
ed
u
led
at
t
h
e
e
n
d
o
f
th
e
s
e
m
ester
.
T
h
e
p
u
r
p
o
s
e
o
f
f
o
cu
s
g
r
o
u
p
s
ac
co
r
d
in
g
to
Kr
u
eg
er
an
d
C
a
s
e
y
[
3
0
]
is
to
“
p
r
o
m
o
te
a
co
m
f
o
r
tab
le
at
m
o
s
p
h
er
e
o
f
d
is
clo
s
u
r
e
i
n
w
h
ic
h
p
eo
p
le
ca
n
s
h
ar
e
id
ea
s
,
e
x
p
er
ien
ce
s
,
a
n
d
attit
u
d
es
ab
o
u
t
a
to
p
ic”.
Qu
alitati
v
e
m
e
th
o
d
s
s
u
c
h
as
f
o
c
u
s
g
r
o
u
p
s
all
o
w
s
t
h
e
r
esear
ch
er
to
h
av
e
a
b
r
o
ad
er
v
ie
w
o
f
t
h
e
r
ea
lit
y
as t
h
e
y
p
a
y
m
o
r
e
atten
t
io
n
to
v
o
ices o
f
t
h
e
p
ar
ticip
an
t
s
in
t
h
e
s
t
u
d
y
[
3
1
,
3
2
]
.
T
o
p
r
o
b
e
in
to
th
e
s
tu
d
e
n
ts
’
i
n
s
ig
h
ts
o
n
R
J
W
,
th
e
f
o
llo
w
i
n
g
q
u
esti
o
n
s
w
er
e
o
u
tl
in
ed
f
o
r
t
h
e
FGD.
1.
W
h
at
ar
e
y
o
u
r
th
o
u
g
h
t
s
an
d
f
e
elin
g
s
ab
o
u
t RJ
W
as a
m
at
h
e
m
atic
s
class
ac
t
iv
it
y
?
2.
Do
y
o
u
f
i
n
d
j
o
u
r
n
al
w
r
iti
n
g
h
e
lp
f
u
l?
I
f
y
e
s
,
in
w
h
at
w
a
y
?
3.
Ho
w
d
o
es r
ef
lectio
n
p
la
y
a
r
o
le
in
o
n
e
’
s
m
ath
e
m
at
ics lea
r
n
in
g
?
An
ex
p
er
ien
ce
d
q
u
alitat
iv
e
r
esear
ch
er
w
a
s
ask
ed
to
f
ac
ilit
a
te
th
e
d
is
cu
s
s
io
n
.
T
h
e
f
ac
ilit
a
to
r
s
tar
ted
th
e
FG
D
w
it
h
a
b
r
ief
i
n
g
o
f
t
h
e
p
ar
ticip
an
ts
a
b
o
u
t t
h
e
p
r
o
ce
s
s
o
f
th
e
d
is
c
u
s
s
io
n
.
T
h
e
y
w
er
e
i
n
f
o
r
m
ed
th
at
t
h
er
e
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
E
R
E
I
SS
N:
2252
-
8822
I
n
ve
s
tig
a
tin
g
C
o
lleg
e
S
tu
d
en
ts
’
V
iew
s
o
n
Ma
th
ema
tics
Lea
r
n
in
…
(
I
ve
e
K
.
Gu
ce
)
41
w
il
l
b
e
n
o
r
i
g
h
t
o
r
w
r
o
n
g
a
n
s
w
er
s
.
W
ith
t
h
e
p
er
m
i
s
s
io
n
o
f
t
h
e
s
tu
d
e
n
t
p
ar
ticip
a
n
ts
,
a
n
o
t
h
e
r
r
esear
ch
a
s
s
i
s
ta
n
t
w
a
s
i
n
v
ited
to
au
d
io
-
r
ec
o
r
d
an
d
n
o
te
-
ta
k
e
d
u
r
in
g
t
h
e
s
es
s
io
n
.
Ho
w
e
v
er
,
it
w
as a
s
s
u
r
ed
to
th
e
s
t
u
d
en
ts
t
h
at
a
n
y
in
f
o
r
m
atio
n
t
h
at
w
i
ll b
e
g
at
h
er
ed
d
u
r
in
g
t
h
e
s
es
s
io
n
w
i
ll b
e
d
ea
lt
w
i
th
co
n
f
id
e
n
tialit
y
a
n
d
an
o
n
y
m
it
y
.
Fo
r
th
e
an
al
y
s
i
s
o
f
d
ata,
a
tr
an
s
cr
ip
tio
n
g
r
id
w
as d
e
v
elo
p
ed
f
o
r
th
e
r
esp
o
n
s
es i
n
ea
ch
o
f
t
h
e
q
u
esti
o
n
s
r
esu
lti
n
g
to
th
r
e
e
g
r
id
s
.
T
h
e
r
esp
o
n
s
es
w
er
e
ca
r
ef
u
ll
y
an
d
o
b
j
ec
tiv
el
y
co
d
ed
.
T
h
e
g
r
id
s
w
er
e
s
t
u
d
ied
all
to
g
eth
er
.
Si
m
ilar
an
d
ass
o
cia
ted
(
p
r
elim
i
n
ar
y
)
co
d
es
w
er
e
n
o
ted
th
en
g
r
o
u
p
ed
to
g
et
h
e
r
to
d
ete
r
m
i
n
e
t
h
e
th
e
m
es
th
a
t
e
m
er
g
ed
.
Fre
q
u
e
n
cies
o
f
r
esp
o
n
s
e
s
w
er
e
r
ec
o
r
d
ed
as
w
e
ll
to
id
en
ti
f
y
t
h
e
s
i
g
n
i
f
ica
n
t
is
s
u
e
s
th
a
t
b
ec
am
e
ap
p
ar
en
t d
u
r
i
n
g
th
e
d
i
alo
g
u
e.
3.
RE
SU
L
T
S
A
ND
AN
AL
Y
SI
S
3
.
1
.
T
ho
ug
hts a
nd
F
ee
lin
g
s
a
bo
ut
RJ
W
B
ased
o
n
th
e
r
esp
o
n
s
e
s
,
d
if
f
er
en
t
y
et
i
n
ter
co
n
n
ec
ted
o
p
in
io
n
s
o
n
R
J
W
s
u
r
f
ac
ed
d
u
r
i
n
g
th
e
d
ialo
g
u
e.
T
h
r
ee
s
tu
d
en
ts
co
n
v
e
y
ed
th
a
t
th
e
ac
tiv
it
y
allo
w
ed
th
e
m
to
ex
p
r
ess
th
e
m
s
elv
e
s
f
r
ee
l
y
.
Stu
d
en
t
1
s
tated
,
“I
ca
n
w
r
ite
w
h
a
t
I
lea
r
n
ed
a
n
d
w
h
a
t
I
feel
esp
ec
ia
lly
if
I
u
n
d
ers
to
o
d
th
e
less
o
n
.
”
Stu
d
en
t
2
s
h
ar
ed
th
e
s
a
m
e
o
p
in
io
n
b
y
s
a
y
in
g
:
“S
in
ce
w
e
a
r
e
to
ld
to
w
r
ite
ex
a
ctly
w
h
a
t
w
e
w
a
n
t,
I
felt
th
a
t
I
co
u
ld
s
p
ea
k
my
min
d
o
n
th
e
jo
u
r
n
a
l
w
h
en
I
fin
d
th
e
less
o
n
d
iffi
cu
lt
fo
r
th
a
t
d
a
y.
”
C
o
n
s
eq
u
en
t
l
y
,
R
J
W
en
ab
led
th
e
s
tu
d
en
ts
to
tell
th
e
teac
h
er
th
i
n
g
s
t
h
e
y
ca
n
n
o
t
s
a
y
v
er
b
all
y
.
St
u
d
en
t
3
r
ec
alled
t
h
at
t
h
r
o
u
g
h
t
h
e
j
o
u
r
n
a
l
s
h
e
i
s
ab
le
to
co
m
m
u
n
icate
to
th
e
teac
h
er
th
e
to
p
ics
s
h
e
f
in
d
co
n
f
u
s
i
n
g
,
o
n
w
h
ic
h
s
h
e
e
m
p
h
asi
ze
d
“I
co
u
ld
n
o
t
tell
d
ir
ec
tly
t
o
th
e
tea
ch
er
th
a
t
I
d
o
n
o
t
u
n
d
ers
ta
n
d
th
e
less
o
n
w
ith
o
u
t wo
r
r
yin
g
a
b
o
u
t b
ei
n
g
o
ffen
s
ive.
”
An
o
th
er
p
er
ce
p
tio
n
ar
ticu
lated
ab
o
u
t
R
J
W
is
th
at
it
is
a
m
ea
n
s
to
ev
al
u
ate
o
n
e
’
s
o
w
n
p
er
f
o
r
m
a
n
ce
i
n
th
e
class
.
St
u
d
en
t
3
r
ec
o
u
n
ted
th
at
th
r
o
u
g
h
t
h
e
j
o
u
r
n
al
s
h
e
c
an
r
ef
lect
o
n
“
w
h
a
t
s
kills
s
h
e
i
s
la
ck
in
g
o
f,
w
h
a
t
to
p
ics
s
h
e
n
ee
d
s
mo
r
e
s
tu
d
yin
g
o
n
,
o
r
w
h
eth
er
s
h
e
lis
ten
ed
en
o
u
g
h
o
r
n
o
t
d
u
r
in
g
d
is
cu
s
s
io
n
.
”
St
u
d
en
t
6
b
eliev
ed
t
h
at
p
er
f
o
r
m
an
ce
is
i
m
p
r
o
v
ed
b
y
clai
m
in
g
:
“I
ca
n
r
ec
a
ll
d
u
r
in
g
a
n
ex
a
m
w
h
a
t
I
h
a
ve
w
r
itten
in
th
e
r
eflec
tio
n
like
my
o
w
n
mis
t
a
ke
s
a
n
d
th
u
s
,
I
a
m
a
b
le
to
co
r
r
ec
t it.
”
I
n
t
h
e
f
o
r
eg
o
in
g
,
w
h
i
le
m
o
s
t
o
f
t
h
e
s
t
u
d
en
t
s
v
ie
w
ed
R
J
W
as
a
m
ed
i
u
m
to
co
m
m
u
n
icate
,
it
i
s
in
ter
esti
n
g
to
n
o
te
t
h
at
o
n
e
p
er
ce
iv
ed
it
as
a
n
aid
to
r
elate
m
a
th
e
m
atics
to
r
ea
l
-
li
f
e
s
ett
in
g
.
Stu
d
e
n
t
4
d
ec
lar
ed
,
“T
h
r
o
u
g
h
r
eflec
tive
jo
u
r
n
a
li
n
g
,
I
feel
th
a
t
I
a
m
in
th
e
r
ea
l
w
o
r
ld
a
s
o
p
p
o
s
ed
w
h
en
d
o
in
g
ju
s
t
p
u
r
e
co
mp
u
ta
tio
n
s
,
w
ith
o
u
t
a
p
p
lica
tio
n
s
.
”
Sh
e
ad
d
s
,
“In
th
e
a
ctivity,
I
a
m
a
b
le
t
o
r
ela
te
ma
th
e
ma
tics
to
r
ea
l life.
”
3
.
2
.
B
enef
it
s
o
f
J
o
urna
l W
rit
ing
T
h
e
n
o
tio
n
t
h
at
R
J
W
is
not
g
r
ad
ed
in
s
t
ig
ated
m
o
s
t
o
f
th
e
b
en
ef
its
o
f
t
h
e
ac
t
iv
it
y
.
Fo
r
o
n
e,
th
e
s
tu
d
e
n
ts
f
el
t
t
h
at
j
o
u
r
n
ali
n
g
s
ets
a
n
a
v
e
n
u
e
f
o
r
t
h
e
m
to
o
p
en
th
e
m
s
e
lv
e
s
u
p
w
it
h
o
u
t
r
estra
in
t.
Stu
d
e
n
t
4
af
f
ir
m
ed
th
i
s
b
y
s
a
y
in
g
,
“Ou
r
p
r
ev
io
u
s
jo
u
r
n
a
l
w
r
itin
g
a
ctiv
ity
w
a
s
g
r
a
d
ed
;
h
en
ce
,
it
s
o
me
h
o
w
limit
w
h
a
t
w
e
ca
n
s
h
a
r
e.
Th
is
(
R
J
W)
,
o
n
th
e
co
n
tr
a
r
y,
a
llo
w
ed
u
s
to
s
h
a
r
e
w
h
a
teve
r
w
e
w
a
n
t
to
s
a
y.
”
An
d
b
ec
au
s
e
t
h
e
s
tu
d
e
n
ts
ca
n
w
r
i
te
in
t
h
e
j
o
u
r
n
al
ev
e
n
th
e
ir
o
w
n
f
r
u
s
tr
atio
n
s
f
r
o
m
p
ast
m
is
tak
e
s
,
R
J
W
s
er
v
ed
as
a
co
n
s
tan
t
r
e
m
in
d
er
f
o
r
th
e
m
to
d
o
b
etter
.
T
h
r
o
u
g
h
t
h
e
j
o
u
r
n
al,
t
h
e
s
t
u
d
en
ts
r
ea
lized
t
h
eir
s
tr
en
g
t
h
s
a
n
d
w
ea
k
n
e
s
s
e
s
a
n
d
u
s
ed
t
h
e
m
to
i
m
p
r
o
v
e
t
h
e
m
s
el
v
es.
As
St
u
d
en
t
5
p
u
t
it,
“Jo
u
r
n
a
l
w
r
itin
g
h
elp
ed
me
a
lo
t.
I
t
ma
d
e
me
d
is
co
ve
r
my
s
tr
en
g
th
a
n
d
w
ea
kn
ess
th
r
o
u
g
h
r
eflec
tio
n
.
”
Asi
d
e
f
r
o
m
t
h
e
“n
o
t
g
r
ad
ed
”
f
ea
t
u
r
e
o
f
t
h
e
ac
tiv
it
y
,
f
ee
d
b
a
ck
s
co
m
i
n
g
f
r
o
m
t
h
e
teac
h
er
p
r
o
v
ed
to
h
av
e
b
r
o
u
g
h
t
ab
o
u
t
o
t
h
er
r
e
w
ar
d
s
o
f
R
J
W
.
Feed
b
ac
k
s
to
o
k
a
w
a
y
t
h
e
an
x
iet
ies
o
f
th
e
s
t
u
d
en
ts
i
n
t
h
e
s
u
b
j
ec
t.
Stu
d
e
n
ts
f
elt
t
h
at
t
h
eir
n
ee
d
s
ar
e
ad
d
r
ess
ed
p
r
o
m
p
t
l
y
s
i
n
ce
th
e
j
o
u
r
n
als
ar
e
co
llected
ev
er
y
m
ee
ti
n
g
b
y
t
h
e
teac
h
er
an
d
g
i
v
en
f
ee
d
b
ac
k
s
r
ig
h
t
a
w
a
y
.
St
u
d
en
t
3
s
h
ar
ed
th
at
s
h
e
w
a
s
en
co
u
r
a
g
ed
to
w
r
ite
b
ec
au
s
e
s
h
e
“r
ec
eive
s
feed
b
a
ck
r
a
th
er
th
a
n
a
co
mmu
n
ica
tio
n
th
a
t
is
o
n
e
-
s
id
ed
.
”
Sh
e
s
tr
es
s
ed
,
“A
co
n
n
ec
tio
n
is
b
u
ilt.”
R
J
W
is
th
u
s
s
e
n
s
ed
as a
t
w
o
-
w
a
y
co
m
m
u
n
ica
tio
n
b
et
w
ee
n
t
h
e
teac
h
er
a
n
d
th
e
s
t
u
d
en
t
s
.
L
ast
l
y
,
t
h
e
s
t
u
d
en
t
s
b
eliev
ed
t
h
at
t
h
e
r
ef
lecti
v
e
ac
ti
v
it
y
en
ab
led
th
e
m
to
co
n
n
ec
t
m
o
r
e
w
i
th
th
eir
p
ast
lear
n
in
g
d
u
e
to
th
e
g
i
v
e
n
j
o
u
r
n
al
p
r
o
m
p
t
s
.
Stu
d
e
n
t
2
m
en
t
i
o
n
ed
th
at
th
r
o
u
g
h
th
e
m
ea
n
i
n
g
-
m
a
k
i
n
g
q
u
e
s
tio
n
s
s
h
e
is
ab
le
to
“r
ela
te
ma
th
to
r
ea
l
-
life”.
Sh
e
f
u
r
t
h
er
ad
d
ed
t
h
at
“T
h
e
u
s
e
o
f
p
r
o
mp
ts
ma
d
e
R
JW
d
iffer
en
t
fr
o
m
th
eir
p
a
s
t
r
eflec
t
ive
a
ctivitie
s
w
h
erein
th
e
ma
in
p
o
in
t
is
o
n
ly
to
w
r
ite
w
h
a
t
h
a
s
b
ee
n
lea
r
n
ed
d
u
r
in
g
th
e
mee
tin
g
a
n
d
n
o
th
i
n
g
mo
r
e.
”
3
.
3
.
Ref
lect
io
n’s r
o
le
in
m
a
t
he
m
a
t
ic
s
lea
rning
Ma
j
o
r
ity
o
f
t
h
e
co
m
m
e
n
t
s
g
i
v
en
b
y
th
e
s
t
u
d
en
t
s
co
n
ce
r
n
i
n
g
t
h
e
r
o
le
p
la
y
ed
b
y
r
e
f
lecti
o
n
i
n
o
n
e
’
s
m
at
h
e
m
a
tics
lear
n
in
g
ten
d
e
d
to
f
all
alo
n
g
its
g
ain
s
.
Gen
er
all
y
s
p
ea
k
i
n
g
,
r
e
f
lecti
o
n
i
m
p
r
o
v
ed
o
n
e’
s
m
at
h
e
m
a
tics
lear
n
i
n
g
.
A
cc
o
r
d
in
g
to
Stu
d
e
n
t
4
,
s
h
e
i
s
ab
le
to
w
r
ite
i
n
t
h
e
j
o
u
r
n
al
t
h
e
to
p
ics
o
r
p
o
in
ts
th
e
y
f
i
n
d
d
if
f
ic
u
lt.
F
u
r
th
er
,
s
h
e
e
x
p
r
ess
ed
th
at
r
e
f
lectio
n
s
er
v
ed
a
s
a
g
u
id
e
f
o
r
t
h
e
s
t
u
d
en
ts
to
k
n
o
w
a
n
d
f
o
cu
s
m
o
r
e
o
n
th
e
ar
ea
s
th
e
y
n
ee
d
to
i
m
p
r
o
v
e
o
n
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
2
5
2
-
8822
I
J
E
R
E
Vo
l.
6
,
No
.
1
,
Ma
r
ch
20
1
7
:
38
–
44
42
“
W
h
en
a
q
u
iz
is
a
n
n
o
u
n
ce
d
fo
r
th
e
n
ex
t m
ee
tin
g
,
I
c
a
n
g
o
o
v
er th
e
jo
u
r
n
a
l.
I
a
m
a
b
le
to
p
o
in
t
o
u
t
w
h
a
t
I
fin
d
d
iffi
cu
lt
a
n
d
it
is
w
h
ere
I
fo
cu
s
w
h
en
I
s
tu
d
y.
My
p
r
ep
a
r
a
tio
n
is
mo
r
e
g
u
id
ed
a
n
d
o
r
g
a
n
iz
ed
.
I
co
n
ce
n
tr
a
te
o
n
w
h
a
t I
fin
d
co
mp
lica
ted
.
I
g
et
t
o
p
a
y
mo
r
e
time
a
n
d
a
tte
n
tio
n
to
s
u
c
h
.
”
(
Stu
d
en
t 4
)
T
h
is
id
ea
is
s
ec
o
n
d
ed
b
y
Stu
d
en
t
2
w
h
o
th
o
u
g
h
t
th
at
w
r
iti
n
g
d
o
w
n
t
h
eir
o
w
n
w
ea
k
n
e
s
s
es
m
ak
e
s
i
t
p
o
s
s
ib
le
f
o
r
th
e
m
to
o
v
er
co
m
e
th
e
m
a
n
d
ex
ce
l
in
t
h
e
s
u
b
j
ec
t.
Hen
ce
,
r
ef
lectio
n
m
o
t
iv
ate
d
th
e
s
tu
d
en
ts
to
d
o
b
etter
.
T
h
eir
jo
u
r
n
al
en
tr
ie
s
s
e
r
v
ed
as
a
co
m
m
u
n
icatio
n
w
i
th
th
e
m
s
el
v
es
as
lear
n
er
s
an
d
a
r
e
m
in
d
er
f
o
r
th
e
m
as
w
ell
to
ap
p
l
y
w
h
at
t
h
e
y
h
a
v
e
w
r
it
ten
.
St
u
d
en
t
3
n
ar
r
ated
th
at
w
h
en
s
h
e
w
r
i
tes
“I
n
ee
d
to
imp
r
o
ve
th
is
”
i
n
th
e
j
o
u
r
n
al,
s
h
e
i
s
“mo
tiva
ted
to
d
o
s
o
.
”
T
h
e
u
s
e
o
f
j
o
u
r
n
al
p
r
o
m
p
ts
b
ec
am
e
a
ch
a
n
n
el
f
o
r
th
e
s
t
u
d
en
t
s
to
lear
n
m
at
h
e
m
a
tics
.
Stu
d
en
t
2
r
ec
alled
w
h
a
t
s
h
e
m
e
n
tio
n
ed
ab
o
u
t
h
o
w
p
r
o
m
p
t
s
h
elp
ed
h
er
to
“r
ela
te
ma
th
to
r
ea
l
-
life”
.
Stu
d
e
n
t
4
i
m
m
ed
iatel
y
g
a
v
e
a
f
o
llo
w
u
p
r
e
ac
tio
n
s
a
y
i
n
g
,
“M
a
th
ema
ti
cs
is
b
ette
r
u
n
d
ers
to
o
d
w
h
en
it
ca
n
b
e
r
ela
ted
to
s
o
meth
in
g
.
”
B
ec
au
s
e
r
ef
lectio
n
p
r
o
m
p
ts
allo
w
ed
th
e
s
t
u
d
en
t
s
to
co
n
n
ec
t
t
h
eir
lear
n
in
g
to
r
ea
l
-
li
f
e
s
it
u
atio
n
s
,
th
e
y
f
elt
t
h
at
Ma
t
h
e
m
atics i
s
b
etter
u
n
d
er
s
to
o
d
an
d
m
a
k
es
m
o
r
e
s
en
s
e
w
h
e
n
it i
s
r
elate
d
to
r
ea
l
-
li
f
e
p
ictu
r
e.
3
.
4
.
Ana
ly
s
is
Fiv
e
t
h
e
m
es
e
m
er
g
ed
a
s
r
eg
ar
d
s
s
tu
d
e
n
ts
’
v
ie
w
s
o
n
R
J
W
as
a
m
at
h
e
m
a
tics
cla
s
s
ac
ti
v
it
y
.
First,
R
J
W
p
r
o
v
id
ed
o
p
p
o
r
tu
n
ities
f
o
r
t
h
e
s
tu
d
e
n
ts
to
co
n
s
tr
u
ct
m
ea
n
in
g
an
d
e
x
p
r
ess
p
er
s
o
n
al
v
ie
w
s
a
n
d
id
ea
s
.
T
h
e
w
r
iti
n
g
ac
ti
v
it
y
allo
w
ed
t
h
e
s
t
u
d
en
t
s
to
f
r
ee
l
y
e
x
p
r
ess
t
h
eir
f
ee
li
n
g
s
.
A
cc
o
r
d
in
g
to
Ya
n
g
[
34
]
,
m
at
h
e
m
atica
l
d
iar
y
w
r
iti
n
g
i
s
n
o
t
o
n
l
y
a
g
o
o
d
w
a
y
f
o
r
s
t
u
d
en
ts
to
r
ep
r
esen
t
t
h
eir
th
i
n
k
i
n
g
,
b
u
t
also
a
c
o
n
s
tr
u
ct
iv
e
w
a
y
f
o
r
th
e
m
to
co
m
m
u
n
icate
w
it
h
th
e
m
s
el
v
es
a
n
d
w
ith
t
h
eir
teac
h
er
s
.
Stu
d
en
ts
’
b
ein
g
ab
le
to
w
r
it
e
in
th
e
j
o
u
r
n
al
th
e
to
p
ics th
e
y
f
i
n
d
d
if
f
ic
u
lt r
elate
s
to
th
e
s
t
u
d
y
m
ad
e
b
y
Am
ir
an
d
L
ia
n
g
u
o
[
20
]
w
h
ich
s
tated
th
at
j
o
u
r
n
al
w
r
iti
n
g
p
r
o
v
id
es a
n
av
e
n
u
e
f
o
r
th
e
s
t
u
d
en
ts
to
ex
p
r
ess
t
h
eir
n
ee
d
s
i
n
ter
m
s
o
f
m
at
h
e
m
atics tea
c
h
i
n
g
an
d
lear
n
in
g
.
T
h
e
p
r
ec
e
d
in
g
clai
m
lead
s
to
th
e
id
ea
th
at
R
J
W
b
u
ilt
a
c
o
n
n
ec
tio
n
b
et
w
ee
n
th
e
teac
h
er
an
d
th
e
s
tu
d
e
n
ts
.
R
e
f
lectio
n
estab
li
s
h
e
s
a
p
er
s
o
n
al
r
elatio
n
s
h
ip
b
et
w
ee
n
th
e
lear
n
er
a
n
d
th
e
teac
h
e
r
[
1
0
]
p
r
o
v
id
in
g
a
n
o
p
p
o
r
tu
n
it
y
f
o
r
th
e
s
t
u
d
en
t
s
to
r
ea
lize
w
h
e
n
to
as
k
f
o
r
h
elp
an
d
co
n
t
r
ib
u
te
to
th
e
teac
h
i
n
g
-
lear
n
i
n
g
en
v
ir
o
n
m
e
n
t
[
3
]
.
I
t
p
r
o
v
id
es
th
e
teac
h
er
w
i
th
i
n
f
o
r
m
atio
n
o
n
w
h
a
t
h
i
s
s
t
u
d
en
t
s
lear
n
ed
a
n
d
w
h
a
t
th
e
s
tu
d
e
n
ts
h
av
e
d
i
f
f
ic
u
lties
lear
n
i
n
g
.
T
h
i
s
c
o
n
f
o
r
m
s
to
w
h
at
Va
n
Der
Mo
len
[
1
]
w
r
o
te
in
h
i
s
s
t
u
d
y
th
at
r
e
f
lecti
v
e
m
a
t
h
j
o
u
r
n
al
in
g
w
it
h
i
n
t
h
e
c
lass
r
o
o
m
w
a
s
b
en
e
f
ic
ial
f
o
r
b
o
th
t
h
e
in
s
tr
u
cto
r
a
n
d
t
h
e
s
tu
d
e
n
ts
.
A
d
d
is
o
n
[
1
5
]
,
in
h
is
s
tu
d
y
,
m
e
n
tio
n
ed
th
at
t
h
e
“
li
n
es
o
f
co
m
m
u
n
icatio
n
b
et
w
e
en
th
e
teac
h
er
an
d
s
tu
d
e
n
ts
s
tr
en
g
t
h
e
n
ed
as
t
h
e
w
r
iti
n
g
ac
ti
v
it
y
p
r
o
ce
ed
ed
”.
I
n
th
is
s
tu
d
y
,
f
ee
d
b
a
ck
s
p
r
o
v
ed
to
h
av
e
w
o
r
k
ed
f
o
r
t
h
e
b
en
ef
it
o
f
th
e
s
tu
d
e
n
ts
,
tak
i
n
g
a
w
a
y
th
e
a
n
x
ieties
t
h
e
y
h
a
v
e
f
o
r
th
e
s
u
b
j
ec
t.
Sad
ler
[
2
1
]
d
escr
ib
ed
it
as
a
“
d
elib
er
ate
d
ialo
g
u
e
th
at
ai
m
s
to
h
elp
s
tu
d
e
n
ts
to
o
v
er
c
o
m
e
t
h
eir
d
i
f
f
icu
l
ties
”.
Hattie
an
d
T
i
m
p
er
l
y
[
2
2]
p
o
in
ted
o
u
t
th
at
s
t
u
d
en
ts
o
f
te
n
s
ee
k
in
f
o
r
m
at
io
n
ab
o
u
t
“h
o
w
t
h
e
y
ar
e
g
o
in
g
”
alt
h
o
u
g
h
t
h
e
y
m
a
y
n
o
t
al
w
a
y
s
w
elco
m
e
t
h
e
an
s
w
er
s
.
R
ec
ei
v
i
n
g
f
ee
d
b
ac
k
s
clea
r
l
y
en
co
u
r
ag
ed
th
e
s
tu
d
e
n
ts
to
w
r
ite.
T
h
is
co
n
n
ec
ts
to
a
r
esear
c
h
m
ad
e
b
y
O
d
af
e
[
3
]
th
at
w
h
e
n
teac
h
er
an
d
s
tu
d
e
n
ts
b
ec
o
m
e
p
ar
tn
er
s
in
m
ath
e
m
at
ics
teac
h
in
g
an
d
lear
n
i
n
g
p
r
o
ce
s
s
,
r
ef
lect
io
n
p
r
o
ce
s
s
r
esu
lt
s
in
i
m
p
r
o
v
ed
s
t
u
d
en
t le
ar
n
i
n
g
en
v
ir
o
n
m
e
n
t.
T
h
ir
d
,
th
r
o
u
g
h
th
e
j
o
u
r
n
al
p
r
o
m
p
ts
,
R
J
W
allo
w
ed
th
e
s
t
u
d
en
ts
to
r
elate
m
at
h
e
m
atics
to
r
ea
l
-
li
f
e
f
ac
t
s
i
m
p
r
o
v
i
n
g
t
h
e
ir
u
n
d
er
s
ta
n
d
in
g
o
f
t
h
e
s
u
b
j
ec
t.
Ke
n
n
e
y
,
Sh
o
f
f
n
er
a
n
d
No
r
r
is
[
2
3]
w
h
o
li
k
e
w
i
s
e
u
s
ed
p
r
o
m
p
t
s
in
t
h
eir
s
t
u
d
y
f
o
u
n
d
o
u
t
t
h
at
t
h
e
m
ea
n
in
g
-
m
a
k
i
n
g
ac
ti
v
it
y
h
el
p
ed
th
e
s
tu
d
e
n
ts
to
r
ef
lect
o
n
t
h
eir
o
w
n
lear
n
in
g
.
C
o
o
k
an
d
C
r
aig
[
1
0
]
w
r
o
te
th
at
th
r
o
u
g
h
w
r
iti
n
g
,
s
tu
d
e
n
ts
ca
n
ex
p
r
ess
h
o
w
t
h
e
y
u
s
e
m
at
h
e
m
atic
s
in
t
h
eir
liv
e
s
an
d
h
o
w
th
e
y
s
o
lv
e
p
r
o
b
lem
.
P
ar
s
o
n
s
[
9
]
d
o
cu
m
e
n
ted
t
h
at
s
tu
d
en
t
s
“m
ad
e
s
i
g
n
i
f
ica
n
t
c
h
a
n
g
e
s
i
n
ap
p
r
o
ac
h
es
to
lear
n
in
g
an
d
also
m
ad
e
d
ee
p
an
d
m
ea
n
i
n
g
f
u
l
co
n
ce
p
tu
a
l
co
n
n
ec
tio
n
s
”
t
h
r
o
u
g
h
w
r
iti
n
g
in
m
at
h
e
m
atic
s
.
He
f
u
r
t
h
er
ad
d
s
th
at
ac
tiv
it
y
ac
tu
all
y
en
ab
led
th
e
s
t
u
d
en
t
s
to
f
ac
ilit
ate
m
ea
n
i
n
g
f
u
l
co
n
n
ec
t
io
n
s
ab
o
u
t
w
h
at
t
h
e
y
lear
n
an
d
th
e
m
s
elv
e
s
as lea
r
n
e
r
s
.
T
h
e
j
o
u
r
n
al
b
ein
g
a
n
aid
f
o
r
th
e
s
t
u
d
en
ts
to
b
u
ild
ass
o
ciat
i
o
n
o
f
id
ea
s
u
s
in
g
t
h
eir
p
r
io
r
k
n
o
w
led
g
e
a
n
d
ex
p
er
ien
ce
s
i
s
t
h
e
f
o
u
r
th
t
h
e
m
e
t
h
at
to
co
m
e
u
p
in
t
h
is
s
tu
d
y
.
R
e
s
u
lts
r
e
v
ea
led
a
clea
r
e
m
p
h
a
s
is
t
h
at
R
J
W
w
a
s
p
er
ce
iv
ed
a
s
a
m
ea
n
s
b
y
w
h
ic
h
s
tu
d
e
n
ts
r
ec
all
t
h
eir
lear
n
i
n
g
.
I
n
t
h
e
s
t
u
d
y
co
n
d
u
cted
b
y
L
e
w
an
d
Sch
m
id
t
[
4
]
,
th
er
e
ar
e
th
r
ee
g
en
er
al
ca
te
g
o
r
ies
o
n
w
h
ich
s
tu
d
e
n
ts
r
ef
lect
o
n
.
On
e
o
f
wh
ich
i
s
s
u
m
m
ar
y
o
f
w
h
at
h
as
b
ee
n
lear
n
ed
.
T
h
e
p
r
o
ce
s
s
o
f
w
r
iti
n
g
f
o
r
ce
s
a
s
t
u
d
en
t
to
s
lo
w
d
o
w
n
an
d
allo
w
h
i
m
to
th
o
r
o
u
g
h
l
y
p
r
o
ce
s
s
in
f
o
r
m
a
tio
n
a
n
d
r
ev
iew
m
at
h
e
m
a
tical
co
n
ce
p
ts
s
t
u
d
ied
[
5
]
.
I
t
d
ee
p
en
s
lear
n
in
g
f
o
r
lo
n
g
er
r
eten
tio
n
o
f
co
n
ce
p
ts
[
1
0
]
.
L
ast
o
f
all,
th
r
o
u
g
h
j
o
u
r
n
al
w
r
iti
n
g
,
s
tu
d
e
n
ts
w
er
e
ab
le
to
d
ev
elo
p
an
a
w
ar
en
e
s
s
o
f
t
h
e
s
el
f
w
h
ile
i
m
m
er
s
ed
in
m
ea
n
in
g
m
ak
i
n
g
p
r
o
m
p
ts
.
B
ec
au
s
e
w
r
iti
n
g
p
r
o
m
o
te
s
s
el
f
-
an
al
y
s
is
a
n
d
r
ef
lectio
n
a
m
o
n
g
s
tu
d
e
n
ts
,
it
r
ev
ea
ls
w
h
er
e
t
h
e
y
h
av
e
m
is
u
n
d
er
s
ta
n
d
i
n
g
s
o
r
p
r
o
b
lem
s
[
1
0
]
.
I
n
th
e
s
t
u
d
y
,
s
tu
d
en
t
s
w
er
e
ab
le
to
r
ef
lect
o
n
t
h
eir
s
tr
e
n
g
th
s
a
n
d
w
ea
k
n
ess
e
s
,
m
ak
in
g
it
p
o
s
s
ib
l
e
f
o
r
t
h
e
m
to
o
v
er
co
m
e
th
e
m
.
T
h
is
co
n
n
ec
ts
to
th
e
s
tu
d
y
m
ad
e
b
y
E
d
w
i
n
s
[
19
]
w
h
ich
co
n
f
ir
m
ed
th
at
w
r
iti
n
g
r
ef
lecti
v
el
y
d
r
iv
e
s
s
t
u
d
en
t
s
to
b
ec
o
m
e
e
n
th
u
s
iast
i
c
an
d
ea
g
er
to
s
et
g
o
als to
b
etter
th
e
m
s
el
v
es.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
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N:
2252
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8822
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K
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43
4.
CO
NCLU
SI
O
N
W
h
ile
R
J
W
w
a
s
s
tr
ictl
y
r
ef
l
ec
tiv
e
w
r
iti
n
g
,
it
w
as
f
o
u
n
d
o
u
t
in
t
h
is
s
t
u
d
y
t
h
at
u
tili
zi
n
g
w
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iti
n
g
ac
tiv
itie
s
w
o
u
ld
ce
r
tai
n
l
y
e
n
h
a
n
ce
th
e
m
at
h
e
m
atics
lear
n
in
g
e
x
p
er
ien
ce
o
f
t
h
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s
t
u
d
en
t
s
.
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h
e
g
o
al
in
t
h
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s
tu
d
y
is
to
ex
p
lo
r
e
h
o
w
co
lleg
e
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d
en
ts
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ie
w
m
ath
e
m
at
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n
in
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th
r
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R
J
W
in
h
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p
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w
w
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m
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lp
b
o
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a
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d
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d
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ex
p
lo
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m
at
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m
a
tics
i
n
a
w
a
y
t
h
e
y
f
i
n
d
p
r
ac
tical
an
d
co
n
s
tr
u
ct
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e.
T
h
is
s
t
u
d
y
s
h
o
w
ed
th
at
b
y
en
g
ag
in
g
s
t
u
d
en
t
s
in
r
ef
lecti
v
e
w
r
i
tin
g
ex
p
er
ien
c
es
w
h
ic
h
ex
p
licitl
y
p
r
o
m
o
te
m
ea
n
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g
-
m
a
k
i
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s
elf
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q
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teac
h
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s
cr
ea
t
e
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o
p
p
o
r
tu
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it
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f
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h
e
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to
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s
s
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f
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ca
n
en
r
ic
h
t
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m
ath
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atic
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I
t
m
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d
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lize
t
h
at
lea
r
n
in
g
m
at
h
e
m
atic
s
d
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es
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o
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al
w
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a
v
e
to
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ab
o
u
t
d
o
in
g
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lc
u
latio
n
s
.
R
a
th
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,
d
o
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g
m
at
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atic
s
m
a
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o
h
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d
in
h
a
n
d
w
it
h
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e
w
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iti
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g
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s
it
ca
n
p
r
o
p
el
d
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p
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th
e
s
t
u
d
en
t
s
’
u
n
d
er
s
tan
d
in
g
o
f
th
e
s
u
b
j
e
ct.
E
s
s
en
tiall
y
,
b
y
u
s
i
n
g
r
ef
lecti
v
e
w
r
iti
n
g
ac
t
iv
i
ties
,
teac
h
er
s
la
y
t
h
e
g
r
o
u
n
d
w
o
r
k
f
o
r
t
h
e
s
tu
d
e
n
ts
to
e
x
p
er
ien
ce
m
at
h
e
m
atics
f
av
o
r
ab
l
y
.
RE
F
E
R
E
NC
E
S
[1
]
V
a
n
De
r
M
o
le
n
L
.
Ac
ti
o
n
re
se
a
rc
h
:
Re
f
lec
ti
v
e
jo
u
rn
a
li
n
g
w
it
h
in
m
id
d
le
g
ra
d
e
s.
Bo
w
li
n
g
G
re
e
n
S
tate
Un
iv
e
rsit
y
;
2
0
1
5
.
[2
]
L
a
tu
li
p
p
e
C,
L
a
tu
li
p
p
e
J.
S
t
u
d
e
n
t
p
e
rc
e
p
ti
o
n
s
o
f
w
rit
in
g
p
ro
jec
ts
in
a
u
n
iv
e
rsity
d
iff
e
re
n
ti
a
l
-
e
q
u
a
ti
o
n
s
c
o
u
rse
.
In
ter
n
a
t
io
n
a
l
J
o
u
rn
a
l
o
f
M
a
t
h
e
ma
ti
c
a
l
Ed
u
c
a
ti
o
n
in
S
c
ien
c
e
a
n
d
T
e
c
h
n
o
l
o
g
y
.
2
0
1
1
;
45
(1
):
1
–
1
1
.
[3
]
Od
a
f
e
V
U.
T
e
a
c
h
in
g
a
n
d
L
e
a
rn
i
n
g
M
a
t
h
e
ma
ti
c
s:
S
t
u
d
e
n
t
Ref
lec
ti
o
n
A
d
d
s
a
Ne
w
Dime
n
sio
n
.
In
D
.
K.
P
u
g
a
lee
,
A
.
Ro
g
e
rso
n
,
a
n
d
A
.
S
c
h
in
c
k
(Ed
s.)
P
r
o
c
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e
d
in
g
s
o
f
th
e
Nin
t
h
In
ter
n
a
ti
o
n
a
l
Co
n
f
e
re
n
c
e
:
M
a
th
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m
a
ti
c
s
Ed
u
c
a
ti
o
n
i
n
a
G
lo
b
a
l
Co
m
m
u
n
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y
(p
p
.
4
8
6
-
4
9
0
)
IS
BN 8
3
-
9
1
9
4
6
5
-
8
-
4
.
2
0
0
7
.
[4
]
L
e
w
M
DN
,
S
c
h
m
id
t
HG
.
S
e
l
f
-
r
e
f
le
c
ti
o
n
a
n
d
a
c
a
d
e
m
i
c
p
e
rf
o
r
m
a
n
c
e
:
Is
th
e
re
a
re
latio
n
sh
ip
?
A
d
v
a
n
c
e
s
in
He
a
lt
h
S
c
ien
c
e
s E
d
u
c
a
ti
o
n
.
2
0
1
1
;
16
(4
),
5
2
9
–
5
4
5
.
[5
]
M
in
t
o
n
C
.
W
rit
in
g
a
n
d
re
a
d
in
g
i
n
m
a
th
e
m
a
ti
c
s,
Un
iv
e
rsit
y
o
f
W
isc
o
n
sin
-
S
to
u
t;
2
0
0
8
.
[6
]
Ril
lero
P
,
Clela
n
d
J,
Zam
b
o
R.
I
n
:
Na
ti
o
n
a
l
A
ss
o
c
iatio
n
o
f
Bio
lo
g
y
Tea
c
h
e
rs
Na
ti
o
n
a
l
Co
n
v
e
n
ti
o
n
.
W
rit
e
f
ro
m
th
e
sta
rt:
W
rit
in
g
-
to
-
lea
rn
sc
ien
c
e
a
n
d
m
a
th
e
m
a
ti
c
s;
1995.
[7
]
M
il
ler L
D.
W
rit
in
g
in
m
a
th
e
m
a
ti
c
s cla
ss
e
s.
Re
p
o
rt
n
u
m
b
e
r:
IS
S
N
-
1
0
3
3
-
3
7
3
8
;
1
9
9
2
.
[8
]
O'
Ke
ll
e
y
S
K.
He
lp
in
g
tea
c
h
e
rs co
n
n
e
c
t
w
rit
in
g
to
d
o
i
n
g
m
a
th
e
m
a
ti
c
s.
S
RA
T
E
J
o
u
r
n
a
l
.
2
0
1
3
;
23
(
1
):
1
8
-
23.
[9
]
P
a
rso
n
s
M
R.
Ef
fe
c
ts
o
f
w
rit
in
g
to
lea
rn
in
p
re
-
c
a
lcu
lu
s
m
a
th
e
m
a
ti
c
s
o
n
a
c
h
iev
e
m
e
n
t
a
n
d
a
ffe
c
ti
v
e
o
u
tco
m
e
s
f
o
r
stu
d
e
n
ts
in
a
c
o
m
m
u
n
it
y
c
o
ll
e
g
e
se
tt
in
g
:
A
m
i
x
e
d
-
me
th
o
d
a
p
p
ro
a
c
h
(Un
p
u
b
li
sh
e
d
d
o
c
to
ra
l
d
isse
rt
a
ti
o
n
).
Co
l
o
ra
d
o
S
tate
Un
iv
e
rsity
;
2
0
1
1
.
[1
0
]
Co
o
k
J,
Cra
ig
C.
W
rit
in
g
m
a
th
e
m
a
ti
c
s.
M
issis
sip
p
i;
1
9
9
1
.
[1
1
]
Ca
rd
R.
T
h
e
e
ff
e
c
ts
o
f
w
rit
in
g
in
m
a
th
e
m
a
ti
c
s
o
n
se
c
o
n
d
-
g
ra
d
e
stu
d
e
n
ts'
a
c
h
iev
e
m
e
n
t
a
n
d
m
e
t
a
c
o
g
n
it
io
n
.
S
a
g
in
a
w
S
ta
te Un
iv
e
sity
;
1
9
9
8
.
[1
2
]
Kle
in
C,
P
f
led
e
re
r
B,
T
ru
c
k
e
n
m
il
ler
M
.
In
c
re
a
sin
g
stu
d
e
n
t
m
o
ti
v
a
ti
o
n
th
r
o
u
g
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c
o
o
p
e
ra
ti
v
e
lea
rn
in
g
,
w
rit
in
g
in
m
a
th
e
m
a
ti
c
s,
a
n
d
m
u
lt
ip
le i
n
tell
i
g
e
n
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e
s.
S
a
in
t
X
a
v
ier Un
iv
e
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;
1
9
9
8
.
[1
3
]
Co
o
p
e
r
A
.
T
o
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a
y
’s
tec
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n
o
lo
g
ies
e
n
h
a
n
c
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w
rit
in
g
in
m
a
t
h
e
m
a
ti
c
s
.
T
h
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Cle
a
rin
g
Ho
u
se
:
A
J
o
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r
n
a
l
o
f
E
d
u
c
a
ti
o
n
a
l
S
tra
teg
ies
,
Iss
u
e
s a
n
d
I
d
e
a
s
.
2
0
1
2
;
85
(
2
):
8
0
–
8
5
.
[1
4
]
P
o
w
e
ll
A
B,
L
ó
p
e
z
J
A
.
W
rit
in
g
a
s
a
v
e
h
icle
to
lea
rn
m
a
th
e
m
a
ti
c
s:
A
c
a
se
stu
d
y
.
In
P
.
Co
n
n
o
ll
y
a
n
d
T
.
V
il
a
rd
i
(Ed
s.)
,
T
h
e
Ro
le
o
f
W
rit
in
g
in
L
e
a
rn
in
g
M
a
th
e
m
a
ti
c
s
a
n
d
S
c
ien
c
e
.
Ne
w
Y
o
rk
:
T
e
a
c
h
e
rs
Co
ll
e
g
e
.
Res
e
a
rc
h
e
rs
J
o
u
rn
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l
o
f
T
e
c
h
n
o
l
o
g
y
E
d
u
c
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ti
o
n
.
1
9
8
9
;
9
:
1
5
7
-
1
7
7
.
[1
5
]
A
d
d
iso
n
L
.
A
tea
c
h
e
r’s
re
f
lec
ti
o
n
o
n
t
h
e
im
p
lem
e
n
tatio
n
o
f
jo
u
r
n
a
l
w
rit
in
g
in
m
a
th
e
m
a
ti
c
s
.
Un
iv
e
rsity
o
f
A
lb
e
rta
;
1
9
7
2
.
[1
6
]
Ba
x
ter
J,
W
o
o
d
w
a
rd
J,
Olso
n
D.
W
rit
in
g
in
m
a
th
e
m
a
ti
c
s:
A
n
a
lt
e
rn
a
ti
v
e
f
o
r
m
o
f
c
o
m
m
u
n
ica
ti
o
n
fo
r
a
c
a
d
e
m
ic
a
ll
y
lo
w
-
a
c
h
iev
in
g
stu
d
e
n
ts.
L
e
a
rn
in
g
Disa
b
il
it
ies
Re
se
a
rc
h
a
n
d
P
ra
c
ti
c
e
,
2
0
0
5
;
20
(2
)
:
1
1
9
–
1
3
5
.
[1
7
]
L
ieb
a
rs
C.
IM
T
E
Co
n
fe
re
n
c
e
P
r
o
c
e
e
d
in
g
s,
In
:
A
n
n
u
a
l
M
e
e
ti
n
g
o
f
th
e
A
ss
o
c
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n
o
f
M
a
th
e
m
a
ti
c
s
T
e
a
c
h
e
rs
Ed
u
c
a
to
rs.
J
o
u
r
n
a
l
writi
n
g
:
A
m
o
d
e
l
fo
r m
a
th
e
ma
ti
c
s tea
c
h
e
r ed
u
c
a
ti
o
n
;
1
9
9
7
.
[1
8
]
Tw
a
r
B.
T
h
e
e
ff
e
c
ts
o
f
u
sin
g
a
n
in
tera
c
ti
v
e
stu
d
e
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o
k
o
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o
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n
d
a
lg
o
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m
s
o
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a
d
d
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o
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a
n
d
su
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ti
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o
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f
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s
a
n
d
m
ix
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d
n
u
m
b
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f
o
r
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i
f
th
g
ra
d
e
m
a
th
e
m
a
ti
c
s
stu
d
e
n
ts.
Un
iv
e
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y
o
f
Ce
n
tral
F
lo
ri
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a
;
2
0
0
3
.
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9
]
Ed
w
in
s
S
.
In
c
re
a
sin
g
re
f
lec
ti
v
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writi
n
g
n
d
g
o
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l
se
tt
in
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sk
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ls
o
n
h
ig
h
a
b
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six
th
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ra
d
e
m
a
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m
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No
v
a
S
o
u
t
h
e
a
ste
rn
Un
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;
1
9
9
5
.
[2
0
]
Am
ir
Y,
L
ian
g
h
u
o
F
.
Ex
p
lo
ri
n
g
h
o
w
to
im
p
le
m
e
n
t
jo
u
r
n
a
l
w
rit
in
g
e
ffe
c
ti
v
e
l
y
in
p
rim
a
r
y
m
a
th
e
m
a
ti
c
s
in
S
in
g
a
p
o
re
;
2
0
0
2
.
[2
1
]
S
a
d
ler R.
F
o
rm
a
ti
v
e
a
s
se
ss
m
e
n
t
a
n
d
t
h
e
d
e
sig
n
o
f
in
stru
c
ti
o
n
a
l
sy
ste
m
s.
In
stru
c
ti
o
n
a
l
S
c
ien
c
e
.
1
9
8
9
;
1
8
:
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1
9
–
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4
4
.
[2
2
]
Ha
tt
ie J,
T
i
m
p
e
rl
e
y
H.
T
h
e
P
o
w
e
r
o
f
F
e
e
d
b
a
c
k
.
Rev
iew o
f
Ed
u
c
a
ti
o
n
a
l
Res
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a
rc
h
.
2
0
0
7
;
77
(
1
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8
1
-
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1
2
.
[2
3
]
Ke
n
n
e
y
R,
S
h
o
ff
n
e
r
M
,
No
rris
D.
Re
f
l
e
c
ti
n
g
to
lea
rn
m
a
th
e
m
a
ti
c
s:
S
u
p
p
o
rti
n
g
P
S
M
T
s'
p
e
d
a
g
o
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ica
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t
Evaluation Warning : The document was created with Spire.PDF for Python.
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Vo
l.
6
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No
.
1
,
Ma
r
ch
20
1
7
:
38
–
44
44
k
n
o
w
led
g
e
with
re
f
l
e
c
ti
o
n
o
n
w
rit
in
g
p
ro
m
p
ts
in
m
a
th
e
m
a
ti
c
s
e
d
u
c
a
ti
o
n
.
[P
re
p
ri
n
t
]
.
A
v
a
il
a
b
le
f
ro
m
:
h
tt
p
s:/
/www
.
m
a
th
.
p
u
rd
u
e
.
e
d
u
/
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rh
k
e
n
n
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y
/Ke
n
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m
e
p
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in
k
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to
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u
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c
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P
ra
c
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_
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n
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e
y
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tal_
F
in
a
l.
p
d
f
[
A
c
c
e
s
se
d
1
st
Ja
n
u
a
ry
2
0
1
7
]
[2
4
]
Hie
m
stra
R.
Us
e
s
a
n
d
b
e
n
e
f
it
s
o
f
jo
u
rn
a
l
w
rit
in
g
.
Ne
w
Dire
c
ti
o
n
s
f
o
r
A
d
u
lt
a
n
d
Co
n
ti
n
u
i
n
g
Ed
u
c
a
ti
o
n
.
2
0
0
1
;
9
0
:
19
-
26.
[2
5
]
Ko
lb
A
,
Ko
lb
D.
L
e
a
rn
in
g
st
y
l
e
s
a
n
d
lea
rn
in
g
sp
a
c
e
s:
En
h
a
n
c
in
g
e
x
p
e
rien
ti
a
l
L
e
a
rn
in
g
in
h
i
g
h
e
r
e
d
u
c
a
ti
o
n
.
A
c
a
d
e
m
y
o
f
M
a
n
a
g
e
m
e
n
t
L
e
a
rn
in
g
a
n
d
E
d
u
c
a
ti
o
n
.
2
0
0
5
;
4
(2
):
1
9
3
-
2
1
2
[2
6
]
S
v
in
ick
i
M
,
Dix
o
n
N.
T
h
e
Ko
l
b
m
o
d
e
l
m
o
d
if
ie
d
f
o
r
c
las
sro
o
m
a
c
t
iv
it
ies
.
Co
ll
e
g
e
T
e
a
c
h
in
g
.
1
9
8
7
;
3
5
(4
):
1
4
1
-
1
4
6
.
[2
7
]
P
a
tt
o
n
M
Q,
Co
c
h
ra
n
M
.
A
G
u
id
e
to
Us
in
g
Qu
a
li
tativ
e
Re
se
a
r
c
h
M
e
th
o
d
o
l
o
g
y
.
Av
a
il
a
b
le
f
ro
m
:
h
tt
p
s:/
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lu
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m
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e
_
to
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u
si
n
g
_
q
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a
li
tativ
e
_
re
se
a
rc
h
_
m
e
th
o
d
o
lo
g
y
.
p
d
f
[
A
c
c
e
ss
e
d
1
st
Ja
n
u
a
ry
2
0
1
7
].
[2
8
]
De
n
z
in
N,
L
in
c
o
l
n
Y.
Ha
n
d
b
o
o
k
o
f
Qu
a
li
tativ
e
Re
se
a
r
c
h
,
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Ca
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2009;
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[3
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[3
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W
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1
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2
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]
[3
4
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Ya
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