In
te
r
n
ation
a
l Jou
rn
al
o
f E
v
al
u
a
t
i
on
a
n
d
R
e
se
arc
h
in
Ed
u
c
ation
(
IJERE
)
V
o
l.6,
N
o.3,
S
eptem
b
er
2
01
7
,
pp. 233~
2
4
2
IS
S
N
: 2252-
88
22
233
Jou
rn
a
l
h
o
me
pa
ge
:
ht
tp:
//i
a
e
sj
ourn
a
l.
com
/
on
line
/
in
dex.
p
h
p
/
I
J
ERE
M
a
th
ema
t
ical Writing Errors
in E
x
positor
y
Writi
ngs of
C
o
llege Mathematics Students
Ivee
K.
Guce
Mathematics
Depart
men
t
, De
La
S
al
le
Lipa,
P
hil
i
ppines
Art
i
cl
e In
fo
ABSTRACT
A
r
tic
le hist
o
r
y
:
Re
ce
i
v
e
d
Ju
l
1
2,
201
7
Re
vise
d A
ug
2
1
,
201
7
Ac
ce
p
t
ed
Au
g
2
8
,
2
017
Des
p
ite
t
he
e
fforts
to
c
onfi
r
m
t
he
e
f
f
ecti
v
ene
s
s
of
w
ri
ting
i
n
learni
ng
m
a
th
ematics,
a
naly
sis
on
c
om
m
o
n
errors
i
n
m
a
th
ematical
w
rit
i
ng
s
h
a
s
n
o
t
receiv
e
d
s
u
ff
ic
i
e
nt
a
tten
ti
on
.
Th
is
s
t
u
d
y
a
i
m
ed
t
o
pro
v
i
d
e
an
a
ccoun
t
of
t
h
e
stu
d
e
n
ts’
pro
c
e
dura
l
e
x
p
l
a
n
a
t
ion
s
i
n
te
rm
s
of
t
he
ir
c
o
mmo
nly
c
o
mm
it
te
d
errors
i
n
m
a
th
em
atical
w
riti
ng.
N
in
e
errors in
m
a
th
emat
ical
w
ri
ting
w
ere pr
e-
def
i
ned
--
nam
e
ly
,
m
i
s
u
s
e
o
f
mat
h
em
ati
cal
t
erm
s
,
m
i
su
se
o
f
m
a
th
em
a
tical
sy
m
bols,
i
nco
r
rect
n
ot
a
t
i
o
n
,
i
ncorre
ct
g
ram
m
a
r,
i
nco
r
rect
cap
it
alizat
io
n,
no
or
i
nco
rre
ct
p
u
n
ct
uati
on,
v
ag
ue
t
erm
,
i
n
c
orrect
t
erm
,
a
nd
l
ack
of
t
erm
o
r
phrase.
This
s
t
u
dy
u
sed
qualitative
method
o
f
r
esearch
t
o
keep
a
reco
rd
o
f
errors
i
n
mat
h
e
m
atical
w
ritin
g.
C
o
ndu
cted
i
n
the
Co
ll
ege
o
f
E
d
u
cati
on
A
r
ts
and
S
c
ien
ces o
f
De La S
a
lle
L
ipa,
t
he
s
tu
dy
i
n
v
o
l
v
e
d
twe
l
ve
B
S
Mathema
tics
st
ud
ents
e
nro
l
l
e
d
in
A
dv
anced
C
al
cul
u
s
1
cla
s
s
as
r
es
pon
den
t
s.
Resul
t
s
reveal
ed
t
h
a
t
the
m
o
st
c
o
mmitted
e
rrors
d
on
e
i
n
m
ath
e
m
a
tical
w
r
it
in
g
are
in
correct
g
ram
m
a
r
an
d
m
i
s
u
s
e
o
f
m
a
th
ema
t
ical
s
y
m
bols
.
C
ert
a
in
ly
,
in
terv
enti
on
pro
gram
s on
m
ath
e
matics
writi
n
g
w
i
ll
bri
ng
f
avo
r
ab
le
o
u
t
com
e
s
.
Lan
g
u
a
ge
c
ou
rs
es
i
n
the
s
t
u
d
en
ts
’
curricu
lu
m
w
h
i
c
h
t
ackl
e
p
rop
e
r
g
r
a
mma
r
us
age
m
a
y
be
i
n
t
egrat
e
d
wi
th
w
rit
i
ng
a
b
o
u
t
m
ath
e
matics
as
p
art
of
t
h
e
st
udent
activit
i
e
s.
S
uch
w
ill
p
ro
vide
t
h
e
s
tu
de
n
t
s
with
w
ri
ting
e
xp
eriences
fitted
to
t
heir
di
s
cipl
i
n
e.
K
eyw
ord
:
Ma
th e
xp
la
n
a
t
i
on
Ma
them
at
ica
l
w
rit
i
n
g
P
r
ocedura
l
er
r
ors
W
r
it
ing
e
r
ro
rs
W
r
it
ing
i
n
ma
t
h
Co
pyri
gh
t © 2
017 In
stit
u
t
e
of Advanced
En
gi
neeri
n
g
an
d
Scien
ce.
All
rights
res
e
rv
ed.
Corres
pon
d
i
n
g
Au
th
or:
Ivee K. Guce,
Ma
them
at
ics De
partm
e
n
t
,
D
e
L
a
S
a
l
l
e
L
ip
a
,
Pr
es
.
J
.
P
.
Laurel
Highw
ay,
Lipa
C
ity,
Bata
n
g
as,
P
h
i
l
ip
p
i
n
e
s.
Em
ail:
i
v
e
e
.
g
u
c
e@d
l
s
l
.
e
du.
p
h
1.
I
N
TR
OD
U
C
TI
O
N
Writ
ing
,
a
s
a
f
o
rm
o
f
l
a
ngua
g
e
,
pl
ay
s
an
e
sse
nti
a
l
rol
e
i
n
ma
th
e
m
a
tic
s
l
e
arni
ng
.
A
p
a
rt
f
ro
m
un
dersta
n
d
i
n
g
c
once
p
ts
a
nd
pri
n
ci
p
l
es,
stu
d
e
n
ts
s
h
o
u
l
d
b
e
ab
le
to
w
ri
t
e
m
at
h
e
ma
ti
ca
l
sol
u
ti
on
s
cl
e
a
r
l
y
a
nd
lo
gic
a
l
l
y.
H
ow
ever
,
stu
d
en
ts
i
n
a
ma
the
m
a
tic
s
cla
ss
te
n
d
n
o
t
t
o
b
e
mi
ndf
ul
o
f
ho
w
t
h
e
y
w
o
u
ld
e
xpl
a
i
n
,
i
n
w
r
itin
g,
t
he
ir
s
olu
t
i
o
n
t
o
a
g
i
v
en
p
rob
l
em
a
s
the
y
t
hi
n
k
t
h
a
t
t
he
t
eac
her
foc
u
se
s
o
n
l
y
o
n
t
he
c
or
rec
t
n
e
ss
o
f
the
i
r
a
n
sw
er.
They
b
e
c
o
m
e
t
oo
c
onc
er
ne
d
w
i
t
h
t
he
c
om
pu
t
a
ti
ons,
pa
y
i
n
g
lit
tl
e
or
n
o
a
tte
nt
i
on
t
o
t
he
c
la
rity
o
f
the
i
r
o
v
e
r
al
l
s
o
l
u
tio
n.
I
t
t
h
e
r
e
f
ore
bec
o
me
s
essen
tia
l
for
t
h
e
m
t
o
re
a
lize
the
im
p
o
rta
n
c
e
of
w
ri
ti
n
g
s
k
ills
i
n
ma
them
at
i
c
s c
l
a
s
s
as it is in
th
e
i
r
Eng
lish
or ot
h
er
sub
jec
t
s.
I
n
t
e
grati
o
n
of
w
r
i
ti
n
g
i
n
m
a
t
h
em
ati
c
s
has
r
ece
i
v
e
d
c
o
n
s
i
d
e
ra
bl
e
a
tt
e
n
ti
on
i
n
t
h
e
rese
a
r
ch
l
it
era
t
u
r
e.
The
r
o
l
e
o
f
c
o
mm
unica
ti
on
c
on
t
i
n
u
es
t
o
be
a
d
d
re
sse
d
i
n
curr
en
t
re
po
r
t
s
an
d
s
t
u
d
ie
s.
I
n
fac
t
,
o
n
e
of
t
he
p
r
o
c
ess
st
and
a
rd
s
id
en
tif
i
ed
b
y
t
h
e
Nat
i
o
n
a
l
C
o
un
c
i
l
of
T
e
a
c
h
e
r
s
of
M
athem
a
tics
[
1
]
f
o
r
students,
a
side
f
r
o
m
b
e
co
ming
m
at
h
e
mat
i
c
a
l
p
robl
e
m
s
ol
v
e
rs,
i
s
c
o
m
mu
ni
c
a
t
i
ng
kno
wl
e
d
ge.
Bef
o
re,
ma
t
h
c
l
a
sses
re
lie
d
o
n
s
kil
l
-
bu
ild
i
ng
a
nd
c
once
p
tua
l
u
n
d
e
r
stan
di
n
g
act
iv
i
t
ies.
B
ut
i
n
the
re
ce
n
t
y
ea
rs,
t
e
a
c
h
ers
a
r
e
re
a
l
i
z
in
g
t
h
a
t
i
n
c
l
u
si
on
o
f
w
r
i
t
i
n
g
i
n
a
m
a
t
h
e
m
a
t
i
c
s
c
l
a
s
s
i
s
“
m
o
r
e
t
h
a
n
j
u
s
t
a
w
a
y
t
o
d
oc
um
en
t
in
for
m
a
t
i
o
n”
(
p.
3
);
i
t
i
s
a
w
a
y
o
f
dee
p
e
n
ing
the
stude
n
t
s’
l
ea
rn
ing
a
nd
a
to
o
l
f
or
h
e
l
p
i
n
g
t
he
m
ga
i
n
n
e
w
i
n
s
i
g
h
t
s
[
2
]
.
I
n
s
p
i
t
e
o
f
t
h
e
f
e
w
s
t
u
d
i
e
s
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SSN: 2252-
8822
IJERE
V
ol
.
6,
N
o.
3,
S
eptem
b
er
20
17 :
2
3
3
– 242
23
4
w
h
ic
h
c
ite
d
s
o
me
a
ppr
eh
en
si
ons
o
f
te
ac
h
e
rs
i
n
u
s
i
n
g
w
r
iti
n
g
i
n
m
a
t
h
em
ati
c
s
(e
.g.
[3]
,
[4]
)
,
a
num
ber
of
rese
arc
h
ers
a
r
ti
cula
te
d
its
b
e
n
efi
t
s (e
.g.
[4]
-
[
9
]
)
.
Wri
t
i
n
g-to-
l
ea
rn
m
e
t
h
odo
l
o
g
i
es
r
equ
i
re
s
t
u
d
e
nt
s
t
o
a
na
lyz
e
,
c
om
pa
re,
a
nd
syn
t
hes
i
ze
i
nfor
ma
t
i
on.
I
t
d
e
v
e
lo
ps
a
s
kill
t
h
a
t
is
g
round
e
d
i
n
t
h
e
cogn
it
i
v
e
do
ma
in
[
10
].
D
aily
w
ri
tin
g
a
c
t
i
v
it
ies
i
n
cre
a
se
m
athe
m
a
tics
a
c
h
i
e
v
e
m
e
n
t
a
n
d
me
t
aco
gni
tio
n
[5
].
R
at
h
e
r
th
a
n
f
o
c
u
s
i
ng
on
s
ol
e
m
at
hemat
i
cal
c
o
m
pu
t
a
ti
on
,
st
ud
e
n
t
s
p
ay
atte
n
t
ion
al
so
t
o
how
t
he
y
expr
ess
idea
s
a
n
d
dee
p
en
t
he
i
r
u
nde
r
s
t
a
n
d
i
ng
w
h
en
t
h
e
y
a
r
e
en
ga
ged
i
n
ma
them
at
i
c
al
w
ritin
g
ta
sks
[11].
The
proce
s
s
o
f
w
riti
ng
s
low
s
t
he
s
t
ude
n
t
s
dow
n
all
o
win
g
t
he
m
to
p
roce
ss
in
form
ation
[6
]
.
I
t
give
s
t
h
e
m
t
he
opp
or
tu
ni
t
y
t
o
com
m
u
n
ica
t
e
t
he
w
a
y
t
he
y
u
nde
rs
t
a
n
d
m
at
he
ma
tics
and
u
s
e
ma
them
at
i
c
s
v
o
ca
b
u
l
a
r
y
i
n
w
ritin
g
[1
2].
Th
i
s
l
e
a
d
s
t
o
t
h
e
deve
l
opm
en
t
o
f
t
he
ir
w
r
i
t
i
n
g
s
k
ills
h
e
l
pi
ng
t
he
m
ma
ke
d
eep
a
n
d
m
e
a
n
in
g
f
ul
c
o
nnec
tio
ns be
t
w
een
m
a
t
he
ma
t
i
c
a
l c
onc
e
pt
s
[
4
],
[
7
].
I
n
a
case
st
ud
y
co
nd
uc
ted
by
D
e
fa
zi
o,
e
t
a
l
.
[10]
,
he
s
tressed
tha
t
s
tude
nts
in
a
ll
l
e
v
e
l
s
sho
u
l
d
n
o
t
on
ly
b
e
go
o
d
i
n
w
r
it
t
e
n
c
o
m
m
unic
a
tio
n
bu
t
a
l
s
o
m
us
t
un
d
e
rsta
n
d
the
im
p
o
rta
n
ce
o
f
g
o
o
d
w
r
i
t
i
n
g
sk
ills.
O
b
s
e
r
v
a
t
i
o
n
s
b
y
S
e
o
[
8
]
s
u
g
g
e
s
t
t
h
a
t
w
r
i
t
i
n
g
s
i
n
a
m
a
t
h
e
m
a
t
i
c
s
cla
ss
com
p
rise
m
ain
l
y
of
s
ym
bo
ls,
us
in
g
minim
a
l
num
b
e
r
of
w
ords.
H
e
p
oin
t
ed
o
u
t
t
hat
c
a
lc
u
l
a
t
i
o
n
s
a
r
e
i
m
p
orta
n
t
b
u
t
t
he
y
s
h
o
u
ld
n
ot
b
e
t
h
e
m
a
in
a
c
t
i
vit
y
i
n
an
y
ma
t
h
e
m
a
t
i
c
s
c
l
as
s.
T
hu
s
,
m
at
h
e
mat
i
cs
t
eac
h
e
rs
s
h
oul
d
p
r
om
ote
w
r
i
tin
g
in
t
he
ir
c
lass
es
b
y
e
x
pli
c
it
ly
a
ssi
gn
ing
wri
t
i
n
g
act
iv
i
t
i
e
s [8
]
to
co
n
s
t
a
nt
ly
st
r
en
gthe
n l
i
n
es o
f com
m
uni
c
a
t
i
o
n
be
t
w
e
e
n
t
he te
a
c
h
er
and stu
d
e
n
ts
a
s
th
e
i
r
a
c
tiv
it
ies
pr
ogr
ess
[
13].
It
h
a
s
b
e
e
n
p
r
o
p
o
sed
th
a
t
u
sing
b
ot
h
l
a
ng
u
a
ge
a
nd
s
y
m
boli
s
m
i
n
com
m
unic
a
t
in
g
m
a
them
at
ica
l
i
de
as
s
h
o
u
l
d
b
e
a
ma
i
n
c
o
n
c
ern
in
a
n
y
m
a
them
a
tics
c
u
rr
i
c
u
l
u
m
[
5].
Wri
tin
g
exe
r
cises
in
t
e
g
r
ated
i
n
pr
ob
l
e
m
solv
i
n
g
gi
ve
t
he
t
ea
cher
a
sse
ss
m
e
nt
o
f
th
e
kn
ow
le
d
g
e
g
a
ine
d
by
the
s
t
ude
n
t
s
[8];
t
hus,
i
n
fl
u
e
nc
in
g
his
m
a
t
h
em
ati
c
s
tea
c
h
in
g
[1
2].
They
c
an
be
u
sed
t
o
g
i
v
e
t
h
e
te
ac
he
r
a
hin
t
o
n
h
o
w
t
o
hel
p
st
u
den
t
s s
u
cc
eed
i
n
t
h
e c
l
ass [14]
.
Wh
ile
it
is
a
p
rim
a
r
y
c
onc
e
r
n
t
h
at
m
a
t
he
m
a
tics
s
t
ude
n
t
s
l
e
arn
t
he
m
a
t
he
ma
ti
c
a
l
p
r
i
nc
iple
s
an
d
conc
e
p
ts
t
a
u
gh
t
t
o
t
hem
,
it
is
v
ery
im
p
o
rta
n
t
also
t
hat
t
h
ey
b
e
co
me
s
ki
ll
e
d
a
t
art
i
c
ul
a
tin
g
t
h
e
i
r
t
h
oug
ht
s
a
n
d
idea
s
effec
t
i
v
el
y
in
w
rit
i
ng.
H
ow
ever
,
w
i
th the
b
irt
h
o
f
te
xt
m
e
ssagi
ng,
s
t
ude
n
t
s
now
a
d
a
y
s
oft
e
n
use
i
n
form
al
l
a
ngu
a
g
e
in
t
ry
ing
t
o
c
o
mmu
ni
cat
e
i
n
w
ri
ti
ng
.
Gra
m
ma
r
,
p
un
c
t
u
a
t
ion,
c
api
t
a
l
i
z
at
i
o
n
and
t
h
e
l
i
ke
a
r
e
o
fte
n
ig
nor
ed
[
1
0
].
I
n
m
a
the
m
a
tica
l
e
x
p
o
si
tor
y
w
rit
i
n
g
s,
s
ym
b
o
ls
a
re
at
t
im
es
m
isuse
d
a
n
d
o
veruse
d.
I
t
is
t
hus
a
cha
lle
ng
e
for
t
e
a
c
hers
t
o h
o
n
e
t
he
i
r
st
ude
n
t
s
to bec
ome
e
f
fec
t
i
ve
c
omm
u
n
i
c
a
t
o
rs.
A
p
a
r
t
from
s
ol
vin
g
m
athem
a
t
i
ca
l
pro
b
l
e
ms,
ma
them
at
i
c
ia
n
s
(
shou
l
d)
actu
a
l
l
y
s
p
en
d
a
grea
t
de
a
l
o
f
t
i
me
w
ri
ti
ng
b
ec
aus
e
m
a
s
t
e
ri
ng
t
h
e
a
bil
ity
t
o
wri
t
e
cl
ear
m
a
t
h
e
ma
t
i
cal
e
xpl
a
n
ati
ons
i
s
v
e
ry
e
ssen
ti
al
.
A
c
c
o
rdi
n
g
to
L
e
e
[
15],
i
f
a
m
a
t
he
ma
tic
i
a
n
w
a
nts
to
c
on
tri
b
ut
e
t
o
t
h
e
g
r
eat
e
r
bod
y
of
m
a
t
h
e
ma
t
i
c
al
kn
ow
le
d
g
e,
h
e
m
u
st
b
e
a
b
le
t
o
c
o
m
m
unica
te
h
is
i
de
as
i
n
a
w
a
y
w
h
ich
i
s
c
ompre
h
en
si
bl
e
to
o
the
r
s.
T
h
u
s,
bei
n
g
ab
le
t
o
w
r
ite
c
lear
ly
i
s
a
n
e
q
u
a
l
l
y
i
m
porta
n
t
m
athem
a
tica
l
sk
il
l
a
l
on
g
w
i
t
h
t
he
a
bil
i
t
y
t
o
sol
v
e
equa
t
i
o
n
s.
Thi
s
s
tu
dy
c
once
r
n
s
t
h
e
e
xp
osit
o
r
y
wri
t
in
g
o
f
t
hi
rd
y
e
a
r
B
S
M
a
t
he
ma
t
i
c
s
s
tu
de
n
t
s,
t
h
a
t
is,
w
r
it
i
n
g
w
h
ic
h
is
i
nt
e
n
ded
t
o
e
x
p
l
ai
n
m
a
them
at
ical
i
dea
s
a
n
d
p
r
o
ce
dures.
T
h
e
a
i
m
o
f
t
h
i
s
p
a
p
e
r
i
s
t
o
p
r
o
v
i
d
e
a
n
a
c
c
o
unt
o
f
t
h
e
st
ud
ent
s
’
pro
c
e
d
u
r
al
e
xpl
a
n
ati
o
n
s
i
n
te
rms
o
f
t
h
e
i
r
com
m
onl
y
c
o
mm
itte
d
er
ro
rs
i
n
ma
them
at
i
c
al
w
r
i
ti
n
g
.
D
e
spi
t
e
the
me
nti
o
ned
st
u
d
i
e
s
s
how
in
g
t
h
e
e
ffe
c
t
i
v
enes
s
of
w
riti
n
g
i
n
le
ar
nin
g
ma
them
at
i
c
s,
a
nal
y
s
i
s
o
n
c
om
mon
err
o
rs
i
n ma
t
h
e
m
a
tic
al
w
r
i
ti
n
g
s
h
a
s
n
ot
r
ece
ive
d
s
u
ffi
c
i
e
n
t
a
t
t
e
n
t
io
n.
L
itt
le
i
s
kno
wn
o
f
ho
w
ma
t
h
e
m
a
t
ic
i
a
ns
a
nd
s
t
ude
n
t
s
un
d
e
rst
a
nd
a
nd
u
se
l
a
ngu
a
g
e
in
t
e
c
hn
ic
al
w
ri
tin
g
[
16]
s
i
n
ce
ma
ny
o
f
t
he
se
r
ese
a
rc
hes
foc
u
se
d
o
n
descr
i
b
i
n
g
t
he
e
ffe
cts
of
u
si
n
g
w
rit
i
ng
in
m
at
he
m
a
tics
lear
n
i
n
g
(
e.
g
.
[
4]
).
D
ocum
en
tin
g
t
h
e
c
o
m
m
on
l
y
c
omm
itte
d
err
o
rs
of
t
he
s
tud
e
nt
s
w
h
en
d
oi
n
g
e
xpos
it
o
r
y
w
r
itin
g
i
n
m
athem
a
ti
cs
i
s
prese
n
t
l
y
n
o
t
m
uc
h
o
f
a
c
once
r
n
of
m
a
ny
re
sea
r
cher
s.
T
he
r
esul
t
of
t
his
st
u
d
y
n
o
t
on
ly
p
rov
i
de
s
aw
a
r
ene
s
s
o
n
t
he
n
at
ure
of
c
ol
le
ge
m
a
t
he
ma
t
i
c
s
s
t
u
d
e
n
t
s
’
e
x
p
o
si
tory
w
r
i
ti
n
g
b
u
t
it
a
l
so
g
iv
e
s
bas
i
s
o
n
h
ow
t
heir
w
r
itte
n
c
o
mm
unica
ti
o
n
s
kil
l
s
in
m
a
t
he
ma
ti
c
s
m
a
y
s
t
i
l
l
b
e
i
m
p
r
o
v
e
d
b
y
t
h
e
t
e
a
c
h
e
r
s
w
h
o
trai
n the
m
.
1.1.
Common
l
y
Comm
itt
e
d Mathem
atica
l
Wri
tin
g
E
rror
s
Di
ff
e
r
e
n
t
ma
th
e
m
a
t
ic
al
w
ri
ti
ng
s
ca
n
b
e
g
ro
up
e
d
i
nt
o
fo
rma
l
a
nd
inf
o
rma
l
w
ritin
g.
T
he
f
orme
r
i
n
v
o
l
v
e
s
wri
t
i
n
g
so
me
t
e
r
ms
o
r
bit
s
o
f
expl
an
a
t
i
o
n
on
a
b
la
ck
bo
ard
du
ring
a
l
e
c
t
u
re
o
r
ex
pl
ai
ni
n
g
s
o
m
et
hi
n
g
to
a
f
r
i
en
d
o
n
a
p
iece
o
f
scra
t
c
h
pa
per.
T
he
l
at
t
e
r,
how
e
v
er
,
i
nc
lu
de
s
t
h
e
ki
n
d
o
f
w
r
it
i
n
g
expe
c
t
e
d
i
n
a
paper
.
There
a
r
e di
ffe
r
e
nce
s
i
n w
h
a
t
is ac
ce
pt
a
b
l
e
[17]
. F
or t
h
i
s s
t
ud
y, n
i
n
e
e
rrors i
n m
a
t
h
em
at
ic
al w
riti
ng ha
v
e
b
e
e
n
rega
rded
b
y th
e
resea
r
che
r
a
s the
mos
t
c
omm
on tha
t
s
tu
de
nt
s c
o
m
m
it.
1.
1.1.
Misuse
o
f
Mat
h
ematical
T
erms
C
o
rre
ct
u
se
o
f
ma
them
ati
c
a
l
t
e
r
ms
i
s
very
i
mporta
nt
w
he
n
do
i
n
g
ex
po
si
to
ry
w
ri
ti
ng
.
Ma
t
h
emat
i
c
s
requ
ire
s
a
v
ery
pre
c
i
s
e
use
of
l
a
n
g
u
ag
e
[1
5].
M
i
s
u
se
o
f
a
m
a
the
m
a
ti
ca
l
term
,
if
n
ot
c
orre
cted,
w
i
l
l
cre
a
te
con
f
usi
o
n
am
o
ng
t
h
e
r
eade
r
o
f
the
te
xt.
In
a
s
t
u
d
y
b
y
Mo
ore
[1
8
],
h
e
n
o
t
e
d
t
hat
“
w
hi
le
a
tt
e
m
p
t
in
g
to
w
rite
form
al
p
ro
ofs,
s
tude
n
t
s
do
n
o
t
n
ece
ssa
r
i
l
y
u
nde
rs
t
a
n
d
t
he
c
on
te
nt
o
f
re
l
e
van
t
d
e
f
i
n
iti
on
s
or
how
t
o
use
t
h
ese
definiti
ons
in
p
roof-writing.
”
A
n
e
x
a
m
p
l
e
o
f
mi
su
s
e
o
f
ma
t
h
e
m
a
t
ical
t
e
r
ms
i
s
to
s
a
y
o
r
w
r
i
t
e
“
t
he
lim
it
Evaluation Warning : The document was created with Spire.PDF for Python.
IJERE
I
S
S
N
:
2252-
88
22
M
a
t
h
e
m
atic
al
Writi
n
g
Error
s
in Ex
posi
t
ory W
r
i
t
i
n
gs
o
f
Colle
ge
Ma
t
h
em
a
tic
s St
u
d
en
ts (
I
ve
e
K
.
G
u
ce
)
23
5
conv
e
r
ges
”
or
t
he
u
se
o
f
t
h
e
w
o
rd
“
e
qua
tio
n”
t
o
m
ean
a
m
athe
ma
tica
l
e
xpr
ess
i
on
w
h
i
c
h
is
n
ot
c
o
n
sid
e
r
e
d
an
equa
t
i
o
n
.
1.
1.2.
Misu
se
o
f
Math
ematic
al
S
ymb
o
l
s
I
n
m
athem
a
tic
s,
s
ym
bol
s
are
in
tegra
l
c
om
pone
n
t
s
o
f
c
omm
unic
a
t
i
o
n
a
n
d
th
us
m
us
t
be
u
se
d
appr
opr
i
a
t
e
l
y
.
F
u
rther,
p
a
r
t
o
f
b
e
i
n
g
a
bl
e
to
w
rit
e
w
e
l
l
i
n
m
at
hem
a
t
i
cs
i
s
to
k
n
o
w
w
h
e
n
t
o
use
sym
b
ol
s
or
w
h
en
t
o
use
w
o
rds
instea
d
[1
5].
A
c
o
mm
on
m
istake
i
s
to
m
isuse
t
he
“
=”
s
ymbo
l.
M
a
n
y
stu
d
e
n
t
s
t
e
nd
to
u
s
e
equa
l
s
i
gns
t
o
co
n
n
e
c
t
se
ver
a
l
l
i
nes
of
s
o
l
u
t
i
o
n
w
h
i
c
h
a
r
e
n
o
t
act
u
a
l
l
y
e
qu
a
l
.
An
ot
h
e
r
co
nce
r
n
on
ma
them
at
i
c
al
s
ym
bo
ls
i
s
the
fac
t
t
hat
m
a
t
h
em
atics
is
c
a
s
e
-
sensi
t
i
ve
[
19
]
.
A
s
a
rule
,
u
ppe
r-c
ase
an
d
low
e
r-
ca
se
v
ersi
ons
o
f
the
sam
e
l
e
t
ter
are
not
u
se
d
i
n
ter
c
han
g
e
a
b
l
y
u
n
l
ess
i
t
h
as
b
ee
n
spe
c
i
fi
ed
t
hat
t
h
ey
r
e
p
resen
t
the sa
me
qua
nt
it
y
.
1.
1.3.
Inco
rr
ec
t No
t
a
tio
n
s
N
o
ta
t
i
o
n
s in m
athe
ma
t
i
cs ar
e
the
s
ymb
o
lic e
xpre
s
s
i
ons
w
hi
ch ha
v
e
pre
c
is
e and
e
s
t
a
b
l
is
h
e
d me
ani
n
g.
Exam
p
l
es
o
f
n
o
ta
t
i
ons in
ca
lc
ul
us w
o
u
l
d be the differe
nce
betw
e
en
{
a
n
}
and a
n
.
The
f
o
rme
r
i
s a not
a
t
ion for a
seq
u
enc
e
w
h
i
l
e
t
he
l
at
ter
refers
t
o
a
te
rm
i
n
a
se
que
nc
e.
U
sin
g
t
he
t
w
o
n
ota
t
i
ons
i
n
t
erc
h
an
gea
b
ly
w
ou
ld
ce
rtai
nly
co
nfu
s
e
some
one r
eadi
n
g in
a
m
ath
e
ma
t
i
cs
c
o
n
te
x
t
.
1.
1.4.
Inco
rr
ec
t G
r
a
m
m
a
r
G
o
o
d
q
ua
l
i
t
y
w
riting
o
b
se
r
v
es
c
or
rec
t
g
r
a
m
m
ar.
Thi
s
a
l
s
o
a
p
plie
s
to
e
x
p
os
i
t
or
y
w
r
it
ing
in
ma
them
at
i
c
s.
L
ee
[
15]
e
mpha
siz
e
d
tha
t
w
hen
o
n
e
w
r
i
t
es
i
n
a
ma
th
c
la
ss
h
e
is
e
xpe
c
t
e
d
t
o
us
e
corr
ect
g
r
a
mma
r
a
n
d
sp
e
l
l
i
n
g
a
s
h
i
s
w
ri
t
i
ng
s
houl
d
b
e
c
l
e
a
r
a
nd
p
ro
f
e
ss
i
o
nal. L
ike
o
t
her
tex
t
s, ma
t
hem
a
t
i
c
s
is
w
r
i
t
t
e
n
w
ith
s
en
te
nc
es
a
nd
para
gra
p
h
s
.
Whi
l
e
form
ulas
a
n
d
e
q
u
a
t
i
o
ns
a
re
e
le
me
nts
of
a
m
ath
pa
per,
one
s
h
o
u
l
d
kee
p
in
m
in
d
tha
t
s
ta
ndar
d
g
ram
m
a
t
i
c
a
l
r
ule
s
a
p
p
l
y
i
n
m
a
t
h
em
atic
s
w
r
i
t
in
g.
A
si
de
f
r
o
m
su
bjec
t-verb
a
gre
e
m
e
nt,
ano
t
her
com
m
o
n
g
ram
m
a
tica
l
e
rror
is m
ist
a
kin
g
a
phr
ase
for
a se
nt
e
n
ce
l
i
ke
writi
n
g
“S
i
n
ce
f
is a
f
unction.”
1.
1.5.
Inco
rr
ec
t Ca
p
i
tal
i
z
a
t
io
n
I
n
t
hi
s
study,
t
his
error
refers
t
o
t
w
o
i
d
e
a
s
n
a
m
e
ly
,
(i
)
c
a
p
i
t
a
l
izi
n
g
w
o
r
d
s
w
h
ic
h
sh
ou
l
d
n
o
t
b
e
ca
pita
l
i
ze
d
an
d
(ii)
no
t
cap
i
t
a
l
izi
n
g
wor
d
s
w
h
ic
h
sh
o
u
ld
b
e
cap
i
t
al
ize
d
.
In
w
r
i
t
i
n
g
,
i
t
i
s
i
m
porta
nt
t
o
o
b
s
er
ve
the
r
u
le
s
in
c
a
p
ita
li
za
ti
on
in
t
hat
it
c
o
n
v
e
y
s
to
t
he
r
e
a
der
s
t
he
i
m
porta
nce
of
s
pec
i
fic
w
o
r
d
s
a
nd
t
h
e
c
h
a
nge
i
n
me
anin
gs
o
f
w
o
rds.
I
n
w
r
it
in
g
a
m
a
them
at
ic
a
l
e
x
p
l
ana
t
i
on,
a
s
tu
de
nt
us
u
a
l
l
y
t
en
ds to
ca
pita
l
i
ze
a
w
or
d
w
h
ic
h
she w
a
nts t
o
h
i
g
h
lig
h
t
a
lth
o
u
g
h
the
w
ord is n
ot
a
pr
ope
r nou
n
a
t a
l
l.
1.
1.6.
No or
Inc
o
rr
ect
Pun
c
t
uation
P
u
n
c
tu
at
ion
is
o
n
e
o
f
th
e
most
i
mpo
r
t
a
nt
a
sp
e
c
t
s
o
n
e
s
houl
d
b
e
co
nsc
i
ou
s
of
w
he
n
w
r
i
t
in
g.
I
n
t
h
e
Im
portance
o
f
P
unc
tua
t
i
on
(htt
ps
://
thew
r
ite
c
o
rner
.w
ordpr
ess.com
),
it
w
a
s
po
in
t
e
d
o
u
t
tha
t
t
h
i
s
fe
a
t
u
r
e
of
w
r
itin
g
“
g
i
v
es
m
eani
n
g
to
t
he
w
r
itte
n
w
o
r
d
s,
m
uch
l
i
ke
p
a
u
se
s
a
nd
c
h
a
ng
e
s
i
n
t
o
n
e
s
of
t
h
e
v
oi
c
e
w
hen
spea
ki
n
g
.”
F
urther,
an
e
rror
in
p
unc
t
u
at
i
o
n
ma
y
c
o
n
v
e
y
a
n
en
t
i
r
el
y
d
i
ffe
r
ent
m
e
a
n
in
g
to
t
he
o
ne
t
ha
t
i
s
in
t
e
nde
d.
A
s
st
a
t
e
d
b
y
S
u
[
17
],
“
all
m
a
t
h
em
atic
s
s
h
o
u
l
d
be
w
rit
te
n
i
n
c
o
m
plete
se
nte
n
c
e
s.”
Thus
e
q
u
a
t
i
o
ns,
eve
n
d
isp
l
a
y
e
d
ones,
sho
ul
d h
a
ve
pu
n
c
t
ua
t
i
o
n
if
use
d
i
n a
sente
nc
e.
1.
1.7.
Va
g
u
e T
e
rm
or
Phra
se
A
v
a
g
u
e
t
e
r
m
o
r
p
h
r
a
s
e
i
n
t
h
i
s
s
t
u
d
y
i
s
o
n
e
t
h
a
t
i
s
n
o
t
c
l
e
a
r
o
r
on
e
tha
t
s
ug
ge
sts
d
i
ffe
r
ent
m
e
a
n
in
g
resul
t
i
ng i
n
lac
k o
f
cer
t
a
int
y
a
nd d
i
st
inc
t
ne
ss in a m
at
hem
a
t
i
c
al sta
te
me
n
t
.
S
i
nc
e ma
t
h
e
m
a
t
i
c
s re
quire
s pr
e
c
ise
lan
g
u
age
,
t
he
u
se
o
f
i
ndi
st
in
gu
is
ha
ble
w
o
r
d
s
o
r
t
er
ms
i
s
hi
g
h
ly
d
isc
o
urage
d
.
A
ve
ry
c
omm
on
mis
t
a
k
e
o
n
e
ma
kes
in
m
at
h
e
m
a
tica
l
w
r
i
t
i
n
g
i
s
t
h
e
use
of
t
he
w
ord
“
i
t
”
i
n
g
i
vin
g
a
n
e
x
pla
n
a
t
i
o
n
t
o
a
s
o
l
u
t
i
o
n.
A
ppr
opr
iate
spec
ific
t
er
min
o
l
ogy or
m
a
t
he
ma
ti
c
a
l
e
x
p
res
s
i
on
m
u
s
t
a
lw
a
y
s be
us
ed
s
o
a
s
t
o
de
l
i
ver the
corre
ct
i
nf
orma
tio
n
to a
r
ea
der
[19].
1.
1.8.
Inco
rr
ec
t T
e
r
m
o
r
Phr
a
se
A
ter
m
o
r
phra
s
e
w
h
ich
is
n
o
t
a
p
p
ro
pria
t
e
ly
u
sed
in
a
m
athem
a
ti
ca
l
statem
en
t
l
e
a
d
i
n
g
to
t
he
inc
o
rre
ctn
e
ss
o
f
t
hou
gh
t
or
i
d
ea
in
t
ha
t
sta
t
e
m
ent
is
r
efe
rre
d
t
o
a
s
i
n
c
o
r
r
e
c
t
t
e
r
m
o
r
p
h
r
a
s
e
.
I
n
t
h
i
s
s
t
u
d
y
,
t
h
i
s
err
o
r
i
s
t
ake
n
a
s
a
c
onse
que
nc
e
of
t
w
o
pos
si
bi
l
i
t
i
es
:
(i)
w
r
it
e
r’s
m
i
s
un
der
s
ta
n
d
in
g
ab
o
u
t
c
e
rta
i
n
m
a
the
m
atica
l
pri
n
ci
p
l
e a
nd (i
i)
h
i
s
l
a
c
k
or
sub
p
ar
o
f
ch
oic
e
o
f w
o
r
d
.
1.
1.9.
Lac
k
o
f Ter
m
or
Ph
rase
Wh
ile
a
g
o
o
d
m
a
them
ati
cal
e
xp
la
nat
i
on
i
s
c
hara
cteriz
e
d
b
y
a
de
t
a
i
le
d
so
l
u
ti
on,
o
n
e
s
hou
ld
n
o
t
di
sc
o
u
n
t
t
h
e
i
m
porta
nc
e
o
f
u
sing
w
o
r
d
s
or
phrases
t
o
c
o
n
n
ect
t
h
e
ide
a
s
be
h
i
nd
t
h
e
m
a
t
h
em
atic
al
e
x
p
re
ssi
ons.
A
ma
t
h
emat
i
cal
p
rob
l
e
m
s
u
r
e
do
e
s
h
a
v
e
a
sp
e
c
i
f
i
c
a
n
s
wer;
h
o
w
ev
e
r,
i
t
m
a
kes
perfec
t
s
e
n
se
t
o
sa
y
t
h
at
t
he
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SSN: 2252-
8822
IJERE
V
ol
.
6,
N
o.
3,
S
eptem
b
er
20
17 :
2
3
3
– 242
23
6
answ
e
r
c
ou
l
d
b
e
o
b
t
a
i
ne
d
fr
om
t
w
o
o
r
m
o
re
d
iffer
e
nt
s
olu
t
i
o
ns.
Th
us
,
present
i
n
g
o
nl
y
the
m
a
t
h
e
m
atic
a
l
so
l
u
t
i
o
n
w
i
t
h
o
u
t
a
n
y
ex
pla
n
a
tio
n
is
like
ass
u
min
g
t
ha
t
the
solu
ti
o
n
w
ill
s
p
eak
f
or
i
t
s
e
l
f
w
h
ic
h,
i
n
esse
nc
e,
i
s
no
t
a
pur
po
se
o
f
e
x
pos
it
ory
w
r
iti
n
g
.
F
u
rthe
r,
not
s
u
p
p
l
yi
n
g
t
he
n
eed
e
d
t
erm
o
r
p
h
r
a
s
e
l
e
a
v
e
s
t
h
e
r
ead
e
r
t
o
b
e
in-c
ha
rge
of
g
ue
ssin
g
t
he
s
t
o
ry
b
eh
i
nd
t
h
e
w
r
iter
’
s
solu
t
i
o
n
w
h
i
c
h
is
d
a
nger
ous
a
n
d
v
e
r
y
like
l
y
to
c
ause
misi
nform
a
ti
o
n
.
A
l
so
u
n
d
er
t
hi
s
ty
pe
o
f
er
ror
is
t
he
w
ri
ter
’
s
f
ail
u
re
t
o
d
e
fi
n
e
a
v
ari
a
bl
e
t
h
at
h
e
d
ecl
a
r
es
i
n
hi
s
so
l
u
t
i
o
n
.
E
x
a
m
ple:
T
o
f
i
n
d
t
h
e
sum
of
a
s
e
r
ies,
one
m
ay
e
va
l
u
a
t
e
li
m
n
n
s
wh
ere
n
s
i
s
t
h
e
n
th
p
art
i
al
s
um
of
t
he
ser
i
e
s.
2.
METHOD
Th
is
s
tu
dy
use
d
q
ual
i
t
a
t
i
v
e
m
e
th
o
d
o
f
re
se
arc
h
t
o
kee
p
a
r
ec
ord
o
f
B
S
M
a
t
h
e
m
a
t
i
c
s
s
t
u
d
e
n
t
s
’
e
r
r
o
r
s
i
n
m
a
t
h
e
m
a
t
i
c
a
l
w
r
i
t
i
n
g
.
T
h
e
s
t
u
d
y
w
a
s
c
o
n
d
u
c
t
e
d
i
n
t
h
e
C
o
l
l
e
g
e
o
f
E
d
uca
t
i
on
A
r
t
s
a
nd
S
c
i
e
nce
s
o
f
D
e
L
a
Sal
l
e Lip
a
.
2.1.
Partic
i
p
ants
The
p
a
r
tic
i
p
an
t
s
i
n
the
st
u
dy
w
e
re
t
w
e
lve
t
h
ird
ye
ar
B
S
M
a
them
a
tics
st
u
d
en
ts
e
nro
lle
d
i
n
A
dva
nc
e
d
C
a
l
c
ulu
s
1
i
n
t
h
e
seco
nd
s
e
m
e
s
t
e
r
o
f
s
ch
ool
y
ea
r
20
16
-20
1
7
.
A
s
mathem
a
tics
ma
jo
r
st
u
d
e
n
ts,
the
y
a
re
expe
c
t
e
d
t
o
ha
ve
a
r
ea
sona
ble
le
ve
l
of
m
as
ter
y
i
n
the
co
n
t
en
t
o
f
th
e
subj
e
c
t
a
n
d
t
h
e
ab
il
i
t
y
t
o
d
eli
v
er
t
h
e
con
t
e
n
t t
o
t
he
i
r
audie
n
c
e
,
bo
t
h
o
ra
l
l
y a
nd i
n
w
r
i
ti
n
g
.
2.2.
Writing
Tas
k
P
roc
e
d
u
re
Thr
o
u
g
h
o
u
t
t
h
e
f
irst
h
a
l
f
of
t
he
s
e
m
e
s
te
r,
e
xpla
n
at
ory
es
say
t
y
pe
o
f
q
u
e
st
io
ns
w
ere
g
i
ve
n
to
t
h
e
stude
n
t
s
as
a
r
egu
l
ar
w
r
itin
g
acti
v
ity
i
n
t
h
e
sub
j
e
c
t
.
Se
o
[8
]
e
xpla
i
ns
h
ow
e
xp
la
nat
o
ry
e
s
s
ays
ca
n
be
u
s
e
d
as
a
w
r
itin
g
a
c
t
i
v
i
t
y
f
or
t
he
m
at
hem
a
ti
cs
c
lass
room
.
A
s
ide
from
s
i
m
p
ly
a
sk
ing
a
st
ude
nt
t
o
e
x
p
l
a
i
n
c
e
rtai
n
conc
e
p
ts
i
n
his
ow
n
w
o
rds,
t
h
i
s
ty
pe
o
f
ac
ti
v
i
t
y
r
e
q
u
i
r
e
d
t
h
e
s
t
u
d
e
n
t
s
t
o
e
x
p
l
a
i
n
a
m
a
t
h
e
m
a
t
i
c
a
l
p
r
o
c
e
s
s
i
n
a
form
o
f
essa
y.
A
t
the
tim
e
of
d
a
t
a
c
o
lle
ct
i
o
n,
n
o
spe
c
i
f
i
c
p
ro
g
ram
or
c
urric
u
l
u
m
i
s
u
se
d
t
o
e
nha
n
c
e
t
h
e
st
u
d
e
nt
s’ ma
t
h
e
ma
t
i
c
al
wri
t
i
ng
ski
ll
s.
A
c
tivi
t
i
es
w
e
r
e
gi
ven
a
t
t
he
e
nd of
cer
ta
i
n
t
o
p
ic
s.
T
o
g
i
ve t
he
s
t
u
d
e
n
t
s
e
n
o
u
g
h
t
i
m
e
,
t
he
y
w
e
r
e
g
i
v
e
n
tw
o da
ys
t
o
fi
n
i
s
h
eac
h a
c
tiv
it
y. In s
o
l
v
in
g a
pro
b
lem
, the
y
w
e
r
e
i
ns
truc
te
d t
o
s
how
t
heir
deta
i
le
d sol
u
ti
on a
n
d
wri
t
e
a
c
l
e
a
r
e
x
p
l
a
n
a
t
i
o
n
a
bou
t
it
.
Th
ey
n
eed
ed
t
o
no
t
e
i
n
wo
rd
s
t
h
e
pr
o
c
ess
or
c
onc
e
p
ts
t
ha
t
t
h
ey
u
s
e
d
in
so
l
v
i
ng t
h
e
pro
b
lem
a
nd t
h
e r
a
ti
o
n
a
l
e for usi
ng su
ch
m
e
t
ho
d.
2.3.
T
h
e
Instrument
Be
low
are
the
five a
ct
i
v
i
t
ies t
h
at
t
he
s
tu
de
nt
s com
p
le
te
d.
A
c
tivi
t
y
1
:
Y
our
f
rie
n
d
Jose
p
h
d
id
n
ot
a
tte
n
d
t
o
d
ay’
s
m
eetin
g
a
s
he
i
s
s
t
ill
proce
ssin
g
h
is
e
n
rol
l
m
e
n
t
.
Y
ou
l
ent
hi
m
your
n
ot
es.
He
t
ol
d
y
o
u
h
e
und
erst
oo
d
mo
st
o
f
wh
at
h
e
re
a
d
e
x
c
ept
fo
r
t
h
e
l
a
tt
er
part
o
f
t
h
e
lec
t
ure
.
H
e
w
a
s
overw
he
l
m
ed
w
it
h
the
pas
t
t
op
i
c
s
(l
ike
inde
t
e
rm
inate
form
s)
t
ha
t
em
erge
d
d
u
r
i
n
g
the le
c
t
ure.
H
e aske
d if
you
co
ul
d
(1) exp
l
a
i
n to h
i
m
t
h
e
c
o
n
v
erge
nc
e of
. A
l
s
o,
t
o p
r
epar
e hi
m
fo
r t
h
e
ne
xt
d
isc
u
ssio
n
, he furt
h
e
r
r
equ
este
d
for
yo
u t
o
(2) di
sc
uss
the
ne
xt
e
xam
p
le
i
n
t
h
e
hand
o
u
t
w
h
ic
h
is
f
i
n
d
i
ng
w
h
e
t
he
r
th
e
se
qu
e
n
ce
c
on
ver
g
es
o
r
di
verge
s
.
A
c
tivi
t
y
2:
Wh
at di
d
I
m
iss today
i
n
A
dvca
l
1
?
W
h
a
t tra
n
sp
i
r
e
d
d
u
r
in
g
th
e
mee
t
i
ng?
Looki
ng
at
yo
u
r
no
tes,
I
c
annot
u
n
d
er
sta
nd
the
S
queez
e
The
o
re
m. Can
y
o
u
e
xp
la
in
to m
e
,
p
l
ease
?
A
c
tivi
t
y
3: Y
o
ur
cla
ssm
a
te
J
en a
tt
e
n
ded
the
l
a
st lec
ture
.
H
o
w
e
v
er, sh
e
co
u
l
d
not
f
ull
y
und
e
r
st
an
d
t
h
e
conc
e
p
t of
g
re
a
t
e
s
t
low
e
r
bo
und
an
d
l
e
as
t u
p
p
er
b
o
u
n
d
. S
he
i
s
h
av
in
g d
i
ffi
c
ul
t
y
(
1
)
de
t
e
r
m
i
ni
n
g
if
a
seq
u
ence
is
b
ou
n
d
ed a
n
d
(2) bou
n
d
in
g
a se
que
nce
usi
n
g its
g
rea
t
es
t
lo
w
e
r
boun
d (GLB)
and le
as
t u
pper
b
o
u
n
d
(LU
B).
H
o
w
w
o
u
l
d
you
exp
l
a
i
n i
t
t
o he
r usi
ng t
h
e
s
e
que
nc
e
1
3
n
n
?
A
c
tivi
t
y
4
:
Y
o
u
w
a
nt
t
o
c
o
me
up
w
i
t
h
a
r
e
v
i
e
w
e
r
for
an
u
pcom
in
g
qui
z
.
H
o
w
w
o
u
ld
y
ou
e
xpl
a
i
n
to
yo
ursel
f
i
n
det
a
il t
h
e
fol
l
ow
i
n
g
?
(1)
D
e
termine
if t
he
s
erie
s
2
1
8
32
n
nn
c
onver
g
e
s
or
div
e
rges.
If it con
v
e
r
ges,
w
hat
i
s
t
he
sum
?
(2)
Write
0
.27
2
7
2
7
… as
a
r
at
io o
f
tw
o i
n
teg
e
rs.
A
c
tivi
t
y
5
:
Y
o
ur
t
e
a
c
h
e
r
s
che
dule
d
f
or
a
g
r
a
ded
r
ecit
a
t
i
on
nex
t
m
ee
ti
n
g
.
S
h
e
g
av
e
a
l
i
st
o
f
po
ssi
ble
que
st
ions
t
o
b
e
a
ske
d
d
uri
n
g
t
h
e
ac
tiv
i
t
y.
Y
ou
w
a
n
t
e
d
t
o
r
ea
ll
y
p
r
e
p
a
r
e
f
o
r
i
t
s
o
y
o
u
d
e
c
i
d
e
d
t
o
w
r
i
t
e
a
det
a
il
e
d
scr
ipt
on
h
o
w
yo
u w
oul
d
exp
l
a
i
n th
e
answ
er
t
o ea
c
h
que
sti
on.
Evaluation Warning : The document was created with Spire.PDF for Python.
IJERE
I
S
S
N
:
2252-
88
22
M
a
t
h
e
m
atic
al
Writi
n
g
Error
s
in Ex
posi
t
ory W
r
i
t
i
n
gs
o
f
Colle
ge
Ma
t
h
em
a
tic
s St
u
d
en
ts (
I
ve
e
K
.
G
u
ce
)
23
7
Q
u
e
s
ti
ons
: A
r
e
the
fo
l
l
ow
in
g
sta
t
e
m
e
n
t
s
t
r
u
e
?
W
hy or
w
hy
not
?
(1)
B
e
cau
s
e
4
1
n
ap
proa
c
h
es
0
as
n
a
ppr
oac
h
es
,
4
1
1
0
n
n
.
(2)
B
e
cau
s
e
4
1
li
m
0
n
n
,
t
h
e
series
4
1
1
n
n
co
nv
erg
e
s.
(3)
The
s
e
ries
1
10
00
(
1
)
n
n
n
di
ver
g
es.
2.4.
Writing
Tas
k
A
ssessmen
t
Al
l
th
e
su
b
m
itte
d
a
c
t
i
vi
ti
e
s
o
f
th
e
st
u
d
e
n
t
s
w
e
re
c
o
m
p
i
l
e
d
pe
r
t
o
p
i
c.
T
he
re wer
e a t
o
ta
l o
f
fifty-
s
eve
n
ou
tpu
t
s
e
x
am
i
n
ed.
T
h
re
e
s
t
u
d
en
ts
w
e
r
e
i
n
ci
den
t
a
l
l
y
a
bse
n
t
d
u
ri
n
g
th
e
l
e
ctu
r
e
fo
r
th
e
se
c
o
nd
a
c
t
i
v
ity
a
n
d
were
exc
l
u
d
e
d
i
n
th
e
cor
r
espo
nd
i
n
g
w
r
it
in
g
tas
k
.
A
ll
o
u
t
p
uts
w
e
re
r
e
a
d
t
horo
u
g
h
l
y
f
o
r
at
l
e
a
s
t
tw
o
tim
es.
The
y
w
e
r
e
c
a
r
efu
l
ly
i
nve
sti
g
ate
d
f
or
o
cc
urr
e
nce
of
m
a
t
he
ma
t
i
c
a
l
w
r
i
t
ing
er
rors.
The
num
b
er
o
f
times
tha
t
e
ach
s
t
u
d
e
n
t
c
o
m
m
i
t
s
a
n
e
r
r
o
r
w
a
s
r
e
c
o
r
d
e
d
a
n
d
t
a
l
l
i
e
d
.
F
o
r
e
a
s
y
r
e
f
ere
n
ce
d
u
r
in
g
t
h
e
ta
l
l
y
i
ng
of
d
a
t
a
,
a
c
o
d
in
g
proce
s
s
w
a
s
d
e
vel
o
ped.
T
he
n
ine
pre-
defi
ned
m
a
them
ati
c
a
l
w
r
i
ti
n
g
err
o
rs
w
ere
give
n
co
des
w
h
ich
a
p
pe
ar
i
n
Table
1.
Table
1. Pre-de
f
ine
d
M
at
hem
a
tical
W
riti
ng
E
rrors
C
ode
M
a
t
h
e
m
a
ti
c
a
l
W
r
i
t
i
ng
E
r
ro
rs
MM
T
Misu
se
o
f
math
e
m
a
t
i
c
a
l
ter
m
s
M
M
S
M
isuse
of m
a
t
h
e
m
a
ti
c
a
l
s
y
m
bol
s
IN
I
nc
or
re
c
t
not
a
tion
IG
I
nc
or
re
c
t
g
r
a
m
m
a
r
I
C
I
nc
or
re
c
t
ca
p
i
t
a
li
z
a
t
i
o
n
N
I
P
No
o
r
inc
o
rre
c
t
punc
tu
a
tion
V
T
V
ag
u
e
t
er
m
IT
I
nc
or
re
c
t
t
e
r
m
L
T
P
L
ack
o
f
t
e
r
m
o
r
p
h
r
a
s
e
3.
RESULT
S
A
N
D
DISCU
SSIO
N
3.1.
Resu
l
t
s
O
f
t
he
n
i
n
e
ma
them
at
i
c
al
w
ritin
g
e
rrors,
i
n
cor
r
ec
t
gr
a
m
m
a
r
(n=3
6)
s
h
o
w
e
d
t
o
b
e
t
h
e
m
o
st
com
m
it
ted
e
rror
in
A
c
t
i
v
i
t
y
1
as
i
l
l
u
s
t
r
a
t
ed
i
n
F
i
g
u
re
1
(
Ma
the
m
a
t
i
ca
l
w
r
itin
g
err
o
r
s
i
n
A
c
t
i
v
ity
1
).
E
le
ven
ou
t
of
t
he
t
w
e
lve
s
t
ude
n
t
s
h
a
d
a
t
l
ea
s
t
o
ne
g
ra
mm
atica
l
e
rror.
M
a
n
y
o
f
the
s
e
error
s
r
elate
t
o
s
u
b
j
ec
t
-
verb
agre
em
ent
(
e
.g
.
“S
ince
th
e
l
i
m
it
exi
st
…”
)
a
n
d
u
s
e
of
p
rep
o
s
i
t
i
ons
(
e.g.
“
raise
2
3
by
n
…”
).
N
ext
to
i
nc
orr
ect
gram
m
a
r
is
t
he
v
a
gue
t
er
m
er
ror
w
h
ich
i
s
a
ls
o
r
e
lat
i
vel
y
h
i
g
h
(
n
=
2
8
)
.
T
h
e
u
s
e
o
f
t
h
e
w
o
r
d
“
i
t
”
o
c
c
u
r
r
e
d
fre
que
n
tly
u
n
d
e
r
thi
s
t
ype
o
f
err
o
r.
T
o
c
i
t
e
t
w
o
i
n
s
ta
nc
es,
S
t
u
d
e
n
t
5
w
r
o
t
e
,
“
T
h
i
s
m
e
a
n
s
t
h
a
t
t
h
e
l
i
m
i
t
e
x
i
s
t
s
and the
r
efore
it
con
ver
g
es” to re
f
er
t
o
t
h
e gi
v
e
n
se
q
u
e
n
ce
t
h
a
t
is be
i
n
g
e
va
lua
t
e
d
fo
r
conve
rg
en
ce
,
an
d
no
t
th
e
li
m
i
t
m
e
n
t
i
o
ne
d
i
n
t
he
s
e
n
t
e
n
ce.
W
hi
l
e
,
bas
e
d
o
n
t
he
w
r
i
t
i
ngs
o
f
S
t
u
d
e
n
t
9
h
e
w
rote,
“
S
ince
2
3
i
s
less
than
1
,
i
t
m
e
a
n
s
t
h
a
t
it
c
on
ver
g
es
t
o
0
”
r
e
fe
rring
act
u
a
ll
y
to
2
3
n
.
Misus
e
o
f
m
a
them
at
i
c
a
l
t
erm
s
r
anke
d
t
h
ird
i
n
t
he
firs
t a
c
tiv
it
y.
I
n
correct
g
rammar
remained
f
irs
t
(
n=
2
0
)
i
n
t
h
e
se
co
nd
a
ctiv
i
t
y
wh
il
e
i
n
c
o
rre
ct
c
api
t
a
l
i
zat
ion
ca
me
ou
t
to
b
e
t
h
e
se
con
d
(
n
=
1
4)
m
ost-com
m
i
t
t
e
d
e
rror
as
p
r
e
s
e
nte
d
i
n
F
i
gure
2
(
Mathem
atical
w
riting
e
rrors
i
n
A
c
ti
vi
ty
2
).
O
f
the
seve
n
s
t
u
d
e
n
ts
(
58
%
)
w
ho
h
a
d
i
n
c
o
rre
ct
c
api
ta
l
i
zat
i
on
e
rror
,
only
o
n
e
go
t
s
i
x
e
rror
m
a
rks
in
t
he
t
a
l
ly.
T
h
e
o
t
her
si
x
h
a
d
o
n
e
or
t
w
o
e
rror
ma
rks
on
l
y
.
Ex
a
m
pl
e
s
o
f
w
h
i
c
h
und
e
r
t
hi
s
acti
v
ity
a
r
e
ca
pita
l
i
zi
n
g
t
he
w
or
ds
“
seque
nc
e”
an
d
“
lim
it”,
an
d
ca
p
i
ta
l
i
zi
n
g
a
w
o
r
d
f
o
l
l
o
w
i
n
g
a
c
o
m
m
a
(e.
g
.
“In
2
3
1
sin
n
n
,
We
w
il
l…”).
I
n
t
he
t
al
l
y
f
or
t
he
t
h
i
r
d
a
c
t
ivi
t
y,
F
ig
ure
3
(Mat
hem
a
t
i
ca
l
w
r
i
ting
e
rrors
i
n
Activity
3
)
shows
that
there
is
a
b
i
g
g
ap
b
e
t
w
een the
f
irst
a
n
d
s
ec
on
d-
rank
in
g
e
rror
s
.
A
s
in
t
he
s
ec
ond
ac
t
i
v
i
t
y
,
so
m
e
s
tu
de
nts
w
h
e
r
e
t
r
i
c
k
e
d
in
to
c
ap
it
al
iz
ing
wo
r
d
s
whi
c
h
a
r
e
actu
a
ll
y
no
t
p
ro
p
e
r
n
oun
s
(e
.g
.
l
e
a
s
t
up
p
e
r
bo
und
a
nd
g
re
a
t
est
l
o
w
e
r
bo
u
nd).
Inc
o
rrec
t
g
r
a
m
m
a
r
a
nd
no
or
i
nc
orr
ect
p
unc
t
u
a
t
i
o
n
b
o
t
h
pla
c
e
d
s
ec
o
n
d
(n
=
11)
i
n
t
h
e
ra
nki
ng.
T
h
i
s
i
s
fo
l
l
ow
e
d
c
lose
ly
b
y
err
o
rs
i
n
m
i
su
se
o
f
m
a
them
at
ica
l
s
ymb
o
ls
(
n
=
1
0).
Som
e
o
f
these
are
wron
g
u
s
e
o
f
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
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N: 2
2
5
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22
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J
ERE
Vol.
6, No.
3
, Se
pt
e
m
b
e
r
2017
: 233
– 242
23
8
e
l
l
i
p
s
i
s
(
e
.g.
“The
f
irst
f
our
t
erm
s
o
f
t
h
e
se
que
nce
a
r
e
1
9
,
2
27
,
3
81
,
4
24
3
,…”
)
a
nd
u
s
e
of
m
a
t
er
ial
im
p
l
ica
tio
n
sym
b
ol
“
”
to
m
ean
a
n
equa
l
i
t
y.
F
i
gur
e
1.
M
a
t
he
ma
ti
c
a
l
w
r
i
t
i
n
g
e
r
r
o
r
s
i
n
A
c
tivi
t
y
1
F
i
gur
e
2.
M
a
t
he
ma
ti
c
a
l
w
r
i
t
i
n
g
e
r
r
o
r
s
i
n
A
c
tivi
t
y
2
F
i
gur
e
3.
M
a
t
he
ma
ti
c
a
l
w
r
i
t
i
n
g
e
r
r
o
r
s
i
n
A
c
tivi
t
y
3
O
f
t
he
f
i
v
e
a
c
t
i
v
it
ies
give
n
to
t
he
s
t
u
den
t
s,
t
he
f
o
u
r
t
h
o
n
e
ca
m
e
out
t
o
ha
ve
t
he
l
eas
t
numbe
r
o
f
erro
rs
i
n
mat
h
emat
i
cal
w
ri
t
i
n
g
co
mmi
tt
ed
b
y
t
h
e
st
ud
en
ts.
On
ly
f
i
f
t
y
pe
r
c
e
n
t
(
n
=6
)
of
t
he
r
e
s
p
o
nde
nts
ac
c
o
unt
t
o
t
h
e
hi
gh
e
s
t
fre
quen
c
y
(
n
=
1
8
)
.
Thi
s
c
an
b
e
see
n
i
n
Fi
g
u
re
4
(
Mat
h
em
at
i
cal
w
ri
tin
g
err
o
rs
i
n
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ERE
I
S
S
N
:
2252-
88
22
Ma
them
at
ic
al Wri
t
i
ng Errors
i
n
Exp
o
si
tory
Wri
tin
gs
o
f
C
o
lle
ge Ma
t
h
ema
tic
s
St
u
d
en
ts
(Ive
e
K. Guce
)
23
9
Ac
tivi
t
y
4
).
B
e
cause
t
he
n
a
t
u
r
e
of
que
st
i
ons
i
n
th
is
a
c
tiv
it
y
r
e
q
ui
r
e
s
le
ng
th
y
s
o
l
u
t
i
o
n
s,
s
t
ude
n
t
t
e
n
ded
t
o
e
r
r
m
u
ch
o
n
the
u
s
e
of
d
if
fer
e
nt
m
athe
ma
t
i
c
a
l
s
ym
bo
l
s
(
e
.
g.
n
o
t
u
s
i
ng
e
q
u
a
l
“
=
”
si
g
n
s
a
nd
pa
r
e
n
t
he
sis
w
h
en
ne
cessar
y
a
nd
u
s
e
of
m
a
t
e
r
ia
l
i
m
pl
i
c
a
t
i
o
n
sy
m
bol
“
”
t
o
i
ndi
ca
t
e
a
s
t
a
t
e
me
n
t
o
r
e
x
pl
an
a
t
io
n
i
n
a
s
ol
uti
o
n
)
.
The
se
c
o
n
d
-
r
ank
i
ng
t
ype
o
f
er
r
o
r
,
i
ncor
r
e
c
t
g
r
a
m
m
ar
,
plac
ed
f
a
r
b
e
l
ow
t
he
f
ir
st
w
i
t
h
j
u
s
t
six
c
o
u
n
ts
d
one
b
y
on
l
y
t
w
o
(
17%
)
of
t
he
t
w
e
l
v
e
r
e
spo
nde
n
t
s.
F
i
gur
e
4.
M
a
t
he
ma
ti
c
a
l
w
r
i
t
i
n
g
e
r
r
o
r
s
i
n
A
c
tivi
t
y
4
Lik
e
i
n
t
h
e
fi
rst
t
w
o
a
c
ti
v
iti
es,
t
h
e
l
a
st
a
ct
iv
it
y
ha
d
in
co
rre
ct
g
r
a
m
m
a
r
as
t
he
m
os
t
c
o
mm
it
t
e
d
t
ype
o
f
e
rror
by
the
st
ude
n
t
s.
I
t
is
w
ort
h
n
o
tin
g
tho
u
g
h
t
ha
t
in
A
ct
iv
i
ty
5
a
s
sh
ow
n
in
F
igur
e
5
(
M
a
t
h
e
m
a
t
i
c
a
l
w
r
i
t
i
n
g
erro
rs
i
n
Ac
tivity
5
),
t
h
e
f
requ
en
cy
(
n=1
0
)
i
n
g
r
ammat
i
cal
erro
rs,
i
s
r
e
l
ati
v
ely
lo
w
c
o
mp
are
d
t
o
Act
i
vi
ty
1
(
n
=3
6)
a
nd
A
c
t
i
v
ity
2
(
n=
2
0
)
.
T
he
s
ec
on
d
t
o
r
ank
i
s
t
he
l
ack
o
f
t
er
m
or
phrase
er
ror
w
i
t
h
a
f
r
e
q
u
en
cy
o
f
8
fro
m
o
n
ly
f
iv
e
stud
en
ts
(
42%)
f
o
llo
wed
by
v
a
g
u
e
t
erm
erro
rs
p
la
ci
ng
thir
d
(
n
=
6
)
.
L
a
c
k
o
f
t
e
r
m
or
phr
a
s
e
e
r
r
o
r
s
m
o
s
tl
y
i
n
v
o
l
ve
d def
i
ni
ng
t
h
e
var
i
a
b
le
s
use
d
.
F
o
r
inst
anc
e
,
S
t
u
d
e
n
t
1
w
r
o
te
,
“
…
i
n
t
h
e
s
e
r
ie
s
4
1
1
n
n
,
w
e
sho
u
l
d
fi
nd
t
h
e
li
m
n
n
s
”
w
i
t
hou
t
d
e
f
i
n
i
ng
w
h
a
t
n
s
is.
F
i
gur
e
5.
M
a
t
he
ma
ti
c
a
l
w
r
i
t
i
n
g
e
r
r
o
r
s
i
n
A
c
tivi
t
y
5
3.
2.
Discu
ssi
on
The
N
C
T
M
s
t
a
nda
r
d
s
imp
lic
it
l
y
i
nvi
te
r
ese
a
r
c
her
s
t
o
e
n
g
a
ge
i
n
s
tud
i
es
t
ha
t
will
pr
o
m
ote
effec
t
i
v
e
ma
t
h
e
m
a
t
i
c
al
c
o
m
mu
n
i
c
a
tio
n
t
o
e
n
h
a
n
c
e
ma
th
emat
ic
al
p
e
d
ag
ogy
[
20
].
T
h
e
n
i
n
e
math
emati
cal
w
riti
ng
er
ro
rs
w
h
ic
h
ser
v
e
d
a
s
t
h
e
fr
a
m
e
w
or
k
o
f
t
he
s
tu
dy
e
me
r
g
ed
t
o
be
v
ia
ble
i
n
t
h
a
t
a
l
l
w
er
e
pr
esen
t
in
t
he
s
t
ude
n
t
s’
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
S
N: 2
2
5
2
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88
22
I
J
ERE
Vol.
6, No.
3
, Se
pt
e
m
b
e
r
2017
: 233
– 242
24
0
pr
o
c
e
dur
a
l
e
xp
l
a
na
t
i
o
n
s.
F
i
g
u
r
e
6
(
A
ver
a
ge
p
er
cen
tage
s
of
m
a
t
he
m
a
t
i
c
a
l
w
ritin
g
e
rrors
i
n
the
fi
ve
a
c
tiv
it
ies
)
pr
esen
ts
t
he
a
v
e
r
a
ge
p
er
c
e
nta
g
e
tha
t
e
ach
t
ype
o
f
er
r
o
r
w
a
s
c
o
m
m
itt
e
d
by
the
s
t
ude
nts
i
n
t
he
f
i
v
e
a
c
ti
vit
i
e
s.
I
n
cor
r
ec
t
gr
am
ma
r
r
a
nked
f
i
r
s
t
am
on
g
t
h
e
ni
ne
e
r
r
o
r
s
w
ith
a
n
a
v
er
a
g
e
of
2
1%.
Lew
a
nd
Mej
i
a
-
Ram
os
[
16]
,
i
n
t
he
i
r
qua
l
i
t
a
t
i
ve
s
tu
d
y
,
f
o
u
nd
t
h
a
t
u
nder
g
r
a
dua
t
e
m
a
t
h
e
m
atics
s
t
u
d
en
t
s
a
r
e
i
nc
l
i
ne
d
t
o
t
hi
n
k
t
h
at
t
h
e
r
ul
e
s
o
f
En
gli
s
h
do
n
o
t
a
ppl
y
in
m
a
t
h
e
ma
ti
c
a
l
sett
i
n
g
s
a
lth
oug
h
fo
r
th
e
mo
st
p
art
,
m
at
h
e
mat
i
c
i
an
s
in
the s
t
ud
y
bel
i
e
v
ed
t
hat gr
a
m
m
a
r
a
nd
pa
r
t
s
of s
peec
h
i
n
m
athe
ma
t
ica
l
w
o
r
ds sh
oul
d be c
ar
eful
l
y
a
tte
nd
e
d
t
o i
n
ma
t
h
e
m
a
t
i
c
al
wri
t
i
ng
.
T
h
e
nee
d
f
o
r
c
o
m
pl
et
e
sen
t
en
ce
s
a
n
d
at
t
e
n
t
i
on
t
o
g
r
a
mma
r
is
s
up
por
te
d
b
y
a
n
u
m
ber
of
m
a
them
at
icia
n
s
[
21]
-
[
25]
.
S
i
milar
to
t
h
i
s
s
t
udy,
d
ela
P
e
ña
[
26]
d
i
d
a
n
a
n
a
l
ysi
s
o
f
e
r
r
o
rs
but
i
n
t
h
o
s
e
essa
y
s
w
r
i
t
t
e
n
b
y
m
a
t
h
,
s
c
i
e
n
c
e
,
a
n
d
e
n
g
i
n
e
e
r
i
n
g
f
a
c
u
l
t
y
.
H
i
s
s
t
u
d
y
r
e
vea
l
e
d
t
ha
t
l
a
ng
ua
ge
p
r
o
b
l
em
s
s
u
ch
a
s
gr
am
ma
t
i
c
a
l er
rors
c
onfro
n
t
p
e
o
p
l
e
i
n
th
e
f
ie
ld
o
f scie
n
c
e
and
engi
n
e
e
r
i
ng.
F
i
gur
e
6.
A
ve
rage
p
er
ce
nta
g
e
s
o
f
m
a
them
at
i
cal
w
r
i
t
i
n
g
e
r
r
or
s
in
t
he
f
i
v
e
acti
v
ities
F
o
ll
ow
i
n
g
e
r
r
o
r
s
on
gr
am
mar
is
m
isuse
of
m
a
t
he
ma
tic
al
s
ym
bo
ls
w
h
i
ch
a
ve
r
a
ge
d
t
o
1
6
%
acr
oss
th
e
f
i
ve
a
c
t
i
v
it
ies.
A
c
c
or
di
n
g
t
o
Rube
nste
in
a
n
d
T
h
o
mps
o
n
[
27]
,
a
c
h
ievem
e
nt
i
n
m
a
them
atic
s
ma
y
im
p
r
ove
i
f
st
ude
n
t
s
u
n
d
er
sta
nd
a
n
d
pr
o
p
er
ly
c
om
mun
i
ca
te
u
s
i
ng
m
a
th
s
ym
b
o
l
s
a
nd
n
o
t
a
tio
n.
C
hi
r
u
me
[
2
8
]
w
h
o
in
ves
t
iga
t
e
d
h
ow
t
he
u
se
o
f
ma
t
h
e
m
a
t
i
c
a
l
s
ymbo
ls
i
nf
l
u
ence
s
und
e
r
s
t
a
n
d
i
ng
of
m
a
t
h
conce
p
t
s
f
o
u
nd
tha
t
m
o
st
s
t
u
de
nts
fa
il
t
o
u
n
d
er
sta
nd
or
i
n
t
er
pr
e
t
t
he
m
ean
in
g
o
f
m
a
th
s
ymbo
l
s
due
t
o
t
h
e
w
a
y
the
y
a
r
e
t
a
u
gh
t
to
r
ead,
pr
on
o
u
n
c
e
a
nd
u
s
e
th
em
.
F
u
r
t
he
r
,
t
his
m
i
s
u
se
(
an
d
a
l
so
a
b
use)
o
f
sym
b
ols
m
a
y
esse
n
tia
l
l
y
hi
nde
r
fo
rmat
i
o
n
,
u
nde
r
s
t
a
nd
i
n
g
a
nd
c
o
mmu
n
i
c
a
ti
on
o
f
co
nc
ept
s
a
n
d
mig
h
t
aff
e
c
t
a
c
h
i
e
vement
i
n
th
e
subj
ect
.
A
s
sta
t
ed
b
y
V
i
nc
ent,
Bar
d
i
ni,
P
i
er
c
e
,
a
nd
P
e
a
r
n
[2
9]
,
un
de
r
s
ta
nd
i
ng
t
ha
t
s
y
m
bol
s
do
h
a
v
e
me
ani
n
g
a
n
d
m
a
kin
g
it
a
ha
bi
t
t
o
c
h
e
ck
t
he
m
eani
n
g
o
f
s
ymb
o
l
s
u
se
d
i
s
a
n
a
s
pect
o
f
m
athem
a
t
i
ca
l
w
r
i
tin
g
tha
t
n
ee
ds
t
o
be
c
u
lt
iva
t
e
d
f
r
o
m
p
r
ima
r
y
to
t
e
r
t
i
ary
leve
l
s
.
Usisk
i
n
[3
0]
(
p.
4
)
a
sserts
t
h
at
“
m
a
the
m
a
tic
s
is
b
o
t
h
a
writt
e
n
lan
g
u
a
g
e
a
nd
a
spoke
n
lan
g
u
age
,
f
or
—
p
ar
t
i
c
u
la
r
l
y
i
n
s
ch
oo
l
m
a
th
ematics—we
h
av
e
wo
rd
s
fo
r
virtu
a
lly
a
ll
the
s
y
m
bols.
F
am
i
l
i
a
r
i
t
y
w
i
t
h
th
is
l
a
n
gua
ge
i
s
a
pr
e
c
ur
sor
t
o
a
l
l
u
nder
s
ta
n
d
i
n
g.”
Be
in
g
fam
ilia
r
wit
h
s
y
m
b
o
l
s
a
nd
f
u
ll
y
u
nder
s
ta
ndi
ng
the
i
r
m
eanin
g
n
e
e
d
t
o
be
c
ul
tiv
ate
d
a
t
all
leve
ls
:
pr
im
ar
y,
s
ec
onda
r
y
a
nd
t
e
r
t
i
a
r
y
.
Tw
o
ty
pes
of
e
r
r
or
p
lace
d
thir
d
i
n
t
he
o
v
e
r
a
ll
r
a
nk
ing-
-
-
i
n
c
or
re
ct
c
a
p
ita
li
za
tio
n
an
d
lac
k
o
f
t
e
rm
o
r
phr
a
s
e.
I
t
w
a
s
not
ice
a
b
l
e
i
n
t
he
d
a
t
a
ga
t
h
er
ed
t
h
a
t
m
a
ny
s
t
ude
n
t
s
c
ap
it
aliz
e
wo
rd
s
t
h
ey
w
an
t
to
put
e
mp
h
a
s
i
s
on.
A
l
s
o,
t
he
s
tu
de
nt
s
se
em
ed
t
o
fi
n
d
i
t
d
i
f
f
icu
l
t
t
o
d
if
fe
r
e
nt
ia
te
c
om
m
o
n
a
nd
pr
o
p
er
n
o
uns
w
he
n
i
t
c
o
m
e
s
t
o
m
a
them
at
ica
l
t
erm
s
.
For
lac
k
o
f
ter
m
o
r
p
h
rase
e
r
r
or,
it
u
s
u
a
l
l
y
ste
m
s
fr
o
m
t
he
s
t
ude
n
t
s’
t
hi
n
k
in
g tha
t
t
her
e
i
s
n
o
n
e
e
d
to
d
efi
n
e
a
c
o
mmo
nl
y
use
d
n
o
t
a
tio
n
or
var
ia
b
l
e s
h
o
u
ld
t
h
e
y
de
c
i
de
to
inc
l
ude suc
h in t
he
ir
pr
o
ce
dur
a
l
e
x
p
l
a
n
at
io
n.
T
hey
ha
ve
t
his
a
ssump
ti
o
n
t
ha
t
i
t
w
i
ll
a
u
tom
a
tica
l
ly
b
e
pi
cked
-up
a
n
d
un
de
rst
o
o
d
b
y
th
e
re
a
d
er
the
w
a
y
the
y
m
ea
nt
i
t
to
b
e
de
live
r
ed,
w
h
ic
h
sho
u
l
d
no
t
be
t
he
c
a
se
i
n
mat
h
e
m
a
t
i
c
a
l
expos
it
o
r
y
wri
t
i
n
g
.
F
our
th
i
n
r
a
n
k
i
s
t
he
v
a
g
ue
t
er
m
er
r
o
r
w
i
th
a
n
aver
a
g
e
o
f
11%
.
Th
is
i
s
fo
llo
w
ed
b
y
no
o
r
in
co
rrect
pu
nc
tua
tio
n
a
n
d
inc
o
r
r
ec
t
ter
m
e
r
r
or
s,
both
w
i
t
h
a
v
e
r
a
g
e
o
f
(
1
0
%
)
.
I
n
t
h
e
s
t
u
d
y
o
f
L
e
w
a
n
d
M
e
j
i
a
-
R
a
m
o
s
[
1
6
]
,
r
e
sul
t
s
s
how
e
d
t
hat
th
er
e
is
a
l
ack
o
f
pu
nc
tua
t
ion
a
nd
ca
p
i
t
a
l
i
z
e
d
l
e
t
t
e
r
s
t
o
i
n
d
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c
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t
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t
h
e
e
n
d
i
n
g
a
n
d
be
g
i
nn
i
ng
o
f
s
e
n
te
nce
s
.
Le
e
[
1
5]
s
ta
te
d
t
h
a
t
t
o
ma
ke
s
ur
e
tha
t
t
h
e
pa
p
e
r
flow
s
sm
o
o
th
l
y
,
equa
t
i
ons
a
nd
f
o
r
m
ula
s
,
a
s
ide
fr
om
t
he
w
or
ds,
shou
ld
h
a
v
e
c
o
r
r
e
c
t
p
u
n
c
t
u
a
ti
o
n
.
The err
o
rs
w
i
t
h
the
lowes
t
a
v
era
g
e
pe
rce
n
ta
ges are
m
i
suse
of ma
them
at
ic
al t
e
r
m
s
(6%)
a
n
d
i
n
c
o
rr
ect
no
ta
ti
o
n
(
2%)
.
E
ve
n
t
h
ou
gh
a
ll
t
h
e
s
o
lu
tio
ns
i
n
t
h
e
f
i
v
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a
c
tiv
i
t
i
e
s
r
e
q
u
i
red
hi
gh
m
at
h
e
mat
i
cal
c
on
t
e
nt
,
it
wa
s
e
v
ide
n
t
from
t
h
e
da
ta
t
ha
t
t
h
e
st
u
d
e
n
t
s
w
e
r
e
ve
ry
car
eful
i
n
u
s
i
n
g
the
c
o
rre
ct
n
ot
a
tio
ns.
In
f
act,
f
o
r
Act
i
v
itie
s
2,
4
,
and
5,
t
he
r
e
w
as
no
r
e
corde
d
e
r
r
or
i
n
no
t
a
t
i
on.
Evaluation Warning : The document was created with Spire.PDF for Python.
IJERE
I
S
S
N
:
2252-
88
22
M
a
t
h
e
m
atic
al
Writi
n
g
Error
s
in Ex
posi
t
ory W
r
i
t
i
n
gs
o
f
Colle
ge
Ma
t
h
em
a
tic
s St
u
d
en
ts (
I
ve
e
K
.
G
u
ce
)
24
1
4.
CONCL
U
S
ION
Th
is
s
t
u
dy
in
v
e
st
iga
t
e
d
t
h
e
f
reque
ncies
a
t
w
hic
h
t
he
n
ine
pre
-
d
efine
d
m
athem
a
tica
l
w
riting
errors
we
re
c
o
m
mi
t
t
e
d
b
y
t
h
e
stud
en
ts.
F
r
o
m
t
h
e
f
i
ndi
ng
s
of
t
h
e
s
t
u
dy
,
t
here
i
s
evi
d
e
n
ce
t
hat
t
h
e
mos
t
c
om
m
i
t
t
e
d
err
o
rs
a
re
i
ncorr
ect
g
ram
m
a
r
and
m
i
s
u
se
o
f
ma
the
m
a
t
i
c
al
s
ym
bo
l
s
.
C
e
r
t
a
i
nl
y,
i
nter
v
e
n
t
i
on
pr
o
g
ram
s
o
n
ma
them
at
i
c
s
w
r
iti
ng
w
i
l
l
b
rin
g
f
a
vorab
le
o
u
t
c
o
m
e
s.
L
an
gua
g
e
c
o
u
r
s
es
i
n
t
h
e
s
t
ude
n
t
s
’
curric
u
lum
which
tack
le
p
ro
pe
r
gr
am
mar
usag
e
m
a
y
b
e
i
n
t
e
g
r
a
te
d
w
ith
w
rit
i
n
g
a
bo
u
t
ma
t
h
e
m
a
t
ic
s
a
s
p
ar
t
o
f
t
he
s
t
ude
n
t
ac
t
i
v
i
ti
e
s
.
S
u
c
h
w
ill
p
r
o
vide
t
he
s
t
u
den
t
s
w
i
t
h
w
riti
ng
e
xper
i
e
n
c
e
s
f
i
t
t
e
d
t
o
th
e
i
r
d
i
sc
i
p
li
n
e
.
A
ma
th
emat
ics
stude
n
t
s
h
o
u
l
d
n
o
t
o
n
l
y
k
n
o
w
t
he
t
he
orie
s
a
nd
pr
inc
i
ple
s
b
e
h
ind
s
o
l
vi
n
g
a
gi
ve
n
pro
b
l
e
m
.
Equa
lly
i
mpor
ta
nt
to fi
n
d
i
ng t
h
e a
n
swer
i
s
articu
l
at
i
n
g c
l
e
a
rl
y
and c
o
here
nt
ly
h
o
w
to
a
r
r
ive a
t
t
ha
t a
n
sw
e
r
.
REFE
RENCES
[1]
Nat
i
on
al
C
ou
nc
il
of
T
eachers
o
f
M
a
t
hem
a
ti
cs,
“
C
u
rricul
u
m
and
ev
al
uati
on
s
tandards
f
or
s
c
h
o
o
l
ma
themat
i
c
s,
”
Rest
on, VA, 200
0.
[2]
V.
U
rq
uh
a
r
t
,
“
Usin
g
writ
ing
in
m
a
t
he
ma
t
i
c
s
t
o
d
e
e
p
e
n
s
tu
de
n
t
l
e
arni
ng
,”
D
en
ver,
C
ol
orad
o,
M
i
d
-con
ti
ne
nt
Research
f
o
r
Educat
io
n an
d L
earnin
g
, 20
0
9
.
[3]
K. M
c
C
o
r
mick
, “
E
x
p
eri
e
n
c
in
g
t
h
e
pow
er of learni
ng
m
athe
m
a
tics
t
hough
wr
iting
,
”
Issu
e
s
in
the
Un
de
rgra
du
ate
Ma
th
e
m
a
tic
s
Pre
p
ara
t
ion
of
Sc
h
o
o
l
Te
a
c
he
rs:
T
h
e
J
o
u
r
na
l
, v
o
l. 4,
pp
. 1
-
8, 2
01
0.
[4]
P.
R
il
le
r
o
,
et a
l
.
,
“Wr
i
t
e
f
rom
the
start
:
W
r
iting-
t
o
-
learn
sci
e
nce
and
mathemat
ics,”
P
hoen
i
x, AZ
,
Natio
nal
As
soc
i
atio
n of Bio
l
o
g
y
T
e
a
c
hers
Nat
i
o
nal Co
nven
tio
n
,
19
95.
[5]
R.
C
ard,
“
The
ef
f
ect
s
of
w
r
iting
i
n
m
at
hema
t
i
cs
o
n
second-g
rade
s
t
uden
t
s
'
a
chi
e
vem
e
nt
a
nd
m
etaco
g
n
i
t
i
on,”
Retriev
e
d fro
m ERIC d
atab
ase, 1
99
8.
[6]
C.
M
i
n
t
o
n,
“
W
r
iting
a
n
d
readi
n
g
i
n
m
at
he
mat
i
c
s
,
”
U
npub
l
i
s
hed
ma
s
t
e
r’s
t
he
si
s,
U
niv
e
rsi
t
y
o
f
W
isc
o
nsin
-
S
tou
t
,
M
e
nom
o
n
i
e
, Wisco
ns
in
, US
A, 200
8.
[7]
M
.
R
.
P
a
rso
n
s
,
“
Eff
ects
of
w
riti
n
g
t
o
learn
in
p
re-c
al
cul
u
s
m
a
t
h
e
mati
c
s
o
n
achi
e
ve
m
e
nt
a
n
d
a
ff
ectiv
e
o
u
t
c
o
m
es
f
o
r
st
ude
n
t
s
i
n
a
communi
t
y
c
ollege
s
etting
:
A
m
i
x
ed-met
hod
appro
a
c
h,”
U
n
p
u
b
lis
h
e
d
d
o
ct
oral
d
issertat
io
n
,
C
ol
ora
d
o
Stat
e
Universi
t
y
,
For
t
C
o
llins
, Co
l
orado,
U
SA
, 20
1
1.
[8]
B.
S
eo,
“
M
a
t
hem
a
t
i
cal
w
r
iting
:
W
hat
is
it
and
how
do
w
e
t
each
i
t?
”
Journal of
H
u
manistic
Mathem
at
ics
,
vol/iss
u
e:
5(2
)
, pp
.
1
33
-145, 2
01
5.
[9]
M.
S
h
i
e
l
d
a
n
d
P.
G
a
l
b
r
a
ith,
“
T
h
e
a
na
ly
sis
o
f
s
tu
d
e
nt
e
xp
osito
r
y
w
riti
ng
in
m
athe
mat
i
cs,”
Ed
uca
tio
nal S
t
u
d
i
e
s i
n
Mathem
atics
, vol. 36
,
p
p. 2
9-5
2
, 19
98.
[10]
J.
D
e
f
azi
o,
et
al
.
,
“A
cadem
ic
l
i
t
e
r
acy
:
The
im
p
o
rtan
ce
an
d
im
pact
o
f
writi
n
g
acro
s
s
t
h
e
c
urri
cul
u
m
–
a
case
st
ud
y,”
Jou
r
na
l
o
f
th
e Sc
h
o
la
rs
h
i
p
o
f
Teachi
ng
an
d
L
e
a
r
ning
,
v
o
l/i
ssu
e
:
1
0(2
)
, p
p.
34-4
7
, 20
10.
[11]
A.
C
ooper,
“
Today’s
T
echnologi
es
E
nhance
Writing
in
M
a
t
hemat
i
cs
,”
T
h
e Clear
i
n
g
House,
v
o
l.
85,
pp.
8
0-85,
20
12
.
[12]
L.
D
.
M
ill
er,
“Writi
ng
i
n
m
a
them
ati
c
s
cl
as
s
e
s
(Rep
ort
No.
ISSN-
10
3
3
-3
73
8),”
A
ustra
lia
,
Ke
y
Ce
n
t
re
f
o
r
S
c
h
o
o
l
S
c
ience
and
Mathem
at
ics, 19
92.
[13]
L.
A
ddis
o
n
,
“
A
teach
er’s
r
ef
lect
io
n
o
n
t
he
i
mpl
e
m
e
nt
atio
n
o
f
j
o
urnal
wr
iting
in
m
a
t
hematics,”
Univers
i
ty
o
f
Al
berta, Le
t
hbridge, Albert
a
, 19
7
2.
[14]
E.
R
e
i
l
l
y
,
“
W
r
iting
to
l
earn
math
ematics
:
A
m
ixed
m
ethod
study,
”
Un
pu
bli
s
h
e
d
d
o
ct
oral
d
is
se
rt
a
tio
n,
I
n
d
i
a
na
Universit
y
of
Pennsylvania, Pi
t
t
s
bur
g
h,
Pennsy
l
v
a
nia, USA, 2007
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[15]
K. Lee, “
A
guide to
writi
n
g
m
a
th
emat
ics,” 20
10
.
[16]
K.
L
ew
a
nd
J
.
P.
M
e
j
i
a
-Ram
os
,
“It
’
s
n
o
t
an
E
nglish
clas
s
’
:
Is
c
orrect
g
ram
m
a
r
an
i
mp
ort
a
n
t
p
art
of
m
a
t
hem
a
ti
cal
pro
o
f
w
r
itin
g
at
t
he
u
n
d
erg
r
ad
uat
e
l
evel?
”
Pr
esen
t
e
d
at t
h
e 1
9
th
Ann
u
a
l
Co
n
f
eren
ce on
R
e
sea
r
ch
in U
n
d
e
rgr
a
d
uate
M
a
them
atics
Educa
tio
n. Pitt
sbu
r
g
h
,
Penn
syl
v
an
ia
,
20
16
.
[17]
F.
S
u,
“Some
gu
i
delines
f
or
good
mat
h
e
m
atical
w
riting,”
MAA
Focus
,
vo
l/
issue
:
3
5
(
4), p
p.
20
–
2
2
,
2
01
5.
[18]
R. M
oo
re, “
M
a
ki
ng
th
e
tran
s
iti
on to f
orm
a
l pro
o
f
,
”
E
d
ucat
i
o
nal
Studies
in Math,
vol. 27
,
p
p. 2
49
–266
, 19
9
4
.
[19]
E.
B
er
ry and J.
Lawson,
“W
r
iting
Ma
th
e
m
a
t
ic
s
,
”
Pur
d
ue
U
n
i
ve
r
s
i
t
y
, Cla
ss ha
n
d
o
u
t
,
19
96
.
[20]
S
.
K
l
i
n
e
a
n
d
D
.
I
s
h
i
i
,
“
P
r
o
c
e
d
u
r
a
l
E
x
p
l
a
n
a
t
i
o
n
s
i
n
M
a
t
h
e
m
a
t
i
c
s
W
r
iti
ng,”
W
r
itten C
o
mmuni
ca
ti
on
,
vo
l/
is
s
u
e
:
2
5
(
4)
,
p
p
. 44
1
-
26
1,
20
0
8.
[21]
L.
G
ill
ma
n
,
“
Writ
in
g
m
a
th
e
m
a
t
i
c
s
we
l
l:
a
ma
nu
a
l
f
or
a
utho
r
s
,”
W
a
s
h
i
ngto
n
,
DC,
M
a
t
h
em
ati
cal
A
s
s
o
c
ia
t
i
on
o
f
Am
erica,
19
87.
[22]
N. J.
H
i
gh
a
m
,
“
H
a
n
db
oo
k
o
f
wr
iti
ng f
o
r th
e m
a
th
e
m
at
ical
sci
ence
s,
” Siam
, 1
9
9
8
.
[23]
K. Hou
st
on
, “How
t
o
Write M
a
t
hem
a
ti
cs,” 2
00
9.
[24]
S
.
G
.
Kran
tz,
“A
p
rim
e
r
of
m
a
t
h
e
m
a
ti
cal
w
rit
i
n
g
:
Bei
n
g
a
d
i
s
q
ui
sition
on
h
av
i
ng
your
i
d
eas
r
ecorded,
typeset
,
pu
bl
ish
e
d,
r
ead an
d
ap
p
reciat
ed,”
American M
a
the
m
at
i
c
al
Soc
i
ety
,
19
97
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[25]
F
.
V
iva
l
di, “
I
ntrod
u
ct
io
n t
o
M
athem
a
ti
cal W
riti
ng,” L
o
n
don, T
he
U
ni
versity
o
f
Lon
d
o
n
,
20
14.
[26]
S
.
D
.
P
e
ñ
a
,
“
A
n
a
l
y
s
i
s
o
f
E
r
r
o
r
s
i
n
E
s
s
a
y
s
W
r
i
t
t
e
n
b
y
M
a
t
h
,
S
c
i
e
nce
an
d
E
ngi
neeri
n
g
F
a
c
u
lt
y,”
Liceo Jo
ur
na
l of
Hi
gher
E
duca
t
i
o
n
Res
e
arch
, v
o
l/issu
e:
6
(1),
pp. 76-85
, 2
00
9.
[27]
R.
R
u
b
enstei
n
a
n
d
D.
T
hom
ps
o
n
,
“Learn
i
n
g
M
at
hem
a
ti
cal
S
y
m
bo
lism
:
Chal
lenges
and
Ins
t
ruc
tional
Stra
t
e
gi
es,
”
Mathem
atics
Teacher
, v
ol/i
ssu
e
:
94(4
)
, 2
00
1
.
[28]
S
.
C
hirum
e
,
“Ho
w
d
o
e
s
th
e
U
s
e
of
M
ath
e
m
a
ti
c
a
l
S
y
m
bols
In
flu
e
nce
U
n
d
erst
and
i
n
g
o
f
M
a
th
ematical
C
o
n
cep
ts
b
y
Se
c
o
n
d
a
r
y
Sc
ho
ol
S
tu
de
n
t
s?
”
In
te
rna
t
io
na
l J
.
So
c
.
S
c
i.
&
Ed
u
c
at
io
n,
vol/issu
e: 3
(2), p
p. 35
-
46, 2012
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[29]
J.
V
incent,
et a
l
.
,
“
M
i
s
us
e
o
f
t
he
E
q
u
al
s
Sig
n
:
An
E
nt
renched
P
r
actice
f
r
o
m
E
arly
P
rim
a
ry
Y
ears
t
o
Tert
i
a
ry
Mathematics
,
”
Au
st
ri
a
l
ian
S
e
nior M
a
t
h
em
a
t
ics Jour
na
l
,
vol/is
s
u
e:
2
9(2),
pp. 31-39,
2015.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SSN: 2252-
8822
IJERE
V
ol
.
6,
N
o.
3,
S
eptem
b
er
20
17 :
2
3
3
– 242
24
2
[30]
Z.
U
s
i
s
k
in,
“Wh
a
t
d
o
es
i
t
m
ean
t
o
u
nders
tan
d
s
om
e
m
a
t
h
em
atics
?
”
Pr
oceedi
n
g
s
of th
e
1
2
th
Interna
tio
nal Co
ng
re
s
s
on
M
a
th
ema
t
ical E
d
u
c
ation,
2
012.
B
I
OG
RAP
HY
O
F
A
U
T
H
O
R
Dr.
Iv
ee
K.
G
uce
is
a
m
athem
a
t
i
cs
t
eacher
a
n
d
a
r
esear
c
her.
S
h
e
i
s
an
a
ss
ociat
e
p
rofes
s
or
o
f
th
e
M
a
t
h
em
a
t
i
c
s
Dep
a
rtm
e
nt
o
f
t
h
e
Coll
ege
of
E
d
u
cati
on,
A
rts
an
d
S
c
iences
o
f
t
h
e
De
L
a
S
a
ll
e
Lip
a
.
S
h
e
fin
i
s
h
ed
h
er
B
S
M
a
t
h
em
ati
c
s
M
a
jo
r
in
A
c
t
u
a
ri
al
Sc
ie
nc
e
a
t
t
he
U
n
i
v
e
rsi
t
y
o
f
S
t
o.
T
om
as.
S
h
e
o
b
t
a
ined
h
er
M
as
ter
i
n
M
athe
mat
i
cs
d
e
g
ree
f
r
om
the
D
e
L
a
Sa
lle
U
n
i
versity,
and
e
a
rn
ed
t
he
d
egree
P
h
i
l
oso
phy
in
E
du
catio
n
M
a
jo
r
in
M
at
he
m
a
t
i
c
s
f
r
o
m
t
h
e
U
n
i
v
e
r
s
i
t
y
o
f
B
a
t
a
n
g
a
s
.
D
r
.
G
u
c
e
i
s
c
u
r
r
e
n
t
l
y
t
h
e
P
r
o
g
r
a
m
C
h
a
i
r
i
n
M
a
t
h
e
m
a
t
i
c
s
.
B
esi
d
es
t
each
ing
,
s
he
i
s
acti
v
el
y
en
gage
d
in
r
esear
ch.
S
h
e
has
pu
bl
ish
e
d
a
n
u
m
b
er
o
f
pap
ers
at
i
n
t
ernat
i
o
n
al
j
ou
rnals
and
pres
ent
e
d
h
e
r
w
o
rk
s
at
s
eve
r
al
c
on
feren
ces
b
oth
lo
call
y
a
nd
i
n
t
ern
a
ti
ona
l
l
y.
M
os
t
of
h
er
wo
rks
are
f
o
cu
s
e
d
o
n
m
athem
a
tics
ed
ucati
o
n
.
H
er
p
ass
i
o
n
f
or
t
ea
ch
in
g
m
a
themat
ics
i
s
h
er
m
o
tiv
ati
on in sharin
g
h
er
k
n
o
wl
edg
e
to
o
t
her ed
ucat
ors
thro
ug
h
th
e
st
ud
ie
s
sh
e
c
o
n
d
u
c
t
s
.
Evaluation Warning : The document was created with Spire.PDF for Python.